Taking Place Value Seriously: Arithmetic, Estimation and Algebra
Roger Howe, Yale University
0. Introduction and Summary
Arithmetic, first of whole numbers, then of decimal and common fractions, and later of rational
expressions and functions, is a central theme in school mathematics. This essay attempts to
point out ways to make the study of arithmetic more unified and more conceptual through
systematic emphasis of place value structure in the decimal number system. The essay is
divided into six sections beyond this introduction. A supplemental set of exercises is planned
for a revised version.
Section one reviews the basic principles of decimal (base ten place value) notation. It points out
the sophisticated structure that underlies the amazing efficiency of decimal notation, and it
introduces the terminology of digit, denomination, and place value number, for the basic
constituents out of which base ten numbers are formed. Standard decimal notation is an extremely
compact way of expressing any whole number as a sum of place value numbers. To enable
discussion of the sizes of base ten numbers, the idea of order of magnitude of place value
numbers, and of general numbers, is introduced.
Section two is in four parts, which discuss how decimal notation permits efficient algorithms for
the four basic operations, of addition and subtraction, multiplication and division, for whole
numbers. The main theme is that the form of decimal numbers, as sums of the very special
numbers called place value numbers, together with the Rules of Arithmetic1, determines the main
outlines of the procedures for all the operations.
The discussion of division plays a pivotal role in the essay. There are two main conceptual
interpretations of the base ten expansion of a whole number, and both are tied up with division.
One interpretation of division involves successive division-with-remainder by 10. The other
interpretation involves approximation by place value numbers. This approximation interpretation
provides the basis for the long division algorithm. Long division is a capstone arithmetic skill: it
requires fluency with multiplication, subtraction, and estimation.
Section three discusses ordering, estimation and approximation of numbers. For comparing
numbers, a crucial idea is that of relative place value. The corresponding idea for approximation is
relative error. Both relative and absolute error can be controlled in terms of decimal expansions.
A key concept is significant digit. It is pointed out that relative accuracy of approximation
improves rapidly with the number of significant digits. It is usually unreasonable to expect to
know a “real-life" number (meaning the result of a measurement) to more than three or four
significant digits, and often one must settle for, and can live with, much less. Failure to appreciate
the limits of accuracy seems to be one of the most pervasive forms of innumeracy: it affects many
people who are for the most part quite comfortable with numbers. Scientific notation, which
focuses attention on the size of numbers and the accuracy to which they are known, is also
discussed.
1
By the “Rules of Arithmetic,” we mean what mathematicians refer to as the Field Axioms, and what are often mentioned in
the mathematics education literature as “number properties,” or just “properties.” (They are, in fact, not properties of numbers
but properties of the operations.) They are nine in number: four (Commutative, Associative, Identity and Inverse Rules) for
addition, four parallel ones for multiplication, and the Distributive Rule to connect addition and multiplication. Other rules
such as “Invert and Multiply,” or the formula for adding fractions, or the rules of signs for dealing with negative numbers, can
be deduced from these nine basic rules.
1
The fourth section discusses how place value notation can be extended from whole numbers to
decimal fractions – that is, fractions whose denominators are powers of 10. The procedures for the
arithmetic operations also extend seamlessly. This is a practical reflection of the fact that the Law
of Exponents can be extended to hold for all integers, not just whole numbers.
The ease of computation with decimals together with the efficient approximation properties
discussed in sections 2D and 3, make decimal fractions an efficient tool for practical computation,
especially since numbers coming from measurements are known only approximately. The key
features of place value notation have led to its very heavy exploitation in machine computation.
The fifth section combines the ideas from sections 2 and 3 in a short discussion of accuracy and
estimation in arithmetic computation. Focusing on decimal components permits a highly effective
treatment of this issue.
The last section explores the connections of decimal notation with algebra. It is suggested that
decimal numbers can be profitably thought of as “polynomials in 10," and the parallels between
decimal computation and computation with polynomials are illustrated. Moreover, the structural
understanding that would come with a study of arithmetic based on decimal components should
promote comfort with algebraic manipulation. Further parallels between decimal arithmetic and
algebra are explored in the accompanying problems.
The reader may wonder, how might the ideas in this essay be implemented? What would a
curriculum that emphasized place value look like? This essay will not have much to say on this
point, because there is not much experience, at least in the US, with such a curriculum. However,
between drafts of the essay, the author was looking at the US edition of the Singapore elementary
textbooks. They pay close attention to many of the points discussed here. Their treatment of the
Rules of Arithmetic is much less explicit than the present discussion, but the basic principles of
place value, and their implications for arithmetic, are developed at considerable length. Thus, the
Singapore curriculum can be said to be aligned with the ideas in this essay.
This essay addresses an issue which the author hopes will be of interest to a broad spectrum of
mathematics educators (a term we use inclusively, to comprise mathematics teachers of all levels,
as well as educators with professional interest in mathematics). It probably will meet with the
same problem which bedevils every teacher of mathematics – variations in sophistication and
background among its intended audience. Substantial effort has been made to make the contents
accessible to a fairly wide audience, but what one reader may find obvious, another may find
obscure, and vice versa. It is hoped that many readers can appreciate the broad message, that place
value can serve as an organizing and unifying principle across a surprising span of the elementary
(and beyond!) mathematics curriculum; and that individual readers will be patient with particular
parts which may seem either over- or under-elaborated, keeping in mind that other readers may
have the opposite view.
Acknowledgements: The author is grateful for comments on earlier drafts of this essay from the
MSSG committee members, from Johnny Lott and his associates in NCTM and ASSM, especially
Gail Englert, Bonny Hagelberger, and Mari Muri, and from Scott Baldridge, Thomas Roby, and
Kristin Ulmann. Thanks to Mel Delvecchio and W. Barker for vital production help.
2
1. The Decimal System: Place Value Numbers, Expanded Form and Order of Magnitude.
Our common decimal system is a highly sophisticated method for writing whole numbers
efficiently. It uses only 10 symbols (the digits: 0,1,2,3,4,5,6,7,8,9), arranged in carefully structured
groups, to express any whole number. Further, it does so with impressive economy. To express the
total human population of the world would require only a ten-digit number, needing just a few
seconds to write.
The efficiency of the decimal system is possible because of its systematic use of mathematical
structure – essentially, it enlists algebra in support of counting. Making this structure explicit may
help to make mathematics instruction more effective, by increasing conceptual understanding and
computational flexibility. It should promote numeracy by making students more sensitive to order
of magnitude, and to the estimation capabilities of place value notation. It should also bring out
the parallels between arithmetic and algebra, thus making arithmetic a preparation for algebra,
rather than the impediment that it now sometimes is [KSF, Ch.8].
We will call a whole number expressed in standard place value form a base 10 number. The first
thing to understand about decimal notation is that the base ten expression of a number implicitly
breaks the number up into a sum of numbers of a very special type. Thus
7,452 = 7,000 + 400 + 50 + 2
This is usually called expanded form. It is mentioned in many state standards documents. This
essay explores what would be entailed in making it central to arithmetic instruction.
We will call the summands in the expanded form of a number the place value components of the
number. Each place value component is a digit (possibly 0) times a power of 10:
7,000 = 7 × 1000 = 7 × (10 × 10 × 10) = 7 × 103
400 = 4 × 100 = 4 × (10 × 10)
50 = 5 × 10
= 5 × 101
2= 2×1
= 2 × 100
= 4 × 102
We will refer to any number of this form, regardless of whether it is being thought of as a place
value component of some particular number, as a place value number2. Also, we will call the
powers of 10 denominations. (In this, we follow [BP].) Thus, a place value number is a digit times
a denomination.
We will call the exponent involved in a denomination the order of magnitude of the denomination.
The order of magnitude also may be described as the number of zero digits used to write the
denomination, or one less than the total number of digits used. By the order of magnitude of a
(non-zero) place value number, we mean the order of magnitude of its denomination. Finally, the
order of magnitude of a base ten number is defined to be the order of magnitude of its largest
2
This term is taken from [Ock]. Remarkably, there does not appear to be a standard term in the literature for these numbers.
The author’s first ideas for a name were decimal monomial (rejected as too clinical) and very round number. This latter term
is still the author’s favorite, but it produced strong negative reactions among some mathematicians. However, it might be the
best term for potential classroom use.
3
(non-zero) place value component. Thus, 7,452 has order of magnitude 3, the same as its largest
decimal component, 7000; and the 400, the 50 and the 2 have orders of magnitude 2, 1 and 0,
respectively.
Note that the smallest number of a given order of magnitude is the denomination of that
magnitude, and the largest is one less than the next larger denomination. For example, the
numbers with order of magnitude 3 start at 1000, and continue consecutively until 9999, with
10,000 being the smallest number with order of magnitude 4. Note that numbers are sorted by
order of magnitude: any number of a given order of magnitude is larger than every number of a
smaller magnitude, and is smaller than every number of a larger magnitude. Thus 1000 > 999 and
9999< 10,000.
For this essay, “order of magnitude" replaces the usual naming of places. We want to replace the
standard terminology involving specific place names with this order-of-magnitude terminology in
order to be able to better discuss the relationship between different orders of magnitude. Often it
will be convenient to abbreviate “order of magnitude" to just “magnitude.”
A key point about the expanded form of a number is that no two of the place value components
have the same order of magnitude. This condition in fact characterizes the expanded forms: a sum
of place value numbers is the expanded form of a base ten number exactly when it involves at
most one summand of any given order of magnitude. The non-zero digits of the decimal
expression of the number are then just the digits coming from the place value number summands,
each digit in its corresponding place. Besides these, for orders of magnitude smaller than the
magnitude of the number, if the place value number of that magnitude is “not there,” i.e., is zero,
then one puts a zero in the corresponding place of the number, to indicate the absence of any
multiple of that power of 10, so that the orders of magnitude corresponding to the non-zero digits
can be read correctly. This is the principle of place value. Thus
82 = 80 + 2
802 = 800 + 2
80,020 = 80,000 + 20
= 8 × 10 + 2 × 1
= 8 × 100 + 0 × 10 + 2 × 1
= 8 × 10,000 + 0 × 1000 + 0 × 100 + 2 × 10 + 0 × 1
There is nothing to prevent writing zeroes in places to the left of the largest non-zero digit. Doing
so does not change the value of the number. Thus
0082 = + 0 × 1000 + 0 × 100 + 8 × 10 + 2 × 1 = 8 × 10 + 2 × 1
On the other hand, there is usually no reason to do so, and it takes extra work. Accordingly, it is
very rarely done.
Although the above describes the base ten expansion of a whole number, the mathematician would
like something more: an interpretation of the base ten expansion, in terms of the structure of
arithmetic. There are (at least) two such interpretations, quite different from each other. The more
obvious one is in terms of division by 10; the other appeals to ideas of approximation. It is quite
appropriate that there should be two such distinct interpretations, reflecting two distinct but
important aspects of base ten notation. It is because both interpretations involve considerations
that are important in long division, that we do not attempt to develop them here, but instead
discuss them in section 2D on long division.
4
2. Decimal Components and the Algorithms of Arithmetic.
The fact that base ten numbers are sums has a pervasive influence on the methods for computing
with them. In fact, the procedures for carrying out the four arithmetic operations with base ten
numbers are largely determined by the fact that base ten numbers are sums of their place value
components, together with the Rules of Arithmetic (see footnote, page 1). The special form of
place value numbers has the consequence that adding or multiplying two of them substantially
reduces to the basic “addition facts" or “multiplication facts" for adding or multiplying one-digit
numbers. This is the reason for the key role those facts play in doing decimal arithmetic.
2A. Addition
The basic strategy for adding two base ten numbers is to break each number (summand) into its
place value components, and, for each order of magnitude, to add the components of that
magnitude. These sums are then recombined into a decimal number. The details of this last step
give rise to the trickier parts of the addition algorithm.
Basic Rules for Addition
One can justify this process of decomposing, adding components, and then recombining, by means
of the two most basic rules of addition: the Commutative Rule and the Associative Rule. We recall
them.
The Commutative Rule: The value of a sum does not depend on the order of the summands: for
any two whole numbers, a and b
a+b =b+a
The Associative Rule: When adding three numbers, the value of the sum does not depend on the
way the numbers are combined into pairwise sums: more precisely, for any three whole numbers
a, b, and c,
(a + b) + c = a + (b + c)
These rules are easy to justify in terms of common models of addition. If we model addition by,
say, concatenation of lengths (sticking bars end to end), the Commutative Rule can be
demonstrated by combining bars of lengths a and b into a bar of length a + b , and then rotating
the bar around its midpoint to reverse the order of the summands. Associativity is even simpler:
one just takes a bar composed of subbars of lengths a, b, and c, and observes that it can be thought
of as being made of subbars of length a + b and c, or, equally easily, of subbars of lengths a and b
+ c.
Figure 1: The Commutative Rule for Addition
a
b
a
b
a+b=b+a
5
a
Figure 2: The Associative Rule for Addition
b+c
a
b
a+b
(a + b) + c = a + (b + c)
c
c
Although the Commutative and Associative Rules are basic principles of arithmetic, one rarely
uses them in isolation. Most practical manipulations call for more or less complicated
combinations of both the Commutative and Associative Rules. Therefore, it is advisable to discuss
explicitly the logical consequence of these rules, which is that, when a list of numbers is to be
added, it does not matter what sequence of pairwise sums we do, or the order of the summands in
any of these intermediate sums: the final result will always be the same. We call this the Any
Which Way Rule, a name which emphasizes the freedom we have in performing additions. (In
[BP], it is called the Any Order Property.) Although a formal demonstration of the Any Which
Way Rule is probably not appropriate when multidigit addition is first discussed, nevertheless an
explicit discussion of the rule, with examples, particularly examples showing how it can simplify
calculations and aid mental math, does seem feasible and advisable.
In any case, the Any Which Way Rule justifies the strategy of adding two base ten numbers by
breaking each summand into its place value components, adding pairs of components of the same
order of magnitude, then adding all the results. The next question is, what does this do for us? That
is, why is it an effective method?
Sum of Two Place Value Numbers
The first part of the answer to this question comes from looking at the sum of two place value
numbers of the same order of magnitude. These addition problems simply amount to applying the
“basic addition facts" (i.e., how to add two single digit numbers), plus keeping track of the order
of magnitude. Thus, for example,
7,000 + 2,000 = 7 × 1,000 + 2 × 1,000 = (7 + 2) × 1,000 = 9 × 1,000 = 9,000
It is always the same: to add two place value numbers of the same order of magnitude, simply add
their digits and then append the order of magnitude number of zeros on the right. The formal
justification for this uses the Distributive Rule, which we will discuss later on. It can also be
thought of as simply keeping track of units: in the sum above, we are adding 7 thousands to 2
thousands, and the result is (7 + 2) thousands, or 9 thousands.
Adding Base Ten Numbers Component-wise
This observation is sufficient by itself for adding many pairs of numbers. For example:
7,452 + 1,326
= (7,000 + 1,000) + (400 + 300) + (50 + 20) + (2 + 6)
by definition of base 10 notation and the Any Which Way Rule
= 8,000 + 700 + 70 + 8
by the procedure for adding numbers of the same magnitude
= 8,778
by definition of base 10 notation
6
However, addition will be this simple only when, for every order of magnitude, the two place
value components of that magnitude have digits summing to less than 10. When this happens, each
sum of components will again be a place value number, of the same magnitude, and these may be
put directly back together to form the sum of the original numbers, as in the example above.
The Need to Regroup
Whenever a digit sum is more than 10, an extra complication arises. For example,
70,000 + 60,000 = 130,000 = 100,000 + 30,000
Again, this is a repetition on a larger scale of a basic addition fact, which is why those facts are so
important. The new problem presented by this sum is that it has two place value components, one
of the same magnitude as the place value numbers being added, and the other with magnitude one
larger.
In adding two numbers, if the sum of the two place value components of a given magnitude has a
component of larger order of magnitude, this must be added to the components of that magnitude.
This is “regrouping" or “renaming" or “carrying.” An example of addition with regrouping is
7,453 + 1,729
= (7,000 + 1,000) + (400 + 700) + (50 + 20) + (3 + 9)
by definition of base 10 notation and the Any Which Way Rule
= 8,000 + 1,100 + 70 + 12
by the procedure for adding numbers of the same magnitude
= 8,000 + (1,000 + 100) + 70 + (10 + 2)
by writing 1,100 and 12 in expanded form
= (8,000 + 1,000) + 100 + (70 + 10) + 2
by the Any Which Way rule
= 9,000 + 100 + 80 + 2
by the procedure for adding numbers of the same magnitude
= 9,182
by definition of base 10 notation
Since (when adding two numbers) the sum of components of a given magnitude contributes
directly at most to this magnitude and to the next larger one, an efficient way of finding any sum
would be to start with order of magnitude zero (the ones place), find the sum of those components,
combine the 10 (if it occurs) with the magnitude one components, find the sum of those
components, and so forth, progressing systematically to larger orders of magnitude, one at a time.
This is exactly what one does in the standard addition algorithm, although it may not be expressed
in these terms, and the relevance of the expanded form may be suppressed in a strictly procedural
approach. For example, students may not appreciate that the effect of putting one number under
the other with the ones digits aligned has the effect of putting the digits of same-magnitude
components under each other, so that adding the columns exactly accomplishes combining the
components of the same order of magnitude. If the role of magnitude is not mentioned, students
may feel that the procedure for adding decimal fractions (i.e., aligning the decimal points) is
different from the procedure for whole numbers (i.e., right justifying). If the process is understood
in terms of combining components of equal magnitude, they will see that addition of whole
numbers and of decimal fractions are exactly the same. (See section 4 for more discussion of
decimal fractions.) We note that the above procedure may also be adapted readily to addition of
many numbers (“adding the columns”).
7
Thinking explicitly about the component-by-component aspect of decimal addition can give
greater flexibility in doing addition. For example, in adding two-digit numbers mentally, it is
usually easier to think of adding the 10s and adding the 1s separately (which is exactly using
expanded form in this situation). Further, one usually adds the 10s first, since this gives the main
part of the sum, and then adds the sum of the 1s to that. If there is a carry from the 1s, it just means
increasing the result from the 10s addition by one more 10, which is easy to do when there are
only two decimal places to worry about.
We will see later on (section 5) that these same considerations lead to quick estimation procedures
for sums, easy enough to be carried out mentally in many cases.
Summary of Addition
In summary: if addition is thought of in terms of expanded form, then the basic strategy in doing
addition of two decimal numbers is this:
i a) For each order of magnitude, take the place value component of that order of magnitude
from each summand, and add them. The sum is the sum of the digits of the two components, times
their common denomination.
ii a) In cases when the digit sums are all less than 10, the component sums will be the place
value components of the sum to be found.
ii b) If, however, a digit sum is 10 or more, the 10 times the relevant denomination produces the
denomination of the next larger magnitude, which must be added to the components of that
magnitude.
These ideas need not be introduced in a single large lump. They can be taught as the decimal
system is now, in a gradual way, starting with two-digit numbers. This would simply involve
recalling that a two digit number means so many 10s and so many 1s, e.g., 47 is four 10s and
seven 1s. (Probably explicit attention to this would be helpful to many students in any case, since
the irregularities of English two-digit number words somewhat impede children to think in term of
place value [KSF, p.167 ].) Then the strategy of “adding the 10s and adding the 1s" should appear
more or less natural, perhaps after discussion. First problems with no carries could be done to
establish the basic principle, then there should be consideration of what to do when one or the
other of the digit sums is greater than 10. This approach should extend easily to numbers with
more digits. This approach is followed in the Singapore textbooks [Sing 1B].
In fact, the basic grouping principle of base ten notation is not something that all students take to
readily. Even before general two-digit addition is considered, it can be valuable to emphasize the
grouping principle involved in base ten notation by using the “make a ten" idea [KSF, p.189] in
learning the addition facts with sums above ten. As explained in [Ma], Chinese teachers think of
the addition facts as not simply as a list to be memorized, but as a place to begin teaching the
structure of the decimal system, and they emphasize the process of forming ten ones into a 10
when a sum of digits is greater than 10, and of decomposing a 10 back into ones in the
corresponding subtraction problems. The combination of this approach to the addition facts with
more discussion of number names that better reflect base ten structure may be a way to get
students off to a better start in mathematics.
2B. Subtraction
Thinking in terms of place value components can also help with subtraction.
Connecting Subtraction with Addition
8
A key principle in dealing with subtraction is to link it with addition at every opportunity, until
students think of it as the undoing of addition. This is important conceptually, to prepare for the
submersion of subtraction into addition (i.e., as addition of the additive inverse) when negative
numbers are introduced. However, it also has practical value for making sense of subtraction
procedures, by linking them to the corresponding addition procedures. This linkage also brings
more insight into addition.
Fact families should not be thought of as applying only to addition of one-digit numbers. Rather,
every subtraction problem c – b = a should be linked to the corresponding addition problem a + b
= c (and the family should be understood also to include c – a = b and b + a = c).
In fact, the details of the computations involved in the two problems c – b = a and a + b = c
correspond closely. The subtraction c – b = a will involve regrouping (borrowing) exactly when
the addition a + b = c involves regrouping (carrying). Indeed, the same orders of magnitude will
be affected in both computations.
For example, in the problems
970,957
+ 50,829
1,021,786
1,021,786
– 50,829
970,957
we see that the addition involves carrying from the 1s to the 10s, and from the 100s to the 1000s,
and that a carry from the 10,000s to the 100,000s causes a further carry to the 1,000,000s.
Correspondingly, in the subtraction, there is a borrowing from the 10s to the 1s, another borrowing
from the 1,000s to the 100s, and a “borrowing across zero,” from the 1,000,000s to the 100,000s
and 10,000s.
Borrowing Across a Zero and Rollover
This example includes one of the most troublesome aspects of subtraction, namely “borrowing
across a zero.” Study of this phenomenon in the context of the parallel addition problem reveals an
interesting feature of addition that may go unnoticed in a purely mechanical approach.
In adding two base ten numbers, when the sum of the digits of a given order of magnitude add to
9, that place would not produce a carry. However, if the next smaller place does produce a carry,
then the 1 from that, added to the 9 already there, will produce a carry. (Also, the digit left in that
place will be zero.) If the digit sum for the next larger order of magnitude is also 9, then the carry
caused by the carry from the lower place will in turn cause a carry at the next larger place. This
could continue for several places, producing a sum in which there are one or more zeroes in a row.
We call this phenomenon rollover. Its extreme form occurs in adding 1 to a number like 99 or 999
or 9999, with all digits equal to 9. Then the result, of course, is the next larger denomination. This
is what happened on car odometers when they were mechanical, and one could see the change of
the 1s place producing a change in the 10s place, which produced a change in the 100s place, and
so forth, until many or even all the digits on the odometer had rolled over.
Borrowing past a zero is the parallel for subtraction to rollover in addition: when an addition a + b
= c involves rollover, the corresponding subtraction c – b = a will require borrowing past a zero,
and vice versa. If rollover is explicitly studied as an interesting and exceptional situation in doing
addition, and if subtraction is consistently connected to addition, then borrowing past a zero may
seem less mysterious to students. Here is an example of rollover in an addition, and the
corresponding borrowing across zeroes in the associated subtraction:
9
35, 375
+24, 648
60, 023
60,023
–24,648
35,375
We see that in the addition, the carry from 10s to 100s in the addition causes two places to roll
over: it causes the 100s to carry to the 1000s, and that makes the 1000s carry to the 10,000s. In the
subtraction, in order to do the subtraction at the 10s place, one must borrow from the 10,000s
place, across both the 1000s and 100s.
Methods of Compensation
The discussion above focuses on the standard algorithm for subtraction. However, thinking in
terms of decimal components also suggests alternative procedures to the standard one for
subtraction. As is well-known, [CRL] and [KSF], there are multiple interpretations for subtraction;
take away, difference, comparison and missing addend. Comparison involves a process of
matching elements in two sets, followed by counting of the unmatched elements of the larger set
after all elements of the smaller set have been used up. Thinking in this way, one can see that, if
the same amount is subtracted from two numbers, the difference of the results is the same as the
difference of the original numbers. In algebraic terms, this is expressed by the identity
a – b = (a – c) – (b – c)
This formula may be illustrated as follows:
Figure 3: Compensation Principle for Subtraction
a
c
c
a-c
a+b
c
b-c
b
a-b
a – b = (a – c) – (b – c)
This principle may be applied in various ways to simplify subtraction problems. Methods using
this principle are sometimes called methods of compensation [ BP ].
Making Change Method
First, let us consider the extreme case of borrowing across a zero, namely, the problem of
subtracting a number from a decimal denomination. We call this a making change problem, since
it occurs when a customer hands a large denomination bill to a cashier, and the cashier must
subtract the amount of the customer’s purchase from the bill to give correct change. For example,
consider
10,000
- 964
9036
10
One way of dealing with this kind of computation is to “count up,” successively producing
numbers divisible by higher and higher powers of 10. Starting from 964, one adds 6 to get 970,
then adds 30 to make 1000, then (noting that one does not need to add any 100s to make a whole
number of thousands), adds 9000 to make the goal of 10,000. Thus all together one has added
9000 + 30 + 6 = 9036. This is sometimes called the making change algorithm, since it is what
cashiers often do (or used to do) to produce the correct change when an amount due is paid by a
large denomination bill. It amounts to a place by place enactment of the missing addend
interpretation of subtraction.
If, instead of a denomination, we are subtracting from a place value number, essentially the same
procedure works. Hence, we will include this slightly more general problem under the rubric of
making change problems.
The making change problem can also be approached by a simple method of compensation3. We
subtract 1 from both the subtrahend and the minuend to get
10,000 →
- 964 →
9036 →
9,999
- 963
9036
The point here is that all the necessary regrouping has been accomplished in the subtraction of 1,
so that in the second problem, no further regrouping (borrowing) is needed.
Another Subtraction Method
Here is another procedure for subtraction that utilizes systematically a place-by-place method of
compensation to reduce a subtraction problem to a simpler problem in which, for each order of
magnitude, the digit of either the subtrahend or the minuend is zero. It turns out that such a
problem decomposes into a sum of making change problems.
In more detail, for this method, given a subtraction a – b of decimal numbers, for each order of
magnitude, one subtracts from both numbers the smaller of the components of a and b of that
magnitude, leaving only one of the numbers with a non-zero component of that magnitude. Doing
this systematically leaves one with an equivalent but simpler problem.
Let us see how this works in two-digit subtraction. First take an example without regrouping, say
89 - 63. If we write this in the usual form:
89
- 63
The procedure says to subtract the smaller of the two digits in each column from both digits in the
column. This gives a simpler problem with the same answer. In this case
89 → 26
- 63 → - 00
Here we can read off the answer as 26. Thus, when there is no regrouping, this method is
essentially the same as the standard one.
Next, take an example with regrouping, say 83 - 26. Then the procedure subtracts 20 from both
10s components, and 3 from both ones components:
3
Thanks to Mari Muri for bringing this to our attention. It has also been taught for many years by Herb Gross.
11
83 → 60
- 26 → - 03
57
We recognize the simplified problem as a making change problem. Assuming that we know how
to do these, we will find that the answer is 57, as indicated.
In two-digit subtraction, we can not get problems which require borrowing past a zero. Here is a
three digit problem where this happens.
204 → 200
- 9 → -5
195
Again, we see that we have turned the original problem into a making change problem. This is
something that one might do when solving this problem mentally.
In general, the simplified problem will not be a single making change problem, but will break up
into a sum of making change problems. For example, in the four digit problem
2047 →
- 95 →
2002
- 50
1952
We see that the simplified problem 2002 - 50 may be broken into a sum (2000 - 50) + (2 - 0). Each
of these is a making change problem (although the second one, 2 - 0, is a rather trivial one). Note
that we can actually break up the problem by blocks, and insert the separate answers back into the
original at the appropriate places. Thus, we can say
2002 – 50 = (2000 – 50) + (2 – 0) = (200 – 5) × 10 + 2 = 195 × 10 + 2 + 1950 + 2 = 1952
This is the way this method always works. Here is the first example of this section from this point
of view:
1,021,786 → 1,001,060
- 50,829 →
- 30,103
Here the simplified problem decomposes into three disjoint collections of “making change"
problems. More precisely,
1,001,060 – 30,103 = (1,000,000 – 30,000) + (1,000 – 100) + (60 – 3)
= (100 – 3) × 10,000 + (10 – 1) × 100 + (60 – 3) = 97 × 10,000 + 9 × 100 + 57
= 970,000 + 900 + 57 = 970,957
From a mechanical point of view, we can just operate on the leftmost three digits, the next two
digits, and the rightmost two digits as if we were doing the three separate making change
problems. If we write the answers again in the same positions, they will combine to give us the
answer to the whole problem.
12
This suggests the possibility of an approach to subtraction in which making change problems are a
subject of focused attention, and after they are well understood, they can be applied to the general
subtraction problem as illustrated above. Of course a flexible understanding of the role of decimal
components would be necessary for such an approach to make sense. It will be helpful whatever
method of subtraction is used.
2C. Multiplication
Thinking in terms of decimal components also sheds light on multiplication and division.
However, the development requires a firmer grasp of the rules of arithmetic than was needed for
addition. We begin with a discussion of the relevant rules.
Commutative Rule for Multiplication
Multiplication satisfies the Commutative Rule and the Associative Rule, just as addition does, but
they are much less transparent than in the case of addition, and justifying them takes more work.
Probably the best approach is through the Array Model, which is in any case an important
interpretation of multiplication. The Array Model also helps to prepare for understanding area.
The Commutative Rule for multiplication says that, for any two whole numbers n and m, the
products n × m and m × n are the same: n × m = m × n. If we model a whole number as a
horizontal row of some sort of object, say small circles, then multiplication involves replicating
the row a certain number of times. Thus, if 3 is pictured as
,
Then 4 × 3 can be represented
|
|
|
.
(The vertical lines are only for purposes of partitioning the collection of 12 circles into sets of size
three – they are not to be counted.)
For the Array Model, instead of arranging all the groups along the same line, we stack them one
above the other, to make a rectangular array:
This is the Array Model for 4 × 3: four horizontal rows of three items each, stacked vertically
above each other.
Using the Array Model, it is easy to justify the Commutative Rule for multiplication: one just flips
an m × n array over its diagonal, turning it into an n × m array:
Thus, n × m = m × n, which is the statement of the Commutative Rule for multiplication.
13
Associative Rule for Multiplication
A justification similar in spirit, although more complicated, may also be given for the Associative
Rule. The Associative Rule for multiplication is about forming products of several numbers. We
do this by multiplying one pair of numbers at a time, and to do this, we have to choose some order
to do it in.
For example, suppose we want to multiply 2, 3, 4 and 5 together. We can multiply the 5 by the 4
to form 4 × 5. Then we can multiply that result by 3, and then multiply the result of that by 2. We
indicate this procedure by using parentheses to show how the numbers were combined: 2 × (3 × (4
× 5)). Since the 4 × 5 is inside the inner parenthesis, we know we multiplied this first. Then the 3
multiplies the 4 × 5, so the 3 × (4 × 5) is contained in parentheses, and finally, this whole product
is multiplied by 2.
Another way to form a product from these factors is to multiply the 5 by 4, and to multiply the 3
by 2, and then to multiply these two products together. This is indicated by this parenthesization:
(2 × 3) × (4 × 5).
Notice that
2 × (3 × (4 × 5)) = 2 × (3 × 20) = 2 × 60 = 120
On the other hand,
(2 × 3) × (4 × 5) = 6 × 20 = 120
We see that the two ways of multiplying give us the same answer, and this is the eventual result of
the Associative Rule. However, formally, the Associative Rule for multiplication is only about the
simplest multiple factor situation: products of three numbers. With three factors, there are just two
ways to parenthesize.
The Associative Rule for multiplication says that for any whole numbers l, m and n,
l × (m × n) = (l × m) × n
The parentheses on the left hand side say that first n should be multiplied by m, and then the result
should be multiplied by l. On the right hand side, the parentheses indicate that m should be first
multiplied by l, and then n should be multiplied by the result. The Associative Rule asserts that
these two different ways to form the three-fold product give the same result.
Building on the Array Model, one thinks of l × (m × n) as a rectangular 3-dimensional array of l
layers of planar arrays of size m × n. One can then re-slice this array by planes perpendicular to the
rows of length n, and exhibit the same array as an assemblage of n layers of l × m planar arrays.
Thus
l × (m × n) = (l × m) × n
This is the Associative Rule for multiplication. We do not attempt the illustration here. However,
students might well benefit from studying it carefully.
It is worthwhile to note that the Associative Rule for multiplication is much less transparent, both
in its justification and in its application, than the Associative Rule for addition. Indeed, if you
write both sides of the rule for almost any triple of numbers, it will produce an equation which
takes some thought to verify. For example, if we take n = 3, m = 4 and l = 5, we get (3 × 4) × 5, =
14
12 × 5 and 3 × (4 × 5) = 3 × 20. So the Associative Rule says 12 × 5 = 3 × 20. This is of course
true, but if we did not know where this equation came from, we probably would have to compute
both sides to be sure.
Law of Exponents
Before going on, we should discuss the Law of Exponents. This elegant and important formula
describes how to multiply powers of a number. Since place value numbers are just digits times
powers of 10, the Law of Exponents is relevant to multiplying base ten numbers.
The Law of Exponents depends only on the Associative Rule for multiplication. The standard
statement of the Associative Rule says that either of the two possible ways to group three numbers
in pairs to form products will yield the same result. Although it is stated only for three-fold
products, it in fact implies that any grouping of a product of any number of factors will give the
same result as any other grouping. For example, there are five ways to group four factors:
((a × b) × c) × d, (a × (b × c)) × d,
(a × b) × (c × d),
a × ((b × c) × d), a × (b × (c × d))
The Associative Rule allows us to connect these with a chain of equalities:
(a × (b × c)) × d = ((a × b) × c) × d = (a × b) × (c × d)) = a × (b × (c × d)) = a × ((b × c) × d)
(4PAR)
In the above chain, each expression is equal to the next one by a single application of the
Associative Rule. In the first and fourth equations, the rule is applied within a three-fold
subproduct of the four-fold product. In the second equality, the product ((a × b) × c) × d is
regarded as a product of the three factors (a × b), c, and d. In the third equality, it is similar: the
product a × (b × (c × d)) is regarded as the three-fold product of the factors a, b, and (c × d).
This chain of equalities shows that all parenthesizations of a four-fold product yield the same
result. It is the same with products involving any number of factors. This means that if we write a
multiple product without any parentheses, the result is unambiguous: any way of inserting the
parentheses will give the same result. For this reason, products may be, and usually are, written
without parentheses. This is part of the genius behind the logic of the Rules of Arithmetic.
Seemingly very simple rules involving only two or three numbers imply much more complex
manipulations involving many numbers.
When all the factors in a multiple product are the same number a, the result is called a power of a.
More precisely, if we multiply together m copies of a, the result is called am. Here m is called the
exponent of a. In this situation the Associative Rule guarantees that all possible ways of forming
the product of m factors equal to a give the same result, so all that matters is the number of factors,
which is the exponent m. Here are some examples:
22 = 2 × 2 = 4,
23 = 2 × 2 × 2 = 2 × 22 = 2 × 4 = 8
24 = 2 × 2 × 2 × 2 = 2 × 23 = 2 × 8 = 16,
32 = 3 × 3 = 9,
25 = 2 × 2 × 2 × 2 × 2 = 2 × 24 = 2 × 16 = 32
33 = 3 × 3 × 3 = 3 × 32 = 3 × 9 = 27
34 = 3 × 3 × 3 × 3 = 3 × 33 = 3 × 27 = 81,
35 = 3 × 3 × 3 × 3 × 3 = 3 × 34 = 3 × 81 = 243
15
From the point of view of decimal arithmetic, the most important powers, and also the simplest to
write, are the powers of 10:
101 = 10,
102 = 10 × 10 = 100,
103 = 10 × 10 × 10 = 10 × 100 = 1,000
and so forth. We can see that 10m is just the denomination of order of magnitude m. Thus, the
number of zeroes, the power of 10, and the order of magnitude of the denomination are all
different terms for the same number.
The Law of Exponents describes the product of two powers of a number a. It says:
Law of Exponents:
(am) × (an) = am+n
Examples: 22 × 23 = 4 × 8 = 32 =25; 102 × 103 = 100 × 1,000 = 100,000 = 105
Its proof is easy from what we have already said. The power am is the product of m factors, all
equal to a. Similarly, the power an is the product of n factors, all equal to a. If we multiply am and
an together, we will have a total of m + n factors, all equal to a, and this is just the power am+n.
The Law of Exponents is the key identity for dealing with expressions involving powers of
numbers. Besides its utility in computations, it reveals a beautiful parallel between multiplication
and addition. The Rules of Arithmetic emphasize the fact that both addition and multiplication are
governed by essentially the same four basic rules. This suggests that there might be some deep
relationship between them. The Law of Exponents is a concrete expression of this kinship.
Any Which Way Rule for Multiplication
Once students firmly believe the Commutative Rule and the Associative Rule for multiplication,
the same logic as used in the case of addition shows that the Any Which Way Rule holds also for
multiplication: not only can the factors in a multiple product be grouped according to any scheme
of parenthesization, they also can be reordered as desired. If this was not discussed very explicitly
when developing addition, it might be well to consider examples of how the two rules combine to
create more extensive rearrangements. Here is one. The equality
(a × b) × (c × d) = (a × c) × (b × d)
may be justified by the steps
(a × b) × (c × d) = a × (b × (c × d)) = a × ((b × c) × d)
= a × ((c × b) × d) = a × (c × (b × d)) = (a × c) × (b × d)
Here the first two equations are part of the chain of equations 4PAR above, as are the last two. The
third equation uses the Commutative Rule for b × c.
If the reader thinks that the above transformations amount to a lot of work to justify a simple
exchange of factors, s/he would be right; but the general symbolic formulation given above may
make the identity seem too transparent. Consider a numerical example:
24 × 56 = (3 × 8) × (7 × 8) = (3 × 7) × (8 × 8) = 21 × 64
The equality of the first and last products may not be immediately obvious to all.
16
Distributive Rule
In addition to the Any Which Way Rule, multidigit multiplication makes heavy use of the
Distributive Rule, which is the one rule of arithmetic involving both addition and multiplication.
More precisely, it tells us how to multiply sums of numbers. The standard formulation is:
Distributive Rule
For any whole numbers a, b, and c, the equation
a × (b + c) = a × c + a × c
holds.
The Distributive Rule may be also justified using the Array Model. The picture is as follows:
Just as the Commutative and Associative Rules combine many times to produce the practical Any
Which Way Rule, the Distributive Rule also has a more practical form, which we will designate
(rather prosaically) as the
Extended Distributive Rule
If A and B are sums of several numbers, then the product AB may be computed by multiplying
each summand of A by each summand of B, and adding all the resulting products.
For example, if A = a + b and B = c + d + e, then
AB = (a + b)(c + d + e) = ac + ad + ae + bc + bd + de
(Here, and in what follows, we use the standard convention to indicate multiplication simply by
juxtaposition, with no operation sign written explicitly.)
Figure 4: Distributive Rule
a) Example:
3
○
○
○
b) Schematic Representation:
6
○ ● ● ● ●
○ ● ● ● ●
○ ● ● ● ●
2
a
b
4
3 X (2+4) = 3 X 2 + 3 X 4
c
a X (b + c) = a X b + a X c
17
The Extended Distributive Rule can be pictured using arrays almost as easily as the basic version:
Figure 5: Extended Distributive Rule (Schematic Form)
a
b
c
e
d
(a + b) X (c + d + e) = a X c + a X d + a X e + b X c + b X d + b X e
Multiplication According to the Rules
These rules, plus, of course, the Any Which Way Rule for addition, provide excellent guidance for
multiplying base ten numbers together. Since each base ten number is the sum of its place value
components, the Extended Distributive Rule tells us that the product of two base ten numbers may
be found by multiplying each place value component of one factor by each place value component
of the other, and then summing all the products. The Any Which Way Rule for addition says that
we have great latitude how we sum them. Differences in multiplication algorithms come mainly
from choosing different summation procedures.
Multiplying Place Value Numbers and the Multiplication Facts
Before getting into details of summation schemes, we should have a good grasp of multiplying
place value numbers. Here we find a situation analogous to that of addition. When we multiply
two place value numbers, the result is equal to the product of the digits, times the appropriate
denomination, which has order of magnitude equal to the sum of the orders of magnitude of the
factors, since according to the Law of Exponents, 10a × 10b = 10a+b. Examples:
20 × 40 = 800
300 × 6,000 = 1,800,000
Thus, the basic multiplication facts, combined with a grasp of place value, allow us to multiply
any two place value numbers.
We should perhaps point out that the formal correctness of a formula like 300 × 6,000 = 1,800,000
depends on the Any Which Way Rule for multiplication. In carrying out the detailed
manipulations, we would write
300 × 6,000 = (3 × 100) × (6 × 1,000) = (3 × 6) × (100 × 1,000) = 18 × 100,000
= 1,800,000
(Note that the crucial middle equality is a case of the equation (a × b) × (c × d) = (a × c) × (b × d)
discussed above.) As we have already mentioned, the multiplication of the denominations is
described by the Law of Exponents: 100 × 1,000 = 102 × 103 = 105 = 100,000.
Just as in one-digit multiplication, the product of two place value numbers can itself have one or
two components in its expanded form. Their magnitudes will be the sums of the magnitudes of the
factors, and possibly one more. Perhaps the trickiest case is when one digit is even, and the other
is 5. Then the product of digits will be a multiple of 10, and so the full product will have only one
18
decimal component, but its order of magnitude is one larger than the sum of the orders of
magnitude of the factors. This case, however, follows the same rules as all other cases: the extra
order of magnitude is supplied by the product of the digits. An example:
50 × 800 = (5 × 10) × (8 × 100) = (5 × 8) × 1,000 = 40 × 1,000 = 40,000
Doing Multidigit Multiplication
Let us combine these observations with the Extended Distributive Rule to do multi-digit
multiplication. For example, consider
36 × 487
= (30 + 6) × (400 + 80 + 7)
= 30 × 400 + 30 × 80 + 30 × 7 + 6 × 400 + 6 × 80 + 6 × 7
= 12,000 + 2,400 + 210 + 2,400 + 480 + 42
= 17,532
In this computation, we have arranged the products of place value components in a 3 by 2 array.
We have given no clue as to how the addition of the array of six numbers was performed. One
method might be to sum the terms along each row of the array, and then sum the results. This
would yield the intermediate sum 14,610 + 2,922 + 17,532. We can recognize this sum as the one
which appears in the standard calculation
487
× 36
2,922
14,610
17,532
(Here we have filled in the often omitted zero in the 14610 in order to make clear the connection
with the row sums.)
On the other hand, if we sum down the columns of the array, then we obtain as an intermediate
step 14,400 + 2,880 + 252 = 17,532. We can recognize these summands as those appearing in the
standard procedure with the factors taken in the other order:
36
× 487
252
2,880
14,400
17,532
Thus from the array of products of place value components, we can produce the standard method
for computing products, in either order, by summing first along rows of the array, or along the
columns. By looking at the rows of the array, one sees that the sum along a row amounts to the
product of a decimal component of the first factor times the second factor. This is indeed what we
calculate to perform the standard algorithm. Interpreting the standard procedure for both orders of
the factors in terms of the array of products of components could help students to understand why
both give the same answer (as they must if multiplication is to be commutative), although the
intermediate steps are so different.
19
However, we have seen that in some sense the most natural way to do an addition is to break each
summand into its place value components, and to add the components of a given magnitude.
Neither of the standard algorithms does this. However, this procedure can be carried out in quite
an elegant way, giving an algorithm known as the lattice method, or array method, or Napier’s
Bones. It refines the observation that, if we write the products of components in an array, as in the
example above, then the order of magnitude of the products decreases from left to right along
rows, and from top to bottom along columns, and products of the same order of magnitude are
lined up along diagonals running from upper right to lower left.
To see how the lattice model works for the example above, we put the place value component
products in a 3 × 2 array of boxes, and we write each one as the product of the digit product times
the appropriate power of 10:
Figure 6: The Lattice Model
36 X 487 :
Column
Totals:
Row Totals :
30 X 400 =
12,000
30 X 80 =
2,400
30 X 7 =
210
14,610
6 X 400 =
2,400
6 X 80 =
480
6 X 7 =
42
2,922
14,400
17,532
Grand Total
252
2,880
We can see that the terms for which the power of 10 is constant are arranged along diagonals
going from upper right to lower left. This can be refined. Break up each digit product into its base
ten components. If we divide each box by a diagonal line, and put the tens digit of the digit
product (times the appropriate denomination) above the line, and the ones digit (times the
appropriate denomination) below, then terms with the same order of magnitude are again arranged
along the diagonals, which are now composed of equilateral isosceles triangles, facing up and
down alternately:
Figure 7: The Elaborated Lattice Model
10,000
2,000
2,000
400
200
2,000
40
400
400
80
20
10
2
Now summing along these diagonals gives allows us to sum numbers all of the same order of
magnitude, and the sum of these sums will give us the product:
Figure 8: The Elaborated Lattice Model,
Summing Along Diagonals
Diagonal Sums:
10,000
6,000
10,000
2,000
2,000
400
200
2,000
40
400
400
1,400
10
80
130
2
2
Grand Total: 10,000 + 6,000 + 1,400 + 130 + 2 = 17,532
This says that
36 × 487
= 10,000 + 6,000 + (1,000 + 400) + (100 + 30) + 2
= 10,000 + (6,000 + 1,000) + (400 + 100) + 30 + 2
= 10,000 + 7,000 + 500 + 30 + 2 = 17,532
We hope that it is clear from this example how to carry out the general case. Napier’s Bones is a
pleasant algorithm for multiplication which better displays the structure of the process than does
the standard algorithm. We remark that the process of summing along the diagonals in Napier’s
Bones is parallel to shifting the partial products to the left in the standard method.
2D. Division
The interactions between place value ideas and division are especially illuminating, both for
division and for place value. Indeed, study of division brings out two quite distinct interpretations
or properties of the base ten expansion of a number. In one direction, the idea of division-withremainder provides us with a systematic way to find the digits in the base ten expansion of any
whole number. In the opposite direction, the effort to find a method for computing the base ten
expansion of the quotient in a division problem forces us to find a quite different interpretation of
the base ten expansion. This second interpretation connects directly with the uses of base ten
expansions for approximation.
Division-with-Remainder
Division, in the strict sense of inverting a multiplication, cannot in general be carried out in the
whole numbers, or even in the system of decimal fractions (finite decimals). This is one of the
least satisfactory aspects of decimal arithmetic, and is a major reason for using fractions, i.e.,
general rational numbers.
Instead of division as the inverse of multiplication, one deals with a somewhat looser substitute,
division-with-remainder. This can always be carried out with any two whole numbers, but it does
not yield a single number as a result. It yields a pair of numbers: the quotient, and the remainder.
Division-with-remainder, both theoretically and practically, depends on the order properties of
21
whole numbers. These are discussed in some detail in section 3. Here we assume a basic
acquaintance.
The usual recipe for division-with-remainder (often called the division algorithm, although it gives
little guidance about how one would do division in practice) depends on the order structure of the
whole numbers. It is contained in the formula
b = qa + r
Here b is being divided by a. The number q is called the quotient of b by a. The number r is the
remainder. The essential property of r is that 0 ≤ r < a.
b
is also called the quotient of b by a. It can cause confusion to
a
use exactly the same term to refer to two different things by the same word. When we want to be
clear that we are talking about the integer r coming from division-with-remainder, we will talk
about the integer quotient, or the Iquotient for short.
Division-with-remainder can be thought of in terms of measurement. It is as if we were measuring
a ribbon of length b into pieces of length a, and we can fit q of them in, and then there is a shorter
piece of length r left over. Here is a picture of that process for b = 17 and a = 3. As can be seen, in
this case q = 5 and r = 2.
Of course, the rational number
Figure 9: Division-with-Remainder
17 = 5 X 3 + 2
17
0
20
10
3
3
3
3
3
2
Since r is non-negative, it follows that qa < b. Secondly, and this is the key property of q, it is also
true that q is the largest whole number such that qa ≤ b. For if we increase q by 1, we see that
(q+ 1)a = qa + a = b – r + a = b + (a – r)
and the inequality r < a means that a – r > 0. Hence (q + 1)a > b. For 17 ÷ 3 the figure above
illustrates this4.
Here are some numerical examples:
17 = 5 × 3 + 2.
Thus 15 ÷ 3 = 5, remainder 2, and 5 × 3 = 15 < 17 < 18 = 6 × 3.
200 = 11 × 17 + 13.
Thus 200 ÷ 17 = 11, remainder 13, and 11 × 17 = 187 < 200 < 204 = 12 × 17.
6484 = 72 × 89 + 76. Thus 6484 ÷ 89 = 72, remainder 76, and 72 × 79 < 6486 < 73 × 89.
Interpretations of the Base 10 Expansion
We will now discuss two important interpretations of the base ten expansion of a number. One of
them involves repeated division by 10. The other is less well known, but is crucial for
4
Just as there are two distinct operations, division and division-with-remainder, there should be two different symbols for
division, so that one could be sure which operation was meant. However, this essay has introduced enough new terminology
and declines the task of inventing this notation.
22
understanding long division, and also for the approximation properties of the base ten expansion to
be discussed in the next section.
The Base 10 Expansion and Successive Division by 10
The base ten expansion of a whole number can be interpreted in terms of division-with-remainder
by 10. This interpretation begins with the observation that the 1s digit of a number is the
remainder when the number is divided by 10. This is because all positive powers of 10 are clearly
divisible by 10. Precisely, we have the formula
10n = 10n-1 × 10
for any positive integer n. From this we can see that, if we take any place value number larger than
10, and divide it by 10, then the quotient is just the place value number with the same digit, but of
order of magnitude one less. For example,
7000 ÷ 10 = 700, and 400 ÷ 10 = 40, and 50 ÷ 10 = 5
We can extend this to a simple rule for dividing (with remainder) any base ten number by 10: to
get the Iquotient, we drop the rightmost digit, and this digit is the remainder. For example
7,452 = 745 × 10 + 2,
and
83,244 = 8,324 × 10 + 4,
7452 ÷ 10 = 745, remainder 2
or
or
83,244 ÷ 10 = 8324, remainder 4
Continuing to divide by 10 produces all the digits of the number in succession:
745 ÷ 10 = 74, remainder 5,
and 74 ÷ 10 = 7, remainder 4, and 7 ÷ 10 = 0, remainder 7
This scheme can be easily extended to dividing by higher powers of 10. For example, to divide by
100, we drop the rightmost two digits, and the dropped digits form the remainder; for dividing by
1000, we drop the three rightmost digits and they form the remainder; and so forth:
7452 ÷ 10 = 745, remainder 2,
7452 ÷ 100 = 74, remainder 52,
7452 ÷ 1000 = 7, remainder 452,
or
or
or
7452 = 745 × 10 + 2
7452 = 74 × 100 + 52
7452 = 7 × 1000 + 452
The Base 10 Expansion and Approximation by Place Value Numbers
There is a second, seemingly quite different, interpretation of the decimal expansion of a number
that connects with the idea of approximation. It is this interpretation that will allow us to find the
digits of the base ten expansion of an Iquotient, i.e., to perform long division.
Given any number, we can try to approximate it by sums of place value numbers. There are of
course many, many ways to do this, but there is one way to do it which deserves to be called
simplest: the greedy algorithm. Suppose we are given some collection of special numbers, and we
want to express an arbitrary number as a sum of the special numbers. This can be thought of in
terms of placing orders at a factory. Imagine that the factory makes widgets, and it packages them
in boxes of certain sizes. If you want to place an order for a certain number of widgets, the greedy
algorithm for doing this would say to do the following: take the largest package smaller than your
order; then subtract it, and repeat, with the difference taking the place of your original order.
Continue as long as you can: either until you have filled your whole order, or until all available
packages are larger than the amount you still want.
23
This will not always succeed. For example, if the packages have sizes 2 or 3, and you want to
order 4, then the greedy algorithm would say, first order a package of 3. This leaves you wanting
one more; but there is no package of size 1. However, if you had been more moderate, you could
have ordered two packages of size 2 to get exactly 4.
If the collection of package sizes is the collection of place value numbers, however, then the
greedy algorithm for approximation will work, and it exactly produces the decimal expansion of
the number. Here is an example. Take the number 7,452. Since
7,000 ≤ 7,452 < 8,000
the greedy approximation algorithm would say to write 7, 452 = 7,000 + (7,452 - 7,000) = 7,000 +
452. The next step in the approximation process is to do the same for the “error term" 452. Since
400 < 452 < 500
we see the greedy algorithm says to write 452 = 400 + (452 - 400) = 400 + 52. Similarly, the next
step is to write 52 = 50 + (52 – 50) = 50 + 2. Finally, the remainder term of this stage, namely 2, is
itself a place value number, so our approximation is exact, we have zero remainder, and the
process stops. We have found successively that
7,452 = 7,000 + 452 = 7,000 + 400 + 52 + 7,000 + 400 + 50 + 2
The end result of this approximation process is exactly the base ten expansion of 7,452.
The reason behind the success of the greedy algorithm in producing the base ten expansion is the
following property of the collection of all place value numbers:
For any place value number, of given order of
magnitude, the next largest place value number is
obtained by adding the denomination of that same
order of magnitude.
Thus, in our example above we saw that for 7000, the next largest place value number is 8000 =
7000 + 1000, and for 400, the next largest place value number is 500 = 400 + 100. This property is
perhaps quite obvious when the digit of the place value number is 1 through 8. Then we get the
next largest place value number by just increasing the digit by 1. The most interesting case is
when the digit is 9. Then the next largest place value number is the denomination of order of
magnitude one more. However, the rule holds in this case also: we get the next place value number
by adding the denomination of the order of magnitude of the starting number. Examples of this are
9 + 1 = 10, and 90 + 10 = 100, and 900 + 100 = 1000, and so forth.
From this key fact, we can give an argument for any number, as follows. Let n be any whole
number, and let m1 be the largest place value number not larger than n. Then the next largest place
value number is m1 + d1, where d1 is the denomination of the same order of magnitude as m1. Since
m1 is chosen to be as large as possible, we know that m1≤ n < m1 + d1. By subtracting m1 from all
three numbers, we can conclude that 0 ≤ n - m1 < d1. Since the next stage of the approximation
process is to approximate the remainder n - m1 with a place value number, we see that the next
stage will use a number of smaller order of magnitude. Hence, in the final sum, there will be at
most one term of any given order of magnitude, so (according to the characterization of base ten
expansions noted in section 1) the sum will be exactly the expanded form of n.
24
An interesting observation comes out of this point of view on the base ten expansion. Since the
difference between a number and its largest place value component has smaller order of
magnitude, it is less than the largest place value component. In other words, in the equation n = c + r,
where c is the leading place value component of n, and r = n - c is the rest of n, we have c > r, and
n
therefore 2c > c + r = n, or c >
> r. We may state this in words: For any whole number n, the
2
largest place value component of n accounts for more than half of n.
We will give a more refined version of this statement in the Decimal Estimation Theorem of
section 3.
Approximation and Long Division
Using the ideas above, we can give a brief description of the basic ingredients in long division.
Our treatment uses symbolism which might not be appropriate for students being introduced to
long division. However, the principle governing the procedure should be both teachable and
plausible, and examples should make it convincing. The theoretical justification might be delayed
until an algebra course.
We remind ourselves that, in dividing-with-remainder b by a, we are seeking the largest whole
number q such that aq ≤ b. Because of the approximation interpretation of the base ten expansion
of a number, this rather bare-bones description actually adapts rather directly to what one does to
find the base ten decomposition of q, when a and b are given as base ten numbers. That is, it tells
us how to do long division.
Basic Fact of Long Division (BFLD): Suppose that q is the Iquotient of b by a, so that b = qa + r,
with 0 ≤ r < a. Then the largest place value component of q is simply the largest place value
number q1 such that q1a ≤ b.
The problem of long division is to find the place value components of q. The BFLD says, simply
look for the largest place value number q1 such that q1a ≤ b, and this will turn out to be the largest
place value component of q. This effectively solves the problem: to find the rest of the place value
expansion of q, subtract to get b - q1a = b2, and repeat the process with b2. This is exactly what one
does in the long division algorithm.
The justification of the BFLD is parallel to the discussion of the approximation interpretation of
the base ten expansion. The argument uses the same fact that guaranteed success of the greedy algorithm
discussed above: that the next place value number larger than q1 is q1 + d1, where d1 is the denomination
with the same order of magnitude as q1. Because of this, if q1 is chosen to be the largest place value
number such that q1a ≤ b, we must have q1a ≤ b < (q1 + d1) a. It follows that (q1 + d1)a > b ≥ qa. Dividing
by a tells us that q1 + d1 > q. Keeping in mind that q1 + d1 is the next largest place value number after
q1 , we conclude that this says that q1 is the largest place value number less than or equal to q.
According to the approximation interpretation of the base ten expansion, this means that q1 is the
largest place value component of q, which is the claim of BFLD.
Using BFLD to do Long Division
Thanks to the Ordering Algorithm given in section 3, finding q1 is a fairly easy thing to do in
practice. This is illustrated in the example below.
The full process of long division proceeds by iteration, using the BFLD repeatedly. Having
determined the largest decimal component of q, we can now work with the difference b - aq1, and
25
look for the largest decimal component q2 such that aq2 ≤ b - aq1. Continuing in this fashion, we
will eventually find the complete decimal expansion of the quotient. This procedure is in fact
essentially the standard long division algorithm. Thanks to the structure of the family of decimal
components, the connection between the theoretical construct and the practical procedure is quite
close.
Here is an example. In it, we will try to emphasize the practical aspects of the determination of q1,
and to make clear the iterative aspect of the calculation.
Compute 93,482 ÷ 274.
First we determine the order of magnitude of the quotient. This is easy to do by looking at the
orders of magnitude of dividend and divisor, and comparing first digits (or occasionally more, if
the first digits are equal). In our case, 274 has magnitude 2 and first digit 2, while 93,482 has
magnitude 4, and first digit 9. We conclude that 274 × 100 < 93,482 < 274 × 1000. Hence 100 < q1
< 1000.
Next we want to find the first digit of q1. This requires estimating the largest multiple of 274 × 100
that does not exceed 93,482. We will see in section 3 that the first two digits of a base ten number
determine it with a fair amount of accuracy. Therefore, if we replace the dividend 93,482 with
93,000, and also replace 274 × 100 with 274 × 100 = 27,000 we will probably get the correct value
if we find the largest digit d such that d × 27,000 < 93,000. This is in fact a problem of estimating
with two-digit numbers; the final zeroes on both numbers do not affect the answer. Hence, the
issue of finding the digit of q1 can in the great majority of cases be reduced to a question of
dividing a two-digit number, or perhaps a three digit number, by a two digit number. If this way of
estimating the digit of q1 fails, then the true digit will be at most one larger or smaller, and is easy
to determine when the guess made by this method is wrong.
In our case, 3 × 27 = 81 < 93 < 4 × 27 = 108. Hence, our candidate for q1 is 300. We form
93,482 – 300 × 274 =93,482 – 82,200 = 11,282 < 27,400.
Hence q1 = 300 is the correct guess.
We next proceed to the second step, which is 11,282 ÷ 274. We already know 274 × 100 = 27,400 >11,282,
and clearly 274 × 10 = 2740 <11,282, so the next term in the base ten expansion of the Iquotient
will be a digit times 10, and out job now is to determine that digit.
In this step, the dividend 11,282 has magnitude 4, while 274 × 10 = 2740 has magnitude 3. We
again round both numbers to two places, and consider the approximate division problem 11,000 ÷
2700. We can forget about the last two zeroes here so effectively we are dealing with the problem
110 ÷ 27. Since 4 × 27 = 108 < 110 < 135 = 5 × 27, our guess for the digit is 4. Then forming the
difference,
11,282 – 40× 274 = 11,282 – 10,960 = 322
we see that it is positive, but less than 2740, so 4 is indeed the right guess.
We are left with the problem 322 ÷274. It should be clear that the quotient in this case is 1. The
remainder then is 322 - 274 = 48.
Summing up, we have found that
93,482 = (300 + 40 + 1) × 274 + 48
or 93,482 ÷ 274 = 341, remainder 48.
26
The reader can check that our steps are exactly the steps of the standard long division algorithm,
written out with commentary. Thus, the standard algorithm provides a compact format for carrying
out the iterative calculation of the successive digits of the Iquotient in division-with-remainder. In
addition, the discussion of estimation in the next session furnishes a rationale for guessing the
digits in the decimal expansion of the quotient by doing division of a two- or three-digit number
by a two-digit number. The guess that this gives will be correct most of the time.
Remark: Although it is beyond the scope of this essay, the approximation interpretation of the
base ten expansion extends directly to finding the decimal expansion of any real number, and this
algorithm extends likewise, allowing one to find the decimal expansion of any rational number
b/a.
In summary, the BFLD makes long division a practical and easy-to-use algorithm. To execute it
does require fluency with multiplication and subtraction. It also demands some skill in estimation,
but what is needed in practice boils down to division of two- or three- digit numbers by two-digit
numbers. Due to its recursive nature, it does have a step-by-step feel to it, and it can seem
somewhat lengthy. However, the overall understanding of what it is doing, as summarized by the
BFLD, can give students confidence in the process. Also, since long division starts out by finding
the largest digits of the result (rather than the smallest digits, as with the algorithms for the other
three operations), one can carry it out until one has a sufficiently accurate answer, and then stop.
Our discussion of the operations of arithmetic has tried to make the case that thinking explicitly in
terms of place value components yields benefits for developing understanding of arithmetic. We
will next attempt to show that doing decimal arithmetic with emphasis on the role of place value
components has advantages beyond computation. It also makes estimation a relatively simple
matter, and it brings out the connections between decimal arithmetic and algebra.
3. Decimal Components, Estimation and Error.
Order and Magnitude
Besides the arithmetic operations, the other main structure on the whole numbers is order: we
know when one whole number is larger than another. (In fact, order can be expressed in terms of
addition: whole number a is greater than whole number b if a = b + c, where c is another (nonzero) whole number. However, the role order plays is quite different from the role of the
operations.)
The decimal system is highly compatible with the ordering of the whole numbers, and makes it
easy to compare numbers. Recall that we defined the order of magnitude for any whole number in
section 1. It is the order of magnitude of the largest place value component of the number, or
equivalently, it is one less than the number of digits in the standard base ten expression of the
number. As we noted there, a basic fact is that order of magnitude sorts numbers according to size:
any number of a given order of magnitude is larger than any number of a smaller order of
magnitude. For example, the largest number of magnitude 2 is 999, and the smallest number of
magnitude 3 is 1000 = 999+1. This simple fact has several important consequences. The first is a
recipe for comparing any two base ten numbers.
Ordering Algorithm
To decide which of two base ten numbers a and b is larger, compare their place value components.
Find the largest order of magnitude for which the place value components of a and b are different.
Then the number with the larger component of that magnitude is larger.
27
In particular, if the order of magnitude of a is larger than the order of magnitude of b, then a is
larger than b. If they have the same order of magnitude, but the leading (i.e., largest) place value
component of a is larger than the leading component of b, then a is larger. If the leading
components agree, proceed downward in orders of magnitude, until you first find a magnitude at
which the components of a and b differ. Then the larger number is the one with the larger
component of that magnitude.
Relative Place Value
Often, we want to do more than say whether one number is larger than another, we want to say
how much larger it is, or that it is very much larger. We also want to say when two numbers are
close to one another. Paying attention to place value components also makes these things easy to
do. The main point here is that multiplying by 10 just increases order of magnitude by 1. This is
true no matter what decimal place we are talking about. Thus, given any decimal place, the place
just to the left represents numbers 10 times as large as the given place, and the place just to the
right represents numbers only 1/10 the size of the given place. Two places to the left, you find
numbers 100 times as large as where you are, and two places to the right, the numbers are only
1/100 of the place where you are. For facility with estimation, it is important that students
understand not only the values of each place, but also these relative values of the places. The grasp
of relative place value also can support the introduction of decimal fractions. We will discuss this
in section 4.
Absolute and Relative Error
Relative place value ideas help us understand that, in representing numbers, it is the largest few
decimal components (the leftmost few decimal places) that account for most of the size of a
number. This is true no matter what the magnitude of the number. These ideas are particularly
important in dealing with error, which is unavoidable whenever we must deal with “real-life"
numbers, meaning numbers which come from measurement. These are always known only
approximately, and the question of how accurately they are known is fundamental. Here the key
idea is relative error: the size of the error involved in approximating a given number, in
comparison to the number itself. If you only have a few dollars, you care quite a bit whether it is
$4 or $6. However, if you have around a thousand dollars, probably you do not care too much if it
is $1004 or $1006. And if you have a million dollars, even a thousand dollars more or less, let
alone $2, probably will not keep you up at night.
Suppose you need to work with a number V, but you use a possibly different value v in its place5.
22
and the decimal fraction 3.14 are often used in place of the number
For example, the fraction
7
355
and 3.14159. Call v an approximation to V. The
π; more accurate approximations are
113
(absolute value of) the difference, | V - v | = e is the absolute error of approximating V by v. For
many purposes, the actual size of the error e is not as important as how it compares to the size of
V. To capture this idea, we define the relative error to be the ratio e÷V. The relative error can be
thought of as measuring the error in the most relevant units, namely units of the correct amount.
Both the absolute error and relative error are controlled easily in terms of decimal components.
5
This could be for a variety of reasons: you may not know the precise value of V, or it simply may be more convenient to use
v (or both). If V is irrational, like
numerical computation.
π
or 2 , we do not know the exact value of V, so we must use an approximation in any
28
The main observation is that, the more alike the decimal expressions (read from the left) of two
numbers are, the closer together the numbers are. To be precise about this, we will suppose that
we have two numbers V and v, of the same order of magnitude, and that their largest place value
components are equal, or the two largest ones, or perhaps more. There are two ways of
formulating this: we can specify the number of decimal components that agree, or we can specify
the order of magnitude of the largest decimal components that differ. It is easy to convert one type
of information into the other, but we distinguish them because they reflect the two points of view relative versus absolute error.
We can use the decompositions afforded by division-with-remainder by some power of 10, as
described in section 2D on division, to describe the relationship between V and v. We suppose that
their common order of magnitude is m, and that their place value components of magnitude greater
than ℓ agree. This means that, if we divide-with-remainder them both by 10ℓ+1, the Iquotients will
be the same. In other words, we can write
V = q10ℓ+1 + R
and v = q10ℓ+1 + r
((m,ℓ)Decomp)
The common Iquotient q will have order of magnitude m - ℓ - 1, which means that it has m - ℓ
decimal places. The remainders R and r have order of magnitude ℓ or less: they are less than
10ℓ+1.
In this situation, the following simple statement holds.
Decimal Estimation Theorem (DET): Suppose that V and v are two numbers of magnitude m,
and suppose their decimal components of magnitude ℓ or larger are equal. (That is, the largest
m - ℓ + 1 decimal components of V are equal to the corresponding components of v.) Let their
common Iquotient when divided-with-remainder by 10ℓ be q. Then:
i) The error e = | V - v | has order of magnitude at most ℓ - 1, and hence is less than 10ℓ.
e 1
1
ii) The relative error < . In particular, regardless of q, it is always less than m−l .
V q
10
An important case of the DET is rounding down, also sometimes called front end estimation.
This involves ignoring, or more precisely, replacing by zero, the smaller place value
components of a number. For example, we might approximate 7452 by 7450 or by 7400, or by
7000, depending on the relative importance of accuracy and simplicity of thought.
The relative error of approximating 7452 by 7450 is
7452 − 7450
2
1
1
=
<
<
7452
7452 3700 745
The last amount is the estimate given by the DET, with m = 3 and ℓ = 1. The reason the actual
relative error is so much smaller than the estimate provided by DET is because the actual ones
digit is 2, but it potentially could be as large as 9, and the DET must work for all possible ones
digits.
29
The relative error of approximating 7452 by 7400 is
7452 − 7400
7452
=
52
1
1
<
<
7452 140 74
The last amount is the estimate provided by DET when m = 3 and ℓ = 2. If we replace 7452 by
7000, the relative error is
7452 − 7000
452
1 1
=
< <
7452
7452 16 7
where the final amount is again provided by DET, with m = 3 and ℓ = 3. We note that the
estimates provided by part i) of DET for our example 7452 are better than the worst case
estimates provided by part ii). These would be 1/100, 1/10 and 1/1, rather than 1/745, 1/74 and
1/7 respectively.
We hope that these examples provide enough hints about how one might prove the DET in
general.
Limits of Accuracy in Practice
The DET expresses in a concise way the sense in which knowing the beginning (the largest
several decimal components) of the decimal expression of a number tells us most of what we
want to know about the number as a magnitude. If we are number theorists, it may be
important to know all the decimal components of a number, in order to tell, for example, if it is
a perfect square, or if it is divisible by 7, or if it is prime. But if it is a number coming from a
measurement, then all we need to know, or can expect to know, is an approximate value, and
the largest decimal components supply this with great efficiency. Significant figures is a
common term that refers to the number of digits we are sure are correct when giving the result
of a measurement. If we are sure that we know the largest k decimal components of a number,
we say that the number is known to k significant figures. The DET (part ii)) tells us that if we
know a number to k significant figures, then the relative error between the measured values and
1
the “true" value is at most k −1 . So, for example, if we know a number to 4 significant
10
1
figures, we know it with a relative error of less than
.
1000
In fact, it is rather rare to know a measured number with such accuracy. Take the example of
the “radius" of Earth. It is approximately 4,000 miles, but sometimes you will see figures such
as 3,928 miles. However, the last digit does not have a clear meaning. In speaking of the radius
of Earth, we are pretending that Earth is a perfect sphere. However, it is not. It deviates from
being a perfect sphere in three ways. First, because of its rotation, it is slightly oblate –
flattened at the poles, and thickened around the equator. Second, it has bumps and dimples Mount Everest and the Challenger Deep. Third, because of the motion of its liquid interior, it is
slightly deformed, with a bulge in the north Pacific. These imperfections mean that it does not
make sense to speak of a “radius" of Earth more accurately than about 10 miles. Thus, the
radius of the Earth is a number defined only to 3 significant figures. Most other “real-life"
numbers have similar limitations. The results of polls are typically accurate only to about +3%,
which is slightly better than one significant figure. Although the U.S. Census reports state
populations as exact numbers of people, in the millions (6 significant figures) or the 10s of
millions (7 significant figures), it is lucky if these numbers are accurate to 3 figures.
30
In summary, the notion of significant figures successfully captures many of the key notions of
error and approximation for decimal numbers. Especially, it provides an easy way to deal with
relative error. For many purposes, it suffices to know a number to one significant figure. For
most purposes, two significant figures are enough, and it is rare to know a “real-life" number to
more than 3 significant figures. (Financial transactions can be exceptions to these rules; but it
is also debatable how “real-life" the numbers involved in them are.)
Scientific Notation
Scientists and others who use numbers resulting from measurements are mainly interested,
first, in their size, and second, in the accuracy with which they are known. The way of writing
numbers known as scientific notation is designed to exhibit these two points very clearly.
Formally, scientific notation is simply a variant way of writing decimal fractions. Instead of
writing a decimal fraction d in the conventional way, with a certain number of places to the left
of the decimal point, and a certain number to the right, scientific notation rewrites it in the
form
d = (d÷10m) × 10m
where m is the order of magnitude of d, so that d÷10m is between 1 and 10. (A slight variant,
used in IBM computers, writes d = (d÷10m+1) × 10m+1.)
In this form, scientific notation separates out a key feature of a number, namely its magnitude,
and displays it prominently. A further convention frequently used by scientists (although it is at
odds with standard mathematical notation) is to report only significant digits when recording a
number in scientific notation, in order to be precise about the accuracy to which the number is
known. (Also, measurements are usually presented rounded to the nearest significant digit,
rather than rounded down, as assumed in the DET. This is a minor issue.) By this convention,
2.3 × 104 means some number known to be between 22,500 and 23,500, while 2.30 × 104
means some number known to be between 22950 and 23050. With this understanding,
scientific notation uses exponents and the idea of significant digits to express in a very direct
way the size and accuracy of a “real-life" number. The DET (modified slightly to account for
the rounding convention usually used in practice) tells us tells the maximum possible relative
error in a number with given significant digits.
4. Decimal Fractions.
If place value is firmly understood, and also the basic ideas of fractions are in place, the
extension of the decimal system from whole numbers to decimal fractions should be
straightforward. We will discuss some details here. In this treatment we will assume a mature
understanding of the nature of fractions and division. Thus, our formulation would probably
not be immediately usable with students first meeting decimal fractions. However, we hope it
provides some guidance for a discussion at that level. As with whole numbers, decimal
fractions are usually introduced a few decimal places at a time, and this makes it easier to be
concrete, discuss particular cases and do examples.
Basic Properties of Fractions
We will assume that the following basic understandings about fractions, multiplication and
division are in place.
1) Fraction notation expresses numbers as the result of a process of division. More precisely,
31
b
signifies “b divided by a,” or, the result of dividing b by a. (Technical
a
terminology: b is the numerator and a is the denominator.) This makes sense for any two
numbers b and a, not just whole numbers. In particular, a and b themselves could be fractions.
b
However, when b and a are whole numbers, this does denote the fraction
in the usual
a
colloquial sense. This is the mature understanding of fraction notation6. It is not clear how
many students achieve this final conceptualization of fractions. However, it seems unlikely that
one can advance very far in algebra without such an understanding, so if this essay can serve to
draw attention to this important stage in understanding fractions, it will be well pleased.
the symbol
2) Division by a number b is the same operation as multiplying by the reciprocal (or
1
multiplicative inverse) of b. The reciprocal is often written as , since this is indeed the result
b
a
1
of dividing 1 by b. Thus = a ⋅ . The exponential notation b-1 is also often used
b
b
1
instead of .
b
1 1 1
= ⋅ . That is, the reciprocal of a product is the product
bc b c
of reciprocals. We will need to use this often. (This is a consequence of the more fundamental
Rules of Arithmetic7, but we will not give the derivation here.)
3) We will also record the identity
Definition of Decimal Fractions
If we understand these points, decimal fractions (a term which unfortunately is usually
a
shortened to “decimals") are easy to define: a decimal fraction is a fraction of the form m
10
where a and m are whole numbers (or zero). That is, a decimal fraction is a fraction in the
colloquial sense, whose denominator is a power of 10.
Although decimal fractions include all whole numbers, there are many numbers which can not
be expressed as decimal fractions. Especially, general fractions are not decimal fractions. The
first example is 1÷3. It follows that the system of decimal fractions is not closed under division
(since it includes whole numbers, so if it were closed under division, it would include all
b
quotients
for any whole numbers b and a).
a
Since decimal fractions are just a restricted kind of fraction, and do not allow us to do
arithmetic freely (since they are not closed under division) why should we pay special attention
to them? There are two redeeming features of decimal fractions which have led to their
b
6
There seem to be at least two earlier conceptions of the fraction a . The first and least productive is that it means “b out of
a". That is, one selects b elements from a set of a elements. At this level, a fraction is not even a number, it is a relationship
between two (whole) numbers. In the next stage of understanding fractions, the fraction is related to a unit, and it is understood
that the a things are equal pieces of the unit. When the fraction is related to a unit, it can be conceived of as a quantity in itself,
that is, as a number, and it can be used in arithmetic. However, this usage may not be very flexible. The rules for operating
with fractions may be somewhat mysterious, and it may be hard to bring the rules of arithmetic to bear. In particular, the
Associative Rule for multiplication is not readily usable.
7
See footnote, page 1.
32
widespread (and since the advent of electronic calculators, more and more dominant) use. They
are
i) The computational algorithms developed for base ten whole numbers extend with almost no
change to allow us to compute with decimal fractions. Since arithmetic with general fractions
seems to cause trouble for large numbers of people, the relative simplicity of computing with
decimal fractions is very attractive.
ii) Any real number can be approximated as closely as one wishes by decimal fractions. This is
often expressed by saying that the decimal fractions are dense in the real numbers.
Furthermore, the approximation process is straightforward, in the sense that the same “greedy
algorithm" that works for whole numbers (see section 2D) applies in general.
This combination of familiar, effective computational methods with the ability to approximate
means that one can do approximate arithmetic to arbitrary accuracy with arbitrary numbers, so
for practical purposes, working with decimal fractions allows us to compute anything we want.
In fact, as we can see from the DET of §4, the accuracy of decimal approximations increases
rapidly with the number of digits of accuracy, so that we can usually get enough accuracy
without incurring an onerous computational load.
Computing with Decimal Fractions, I
Let us look at computing with decimal fractions. We will only discuss addition and
multiplication. Subtraction and division are similar.
c
d
and f = l . Multiplying e and f can be done using the Any Which
k
10
10
Way Rule for multiplication, combined with the properties 2) and 3) above. We see that
Suppose we have e =
e×f=
c
1
1
1
1
d
× l = (c × k ) × (d × l ) = (c × d ) × ( k × l )
k
10
10
10
10
10 10
1
1
c×d
= (c × d ) × ( k
) = (c × d ) × ( k +l ) =
l
10
10k + l
10 ×10
(MultDec)
Here we have inserted times signs to be very explicit that we are doing multiplication.
To add two decimal fractions, as for any two fractions, we need to put them over a common
denominator. This is easier than for two arbitrary fractions, since one of the powers 10k and 10ℓ
divides the other. More precisely, suppose that k ≤ ℓ. Then by the Law of Exponents, we know
c
c ×10l − k
c ×10l − k
that 10ℓ = 10k×10ℓ-k, so that we may write k = k
c÷10k = (c × 10ℓ-k ) ÷
=
l−k
l
10 10 ×10
10
(10k×10ℓ-k ) = (c × 10k )÷10ℓ . This allows us to add:
e+ f =
c
d
c ×10l − k
d
c ×10l − k
+
=
+
=
10k 10l
10l
10l
10l
(AddDec)
Since we know how to do the integer arithmetic to compute the numerators of the final
expressions in equations (MultDec) and (AddDec), these formulas do in fact provide practical
recipes to add and multiply decimal fractions. Here is an example of how they work:
33
take
e=
47
10
then
e× f =
and
e+ f =
and
f =
7452
100
47 × 7452 350244
=
1000
1000
47 ×10 + 7452 470 + 7452
.
=
100
100
Decimal Expansions
However, as they are stated above, the formulas MultDec and AddDec are still somewhat
abstract. They do not tell us what is special about decimal fractions. We could establish quite
similar formulas for fractions with denominators which are powers of any number.
The reason for singling out denominators which are powers of 10 is of course because we use
base ten expansions to write whole numbers, and dividing by powers of 10 is highly
compatible with this system. We know that the base ten system writes every number as a sum
of place value numbers – digits times powers of 10. When we divide by powers of ten, the
numbers we get will still be writable in this way, except that to express the fractional part of
the quotient, we will have to use digits divided by powers of 10. Thus for example, the
numbers used in the calculations above are
47 40 + 7 40 7
7
=
=
+ = 4+
10
10
10 10
10
and
7452 7000 + 400 + 50 + 2 7000 400 50
2
5 × 10
2
5
2
=
=
+
+
+
= 70 + 4 +
+
= 70 + 4 + +
100
100
100 100 100 100
10 ×10 100
10 100
d0
1
100
The base ten system for writing whole numbers is based on the denominations
d1
d2
d3
d4
d5
d6
10
100
1000
10,000
100,000
1,000,000
101
102
103
104
105
106
…
…
…
To express a given whole number, we multiply these denominations by appropriate digits, and
sum up.
As the above examples illustrate, we can extend this system to decimal fractions by extending
the system of denominations to include reciprocals of powers of 10:
…
d-3
d-2
d-1
d0
d1
d2
d3
…
…
1
10
100
1000
…
1
1
1
1000
100
10
1
100
101
102
103
…
…
1
1
3
10
2
10
10
34
Just as in the whole number context, to express an arbitrary decimal fraction, we multiply
certain of these denominations by appropriate digits, and sum up. We will call the numbers u × dk,
where u is a digit, place value numbers, as before, but now we will let k be negative as well.
(When k is negative, it would be appropriate to refer to “fractional place value numbers.”)
Then, as with whole numbers, every decimal fraction is a sum of place value numbers. This is
what allows us to extend our calculation procedures in a straightforward fashion from whole
numbers to decimal fractions.
We started this essay with place value notation for whole numbers. Since the base ten
expansion of a number expressed it as a sum of place value numbers, and the place value
numbers are digits times powers of 10, to know the number it is enough to know the digit for
each power of 10, and this is what the place value notation tells us: it writes the sequence of
digits, from right to left, beginning with the digit for 1 = 100. The location of a digit in this
sequence tells us the power of 10 by which it should be multiplied.
We now see that decimal fractions have a similar expansion: they can be written as sums of
digits times powers of 10, possibly using negative as well as positive powers. A given decimal
fraction is determined by the digits which multiply the powers of 10. (Of course, these digits
are non-zero only for finitely many powers.) These digits naturally make a sequence, with the
position of a digit in the sequence determined by which power of 10 it multiplies. When we
were dealing with integers, this sequence had a natural start, namely the ones place (order of
magnitude zero), and progressed to infinity to the left. When we deal with fractions, however,
the sequence can extend arbitrarily far in both directions. There is no longer any starting point.
So, to know which digits go with which powers of 10, we have to show explicitly where to
find the order of magnitude zero (the ones place). Our standard convention for doing this is to
1
place a mark, the decimal point, between the digit for 100 = 1 and the digit for 10−1 = . This
10
locates the digits. The digits to the left of the decimal place multiply non-negative powers of
10, as before, and the digits to the right multiply the negative powers. For our example
numbers from above, we have
47
7452
= 4.7 and
= 74.52
10
100
When we write a number like this, multiplying it by 10 is extremely easy to do. Multiplying by
10 turns each denomination into the next larger one, so each digit now should multiply the next
larger denomination, so it should get shifted one space to the left, relative to the decimal point.
This amounts just to shifting the decimal point one space to the right. Similarly, division by 10
can be recorded by shifting the decimal place one space to the left.
Decimal Expansions as an Address System on the Number Line
A good way to think about the decimal expansion is that it gives an address system for locating
numbers on the number line. This gives a geometric way to picture how a number is
approximated by the sum of its leading decimal components. Consider a number
4
5
2
f 7452
, which is 7.452 in conventional notation. Let’s
like f ′ = =
= 7+ +
+
10 1000
10 100 1000
locate f′ on the number line. Here is a piece of the number line that goes from 0 to 10:
35
7.452
Figure 10: The Number Line (from 0 to 10)
0
1
2
3
4
5
6
7
8
9
10
The 7 in 7.452 tell us that that f′ is between 7 and 8. Thus, it is somewhere in the 8th unit
interval in this piece of the number line.
To locate f′ more precisely, we look at the next place in its decimal expansion. We divide the
interval between 7 and 8 into ten equal subintervals.
Figure 11: The Number Line from 0 to 10,
showing 1/10ths between 7 and 8
0
1
2
3
4
7.4
5
6
7.452
7.5
7
8
9
10
The 4 in 7.452, representing 4÷10, tells us that f′ is located in the fifth of these subintervals,
between 7.4 and 7.5.
To locate f′ still more accurately, we should next divide up the interval between 7.4 and 7.5
into 10 equal subintervals.
Figure 12: The Number Line from 7 to 8, showing 1/10ths between 7 and 8,
and 1/100ths between 7.4 and 7.5
7.452
7
7.1
7.2
7.3
7.45
7.46
7.4
7.5
7.6
7.7
7.8
7.9
8
We see that the divisions are getting rather fine, almost too fine for us to see the subintervals!
This is graphic proof of the rapidity with which the decimal expansion approximates a number.
If, however, we magnify the interval between 7.4 and 7.5, then we can see what happens at the
next step.
Figure 13:
The Number Line from 7.4 to 7.5
7.4
7.41
7.42
7.43
7.452
7.44
7.45
36
7.46
7.47
7.48
7.49
7.5
Notice that when we expand the interval from 7.4 to 7.5 by a factor of 100, it looks just like the
interval from 0 to 10, except that the labels are changed. The division points 7.40, 7.41, 7.42,
etc., are seen to divide the interval into 10 equal subintervals. The next place value component,
that is the 5 in 7.45, tells us that f′ is in the subinterval between 7.45 and 7.46. Finally, the last
decimal component locates f′ in one of the 10 equal subintervals between 7.45 and 7.46. We
are now working at distances which we cannot distinguish visually. We would need a
microscope to really see what is going on. For practical purposes, knowing four terms of the
decimal expansion of a number locates it to a high degree of precision. However, in principle,
decimal fractions can be carried out to any number of places. The number π has been
computed to several billions of decimal places. This is a degree of accuracy that is
unimaginable. Because of the arbitrary fineness of the divisions produced by the decimal
fractions, any number can be located with arbitrary accuracy on the number line, by a decimal
address, using the procedure described above. More precisely, we can consider that the decimal
expansion 7.452 does not locate a given number, but serves as an approximate address for any
number in the interval between 7.452 and 7.453. This is very much the way the idea of
significant digits is used for recording “real life" numbers.
Computing with Decimal Fractions, II
Now let’s discuss some more the methods for calculation with decimal fractions. The key facts
which shape decimal arithmetic are the formulas for addition of two place value numbers of
the same order of magnitude, and for the product of two place value numbers. We may
summarize them as follows:
u × 10k + v × 10k = (u + v) × 10k ,
and
(u × 10k ) × (v × 10ℓ ) = (u × v) × 10k+ ℓ
In these formulas, u and v are digits, and k and ℓ are whole numbers. In performing addition,
the next step is to decompose u + v into its place value components, and use this to rewrite
(u + v) × 10k as a sum of place value numbers. Similarly, in performing multiplication, the next
task is to decompose the product u × v into its place value components, and use this to rewrite
(u × v) × 10k+ ℓ in terms of its place value components. These procedures both look exactly
the same when we allow negative exponents for the powers of 10. As a consequence, the
arithmetic of decimal fractions is essentially the same as whole number decimal arithmetic.
We will look at each operation in a little more detail. We start with addition. The key fact
about the denominations that gives shape to addition of base ten numbers is that each
1
successive denomination is 10 times the previous one, and, reciprocally, is
of the next one:
10
1
dk+1 = 10dk and dk-1 = d k , for all k ≥ 0.
10
This basic relationship between successive denominations persists when we enlarge the list of
denominations to include the dk for k < 0. This is what guarantees that the procedure for
addition of decimal fractions will be essentially the same as for whole numbers. To be able to
formulate the recipe in general, we should extend the terminology of order of magnitude to the
enlarged system. Thus, we will say that any place value number a × dk has order of magnitude
k, for k positive or negative (or zero). With this agreement about language, we can give exactly
the same recipe for adding decimal fractions as for adding base ten whole numbers, except that
we can no longer assume that magnitude zero is the smallest order of magnitude.
37
1. For each order of magnitude, add the place value numbers of
that order of magnitude from each summand.
2. If the sum of the digits for a given order k of magnitude is 10
or more, break the sum into its place value components:
(u + v) × d k = (10 + w) × d k = d k +1 + g × d k .
Combine the dk+1 with the components of magnitude k + 1.
If one wishes to minimize the amount of rewriting, one would start with the smallest order of
magnitude for which there are non-zero components, and proceed upwards one order of
magnitude at a time. This gives the standard procedure. However, in mental math, dealing with
numbers with only a few non-zero place value components, the additions might be done in a
different order. If one only wants an approximate answer, one would start at the other end, with
the largest components, and stop after a few places.
For multiplication, the basic facts governing the form of the computational algorithm were the
(extended) Distributive Rule and the formula for multiplying two place value numbers, as
given above. That formula has two basic ingredients. First we need to know how to compute
the product u × v of digits. These are the multiplication facts. The other part of the product
involves multiplying two denominations. This is governed by the Law of Exponents:
dk d ℓ = d k+ℓ , or 10k10ℓ = 10k+ℓ for all k ≥ 0 and ℓ ≥ 0
In fact, this formula describing the behavior of the dk under multiplication extends to all k and
ℓ, positive or negative. For example,
d-2d4 =
1
10, 000
× 10,000 =
= 100 = d2 = d −2+ 4 .
100
100
A very elegant way of saying this is to extend the exponential notation to all integers, negative
as well as non-negative. Assuming that we already understand 10k as the product of k tens for k > 0,
we also define
1
100 = 1 and 10-k = k for k > 0.
10
With these definitions, we can check that
The Law of Exponents is valid for all integers.
Thus, the analogy between addition and multiplication displayed by exponential notation
extends to include a parallel between subtraction and division.
With these observations, we have essentially arrived at the usual recipes for decimal
arithmetic. For addition (or subtraction) do the usual thing of placing one number under the
other, making sure to line up the decimal points. This will ensure that you are adding like
powers of 10, and then you can proceed as if dealing with whole numbers. The factor of 10ℓ - k
appearing in the final formula of (AddDec) is what is accomplishing the lining up of decimal
points in that computation.
38
For multiplication, shift the decimal place to the far right of each number. This amounts to
turning the decimal fractions into whole numbers by multiplying by a suitable power of 10.
Then do your computation using the normal procedure for whole numbers. Finally, divide by
the product of the powers of 10 you previously multiplied by. The Law of Exponents says that
this amounts to shifting the decimal place in the product by the sum of the number of places
you shifted each factor. This is exactly what is reflected in the formula (MultDec). The
denominators c and d are the original two numbers with their decimal points shifted to the far
right; dividing their product by 10k+ℓ then shifts the decimal point back to where it should be.
Let’s illustrate these ideas by doing the example calculations above using place value
components rather than converting to whole number computations, as in (AddDec) and
(MultDec). We can do addition by combining components of the same order of magnitude and
regrouping/carrying just as when dealing with whole numbers:
47 7452
7
5
2
7 5
2
) = 70 + (4 + 4) + ( + ) +
+
= (4 + ) + (70 + 4 + +
10 100
10
10 100
10 10 100
12 2
10 + 2 2
2
2
= 70 + 8 + +
= 70 + 8 +
+
= 70 + 8 + (1 + ) +
10 100
10
100
10 100
2
2
2
2
7000 900 20
2
= 70 + 8 + 1 + +
= 70 + 9 + +
=
+
+
+
10 100
10 100 100 100 100 100
7000 + 900 + 20 + 2 7922
.
=
=
100
100
First, notice that the answer is the same as we got using the general formula of (AddDec). What
is more important, however, is that the details of the computation are essentially the same as in
the general formula; for to compute the numerator of the formula in (AddDec) by the usual
rules, we proceed as follows:
47 × 10 + 7452 = 470 + 7452 = (400 + 70) + (7000 + 400 + 50 + 2) = 7000 + (400 + 400) + (70 + 50) + 2
= 7000 + 800 + 120 + 2 = 7000 + 800 + (100 + 20) + 2 = 7000 + (800 + 100) + 20 + 2
= 7000 + 900 + 20 + 2 = 7922.
If we compare this computation with the previous one, we see that the steps of one correspond
closely to the steps of the other. The only differences are those used to change the form of
some power of 10. Thus, in the second calculation, we take a step at the beginning to convert
2
as
47×10 to 470, and in the first calculation, we take a step to rewrite the terms 70, 9 and
10
fractions with denominator 100, and another to combine them into one sum. However,
additions of the terms of the same order of magnitude, and the regroupings of the sums which
add to 10 or more (there is only one such sum in this example) correspond precisely.
39
Here is the multiplication:
47 7452
7
5
2
)
×
= (4 + ) × (70 + 4 + +
10 100
10
10 100
5
2
= 4 × 70 + 4 × 4 + 4 × + 4 ×
10
100
7
7
7 5 7
2
+ × 70 + × 4 + × + ×
10
10
10 10 10 100
20 8
= 280 + 16 + +
10 100
28 35
14
+49 + +
+
10 100 1000
48 43
14
= 280 + 65 + +
+
10 100 1000
280, 000 + 65, 000 + 4,800 + 430 + 14 350, 244
.
=
=
1000
1000
In this computation, the second and third lines are the expansion of the product into an array
according to the Extended Distributive Rule. The fourth and fifth lines compute the indicated
products of place value numbers from lines two and three. Note that the powers of 10 in this
array are constant along the upward slanting diagonals. The sixth line displays the sums along
these diagonals. Finally, the seventh line puts all terms over the common denominator of 1000,
and adds all the summands. Note that this calculation is completely parallel to the
multiplication of the numerators (i.e., 47×7452), but is slightly more complicated due to the
appearance of decimal components for negative powers of 10. The alternative computation, to
just compute the product of the numerators, then divide by the product of the denominators
(i.e., by 1000), would have been simpler. This latter computation is the usual recipe. The final
step, of dividing by 1000, corresponds to the recipe of where to put the decimal point.
5. Decimal components and estimation in arithmetic.
Thinking in terms of relative place value also provides efficient ways of estimating sums and
products. We will avoid exact details here, to keep the discussion short, but the principle is
clear: when estimating, focus on the largest decimal components. With our understanding of
how decimal components enter into arithmetic, we can describe what to do to get an estimate
with desired accuracy for a sum or product.
Estimating Sums
For a sum, simply retain the sums of the largest decimal components. The three largest
components of the larger summand, and the components of the same magnitudes for the other
summand, will give an approximation to the sum with relative error small enough for most
purposes. For example,
83, 244 + 5, 293 83, 200 + 5, 200 88, 200
Here the exact sum is 88, 537, so the relative error is
40
88,537 − 88, 400
88,537
=
137
1
<
.
88,537 600
Error Propagation in Addition
Absolute error at worst adds under addition. That is, if u is an approximation to U, with
absolute error e1=|U-u|, and v is an approximation to V, with absolute error e2=|V-v|, then the
absolute error of u+v as an approximation to U+V is at most e1+ e2. This follows from the
Triangle Inequality for absolute value:
|(U+V) – (u+v)| = |(U-u) + (V-v)| ≤ |U-u| + |V-v| = e1+ e2.
A very pleasant fact is that relative error does not increase under addition, provided that all
summands are positive. We assume that we are adding positive numbers U and V. Suppose that
u approximates U with absolute error e and relative error δ. Similarly, suppose that v
approximates V with (at most) the same relative error. One can show that relative error is
preserved in addition. That is, the relative error with which u+v approximates U+V is also at
most δ. We will leave the verification of this as an exercise.
Error Growth in Subtraction
When working with positive numbers, the operation in which relative error may increase is
subtraction. In fact, in subtraction, one may completely lose control over relative error: it can
become arbitrarily large. This is because the difference of two large numbers may be small,
much smaller than either of the numbers; but when this happens, there is no reason for the
error of approximation to have shrunk. Since to compute relative error, we divide by the
number being approximated, the quotient defining relative error may be quite large. In fact, if
we take two different approximations u1 and u2 to the same number U, then the difference u1 u2 is supposed to be an approximation to U - U = 0. Clearly, if u ≠ u2, the relative error is
infinite. More generally, if u and v are approximations of U and V, each with absolute error e
or less, and if the difference between U and V is less than e, then the number u - v can be any
sufficiently small number. In particular, it could be any number of absolute value not more
than e. Thus, one has essentially no control over U - V, except for a bound on how large it can
be. For this reason, it would be a very dicey proposition to use u - v in any further calculation
involving U - V, particularly one that involved dividing by U - V. In less extreme situations,
one may retain some information about U - V, but with a substantial loss of significant digits.
Estimating Products
To discuss estimation of products, we recall that a product of decimal numbers is the sum of all
the products, of any place value component of one factor times any component of the other
factor. Of these summands, the largest will be the product of the largest decimal components of
each factor. The next largest will be the two terms gotten by taking the product of the second
largest component of one factor, with the largest component of the other. To illustrate this,
look again at the example from section 2C, the product 36×487. The largest product of place
value components is 12,000 = 30×400. The two next largest summands are 30×80 = 2400 and
6×400 = 2400. The sum of these three terms is 16,800. If we use this as an estimate for the
17,532 − 16,800 1
product, which is 17, 532 the relative error is
<
, or less than 5%. If we want
17,532
20
more accuracy, we would take the next largest summands, namely 30 × 7 = 210 and 6 × 80 =
480. Adding these to the 16,800 gives 17,490. The relative error is now
17,532 − 17, 490
1
<
.
17,532
400
41
The situation can be understood nicely by using the lattice model for multiplication. Imagine
the products of decimal components arranged in an array of boxes, as in section 2C. The
largest product is in the upper left hand box. The next largest products are on the diagonal
adjacent to this box. The products next again in size are located on the next diagonal down or
to the right, and so forth. For many purposes, taking the largest six terms, on the three upper
left diagonals, will provide a sufficiently accurate approximation to the exact product.
Figure 14: Approximating Products Using Lattice Multiplication
(Example)
36 X 487 :
12,000
2,400
210
2,400
480
42
First Cut: 12,000
Second Cut: 12,000 + 4,800 = 16,800
Third Cut: 12,000 + 4,800 + 690 = 17, 490
When more accuracy is needed, a straightforward continuation of the above procedure
prescribes the next terms to add. The upshot is a method to get a (relatively) very accurate
approximation to a multidigit multiplication with fairly little work.
Figure 15: Approximating Products Using Lattice Multiplication
(General Scheme)
First
Cut
2nd
Cut
3rd
Cut
2nd
Cut
3rd
Cut
4th
Cut
3rd
Cut
4th
Cut
4th
Cut
4th
Cut
Caveat: because of the rollover phenomenon (see section 2B), we cannot guarantee that the
approximate answers we compute by the above methods will agree in any decimal places with
the true answer. However, we can show that they do approximate well the true answer, in the
sense that they produce a small relative error. Usually, in a sense that can be made precise, they
will have the same leading decimal components as the exact answer.
42
6. Decimal Components and Algebra.
The Rules of Arithmetic Rule in Algebra
Discussing arithmetic in terms of decimal components should help prepare students for algebra
in several ways. One way is that it would create increased awareness of and facility with the
Rules of Arithmetic, which are also the rules for manipulating algebraic expressions. (And
essentially, they are the only rules, so they provide a remarkably compact summary of what
constitutes legitimate algebraic manipulation.) Also, the Rules themselves are most succinctly
expressed via symbolic equations, so they would give students exposure to the ideas of literal
symbolism and variables. In these ways, this approach should promote increased confidence
and facility with algebraic manipulation.
Base Ten Numbers as Polynomials in 10
Another way in which working with decimal components would prepare students for algebra is
that it would bring out the deep analogy between decimal arithmetic and algebra. As we
remarked at the outset, decimal notation exploits algebra in the service of arithmetic. Decimal
numbers can be usefully thought of as “polynomials in 10.” Emphasizing this connection
should shed light on arithmetic, and also make algebra more familiar and learnable.
Here are some examples of the analogy. Consider the numbers 21 and 13. We can compute
21 + 13 = 34, and 21 × 13 = 273.
Now consider the expressions 2x + 1 and x + 3. We can also compute, using the same rules as
we did for regular arithmetic, that
(2x + 1) + ( x + 3) = 3x + 4 and (2x + 1) × ( x + 3) = 2x2 + 7x + 3.
In fact, if we do these computations step-by-step, using the Rules of Arithmetic carefully, we
will see that they are completely parallel:
21×13 = (20 + 1 )× (10 + 3) = (2 × 10 + 1) × (10 + 3)
= (2×10) × 10 + (2 × 10) × 3 + 1 × 10 + 1 × 3
= 2 × (10 × 10) + (2 × 3) × 10 + 10 + 3
= 2 × 100 + 6 × 10 + 10 + 3 = 200 + (6 + 1) × 10 + 3 = 200 + 7 × 10 + 3 = 200 + 70 + 3 = 273.
(2x + 1) × ( x + 3) = (2 × x + 1) × ( x + 3)
= (2 × x) × x + (2 × x) × 3 + 1 × x + 1 × 3
= 2 × (x × ) + (2 × 3) × x + x + 3
= 2 × x2 + 6 × x + x +3 = 2x2 + (6 + 1) × x +3 = 2x2 + 7 × x + 3 = 2x2 + 7x + 3.
In these computations, the first lines translate the expression into less compact form, making
the implicit multiplications and additions explicit. There is one more step in the case of the
base ten numbers, since the addition of terms is already explicit in the polynomial, but not in
43
the base ten numbers. The second line expands the product using the Extended Distributive
Rule. The third line simplifies the products of individual terms, using the Any Which Way
Rule for multiplication. The last line is then devoted to recompactifying the expressions. In the
middle step, the terms involving 10 or x are combined using the Distributive Rule. Again, there
is one extra step for the base ten number.
If students do some calculations like these, they should see parallels between the calculations
with decimal numbers and the calculations with polynomials. Discussion could bring out the
key to the analogy: that if we set x = 10, then the polynomial computations become the
arithmetic computations. Further investigation will show that the analogy is imperfect. For
example (2x + 4) + (3x + 7) = 5x + 11, whereas 24 + 37 = 61. Nevertheless, if we plug x = 10
into 5x + 11, we do get 50 + 11 = 61 as the value. Students who are used to thinking in terms
of decimal components should recognize 50 + 11 as the intermediate result we obtain in adding
24 + 37, before we combine ten 1s into a 10 to get the standard decimal form of 50 + 11. In
other words, polynomial arithmetic is decimal arithmetic without the regrouping process (and
hence, with arbitrarily large coefficients).
In this analogy, the order of magnitude of decimal numbers translates into the degree of a
polynomial: the largest power of x which appears in the polynomial with a non-zero
coefficient. Because of the absence of carrying in polynomial arithmetic, degree is in several
ways better behaved than order of magnitude. For example, the degree of the sum of two
polynomials is always less than or equal to the larger of their degrees. The degree of a product
of two non-zero polynomials is (exactly) the sum of the degrees of the two factors.
Just as order of magnitude guides us in dividing decimal numbers, degree is what we use to
find the quotient of two polynomials. The recursive procedure for finding the quotient of two
polynomials is quite analogous to the one described above for long division of decimals. Given
a ( x)
two polynomials a(x) and b(x), the highest degree term of the quotient
is the monomial
b( x )
c1xm such that a(x) - c1xm b(x) = a1x has degree lower than a(x). This term is quite easy to find:
it is exactly the quotient of the highest degree terms of a(x) and b(x). To find the next term in
the quotient, one uses the same procedure for a1x and b(x). This continues until one gets a
difference of degree less than the degree of b(x); this is then the remainder.
Identities in Arithmetic
The connection between polynomial arithmetic and ordinary arithmetic can be developed and
exploited in other ways. For example, the famous identity
x2 – y2 = (x + y)(x – y)
has many applications in arithmetic. One very serious contemporary application is its use as a
basis for algorithms to factor large numbers. On a more mundane level, it can be used to do
mental math. Take the product 43 × 7. This can be computed mentally by thinking of 43 = 45 - 2 and
47 = 45 + 2. Hence 43 × 47 = (45 + 2) × (45 - 2) = 452 - 22. To compute 452, we can use the
same trick: Write 40 = 45 - 5 and 50 = 45 + 5. Therefore, 452 = 40 × 50 + 52 = 2000 + 25, and
so, 43 × 47 = 40 × 50 + 25 – 4 = 2021. Similar tricks can be used to develop mental math skills,
and to find factorizations of fairly large numbers whose prime factors also are fairly large (i.e.,
no factors less than 20). Since this essay is already rather long, we will curtail further
examples. However, a variety of calculations revealing parallels between decimal arithmetic
and algebra are given in the Exercises.
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On Beyond . . .
Regarding decimal numbers as “polynomials in 10” amounts to giving the variable x the value
10. This is called specialization. Specialization can be used to interpret other number systems
in terms of polynomials. The specialization which turns polynomial arithmetic into decimal
arithmetic amounts to requiring x to satisfy the equation x – 10 = 0. Polynomials can be made
to mirror other number systems by positing that x satisfies other equations. For example,
requiring x2 – 2 = 0 would make polynomials act like numbers of the form a + b 2 ; in other
words, x is imitating the irrational number 2 . Requiring x to satisfy x2 + 1 = 0 will essentially
create a copy of the complex numbers.
Here we are getting into fairly advanced ideas, drawing close to abstract algebra. It is
questionable whether the constructions of the last paragraph could be productively inserted
into the K-12 curriculum. However, it does seem desirable that high school teachers should be
conversant with these issues. Whether or not they get used in school mathematics, these
possibilities demonstrate the influence that place value has throughout the K-12 curriculum,
and point to the desirability of treating it more explicitly, and making it one of the “big ideas"
that help students arrange the mathematics they learn into a coherent whole.
References
[BP] S. Baldridge and T. Parker, Elementary Mathematics for Teachers, Septon Ash
Publishers, 2004.
[CFL] T. Carpenter, M. Franke and L. Levi, Thinking Mathematically: Integrating
Arithmetic and Algebra in Elementary School, Heinemann, Portsmouth NH 2003.
[KSF] J. Kilpatrick, J.Swafford and B.Findell, eds., Adding It Up: Helping Children
Learn Mathematics, National Academies Press, Washington, DC, 2001.
[Ma] L.Ma, Knowing and Teaching Elementary Mathematics, Lawrence Erlbaum Associates,
Mahwah, NJ, 1999.
[Ock] S. Ocken, Algorithms, Algebra and Access, www.nychold.com/ocken-aaa01.pdf
[Sing] Curriculum Planning and Development Division, Ministry of Education, Singapore,
Primary Mathematics 1A, 1B, US Edition, Federal Publications, Singapore.
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