INT. J. MATH. EDUC. SCI. TECHNOL., 1993, VOL. 24, NO. 1, 77-83
Non-decimals:
fractions in bases other than ten
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by RINA ZAZKIS
Faculty of Education, Simon Fraser University,
Burnaby, B.C., V5A 1S6, Canada
and DARLENE WHITKANACK
Department of Mathematics, Northern Illinois University,
DeKalb, IL 60115, U.S.A.
(Received 29 November 1991)
This paper generalizes the idea of representing numbers in numeration
systems in bases other than ten to representing fractions. An algorithm for
converting fractions to representations in bases other than ten is suggested.
Various classroom activities at different levels -- junior high school through
college -- are proposed.
1. Introduction
We believe that a unit of instruction on numeration systems with bases other than
ten increases students' understanding of the structure and computational algorithms
of the base ten numbers, as well as their appreciation of the invention of the HinduArabic numeration system. This paper generalizes the idea of representing numbers
in bases other than ten to representing fractions. The process of solving the problems
posed invites the students to 'create mathematical knowledge' and to deepen the
understanding of various representations of rational numbers. Our approach assists
students in establishing connections between various mathematical topics: placevalue representation and decimal fractions, geometric sequences and their sums and
notions from number theory such as primes, relatively prime numbers, prime
decomposition and, in particular, use of Fermat's Little Theorem.
2. Meet non-decimals
Let us start by looking at the following number representations: 302.14five,
0.222four or 43.52,,,.
Following the metaphor of 'soapless soap', we decided to call these numerals
'non-decimal decimals' or, more briefly, 'non-decimals'. Soapless soap serves the
main purpose of soap, but lacks its basic ingredients. In a similar way, non-decimal
imitate decimals in their form and shape. They are written in one line, with a point
separating the integer and the fractional part of the number, but they lack the 'main
ingredient' of decimals—a base often. In the discussion of non-decimals we will use
the term 'separating point' for the point that is usually called 'decimal point'.
How big or how small are the above non-decimals? What does each numeral
mean? In order to clarify the confusion, it helps to find the value of the above
numerals in the decimal system.
0020-739X/93 $10-00 © 1993 Taylor & Francis Ltd.
78
R. Zazkis and D. Whitkanack
Let us take for example 302.14. In the 'regular' decimal system representation
the digits are placed in the following place-value chart.
...
104 103 102 101 10°
3
0
KT1 KT2 1(T3 ...
.
2
.
1
4
The digit 3 represents here three hundreds, 0 represents no tens, 2 represents two
ones, 1 represents one tenth and 4 represents four hundredths. We can summarize
this representation in writing the number in 'expanded notation':
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302.14=3 x 100 + 0 x 10 + 2 x 1 +1 x (l/10) + 4x (1/100)
Now let us look at 302.14fWe. Similarly, the digits can be put in the place value
chart for base five.
51
5°
:-»
c-2
c-3
In the integer part, 2 represents two ones, 0 represents no groups of five and 3
represents three groups of twenty five or three groups of 5 2 . Expanding the idea of
place value into the fractional part, students can justify the convention that the 1
represents one fifth and the 4 stands for four twenty-fifths. Therefore, a base ten
representation of 302.14five is as follows:
302.14five = 3x 25+0 x 5 + 2 x 1 + 1 x(l/5)+4x(l/5 2 )
= 75+0 + 2 + 1/5 + 4/25 = 779/25=77.36 or 77.36ten
In the decimal representation of fractions in which the place value of the digit in
the k th place to the right of the decimal point is 10~*. Similarly, in base b it would be
b~k.
Therefore,
43.52 s!x =4x6 + 3 x l + 5x(l/6) + 2x(l/6 2 )
= 24 + 3 + 5/6 + 2/36 = 27 8/9 = 27.8 or 27.5,en
The above examples illustrate how any non-decimal can be converted into a
decimal or a common fraction.
In this article we will discuss the conversion in the opposite direction, i.e.
changing common fractions or decimals to non-decimals. For example, we will show
how it is possible to express the fraction 1/7 as a non-decimal in base four.
3. Decimal fractions and common fractions—conversion between
representations
Before we discuss conversion into non-decimals, let us look in detail at the
conversion from common fraction representation to decimal representation.
Every rational number can be expressed as either a finite or a repeating decimal.
In order to predict whether the given rational number would appear as a finite or as a
repeating decimal, we consider the prime decomposition of its denominator in
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Non-decimals: fractions in bases other than ten
79
reduced form. If the prime decomposition consists only of powers of 5 and 2 (the
prime factors of 10), the rational number will appear as a finite decimal. This
property means that there must exist an equivalent fraction with a denominator of a
power of 10. So decimals can be thought of as suitably adjusted numerators. For
example, 1/4 = 25/100 = 0-25, 1/20 = 5/100 = 0.05, 1/16 = 625/10000 = 0.0625.
Any factor of the denominator other than powers of 2 or 5 results in a
representation of the rational number as a repeating decimal.
Now, let us express a common fraction with a prime denominator different from
2 or 5 as a decimal fraction. For example, we expect 1/11 to be represented as a
repeating decimal. Performing the division with the help of a calculator, we get
0.09090909091 and may suppose that 1/11=0.09.
We expect 1 /7 to appear as a repeating decimal as well. Dividing 1 by 7 the display
reads 0.14285714286. Again, we may suppose that 1/7 is equal to the repeating
decimal 0.142857.
With 1/17 the problem is more complicated. Performing the division we get
0.05882352941, which doesn't hint at any repeating pattern.
These examples suggest two questions. One, how can we be sure that the digits
that seem to repeat themselves indeed have that repeating pattern, and are not a part
of a larger repeating period? Two, how can we find the repeating period when it is
larger than the display given us by the calculator?
The standard approach to these questions focuses on performing long division
and considering the remainders. Once the same remainder appears a second time, the
repeating pattern is established.
In what follows we suggest an alternative algorithm for conversion into the
repeating decimal form, without carrying out the division of numerator by
denominator. The main advantage of this algorithm, as opposed to performing long
division, is that it can be easily extended to conversion of fractions into repeating
non-decimals, since performing division in bases other than ten is more difficult.
As a basis for developing this algorithm, we begin with a closer look at repeating
decimals. Let us start with 0.3. It can be expressed as an infinite sum:
0.3 = 3/10 + 3/100 + 3/1000 + . . . = £ 3/10"
n=l
As a sum of a geometric progression, where the first element a= 3/10 and the ratio
r=l/10, the sum S is computed by
Let us proceed with the example of 0.123. This can be expressed as an infinite
sum
123/1000+123/10002 + 123/10003 + . . . = £ 123/1000"
n=l
As a sum of a geometric progression, the above expression is equal to 123/999 which
can be reduced to 41/333.
In general, the repeating decimal O.a1a2 ... a,,1, where n is the number of digits in
the period, can be expressed as a sum of infinite geometric sequence, where the
constant ratio is 10~". The sum equals axa2 .. . an/(10" —1).
'The notation a±ai...an means here concatenation of the digits alt a2,...,an, not
multiplication.
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R. Zazkis and D. Whitkanack
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Returning to 1/7, we seek to reverse the process in order to express it as
a repeating decimal. We are looking for the expression of the form
aja ? • • • aJ(lO"— 1), that can be reduced to 1/7, i.e. we are looking for a number of a
form 10"-1 that is divisible by 7. Checking the numbers 9,99,999,9999,... we find
that the smallest number of the form 10"—1 that 7 divides is 999999 or 1 0 6 - l .
Therefore we claim that there are 6 digits in a period of a repeating decimal
equivalent to 1/7. What is left is to express 1/7 with 999999 as the denominator. We
find that 1/7 = 142857/999999 and 1/7 = 0.142857.
In a similar way, for 1/11 we find that 99 is the smallest number of the form
10"—1 that is divisible by 11. Here « = 2, therefore there are 2 digits in the period
1/11=9/99 = 09/99 = 0.09.
4. Why does it work?
In this section we provide some theoretical background using number theory
notation.
Were we just lucky in finding a number of the form 10" — 1 that was divisible by 7 ?
Were we just lucky once again by finding a number of the form 10" —1 that was
divisible by 11? Will we have such luck with all the other fractions?
Actually this 'luck' is guaranteed for prime denominators by Fermat's Little
Theorem:
For any integer b and any prime p, bp = b(mod p)
An equivalent statement is:
For any prime p and integer b, where p and b are
relatively prime, V~l = \(mod p) or bv~1 — 1 = 0(mod p)
The form useful for our algorithm is:
For any prime p and integer b, where p and b are
relatively prime, p divides (bp~l — 1).
In the example of the conversion of 1/7, b= 10 andp = 7. 7 is prime and relatively
prime to 10. Fermat's Little Theorem guarantees in that case that 7 divides 106 — 1.
In the example of the conversion of 1/11, 6=10 and />=11. 11 is prime and
relatively prime to 10. Fermat's Little Theorem guarantees in that case that 11
divides 1010 — 1. Actually Fermat guarantees the existence of a number of the form
10" — 1, but doesn't show the minimal one. As we found 102 — 1 is also divisible by 11,
which gives us a period of 2 digits and not 10 and lets us express 1/11 as 0.09 and not
0.0909090909. We conclude that the number of digits in the period of the repeating
decimal representation of a fraction with denominator/) is either/) — 1, or a divisor of
p — 1. The latter is achieved when a period of length p — \ itself contains a repeating
cycle.
As a summarizing example let us represent 1/13 as a repeating decimal. We look
for a number of the form 10" — 1 that is divisible by 13. The Fermat's Little Theorem
guarantees the period of 12 digits because 13 divides 1012 — 1. Looking for a possible
smaller period, we may try the divisors of 12. 13 doesn't divide 99, 999 or 9999,
which precludes the possibility of 2, 3 or 4 digit period. But 13 divides 999999 (or
106 — 1), and this tives us a six-digit period in the repeating decimal representation of
1/13:
1/13 = 76923/9999989 = 0.076923.
Non-decimals: fractions in bases other than ten
81
5. What about non-prime denominators?
Let us start with 1/6. We cannot find a fraction equivalent to 1/6 with a
denominator either of the form 10" or 10" —1. In this case it is useful to note that
1/6 = 1/2-1/3 or 1/6 = 2/3-1/2. That lets us represent 1/6 as follows:
1/6 = 0.5-0.3=0.16 or 1/6 = 0.6-0.5 = 0.16
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As another example let us take 1/21. Representing 1/21 as a difference of 1 /3 and
2/7 reduces the problem of conversion to the two previously solved problems:
1/21 = 1/3-2/7 = 0.3-0.285714 = 0.047619
It can be shown that each fraction with a composite denominator can be
expressed as a sum of fractions whose denominators are prime or powers of primes.
Suppose we are given a fraction 1 lab, where a and b are relatively prime. Then we can
write ax + by = 1. (The last statement is suggested by the Euclidean algorithm: if d is
the greatest common divisor of a and b, then there exist integers x and y such that
d=ax + by). Dividing through by ab, we get \jab = x\b-\-yla.
6. From fractions to non-decimals
As with decimals, by looking at the prime decompositions of the base and of the
denominator of the common fraction, we are able to predict whether we should
expect a finite or repeating non-decimal. If the prime factors of a denominator are a
subset of the prime factors of the base, we expect a finite representation. Otherwise
we get a repeating non-decimal.
Let us write 5/8 in base six. The prime decomposition of 8 is 23. The prime
decomposition of 6 is 2 x 3 . Therefore we will get a finite non-decimal.
Looking for a fraction equivalent to 5/8 with the denominator expressed as a
power of 6, we find that 5/8 = 135/216 = 135/63. Therefore there will be 3 places used
after the separating point. Now we need to express 135/63 as non-decimal in base six.
Since 135 = 343sixor
135/63 = ( 3 x 6 2 + 4 x 6 + 3)/63 = (3x6 2 )/6 3 + (4x6)/6 3 + 3/63
= 3/6 + 4/62 + 3/63
we find that
5/8 = 0.343six
The result is easy to check, working out the reverse conversion:
0.343six = 3/6 + 4/36 + 3/216 = 108/216 + 24/216 +3/216 = 135/216 = 5/8
After expressing 1/7 as a repeating decimal, we are ready now to express it as a
repeating non-decimal, for example in base four. We are looking for a number of the
form 4" — 1, that is divisible by 7. We find that 7 divides (43 — 1) as 7 divides 63, and it
is the smallest power of 4 that satisfies the divisibility. We may presently conclude
that there will be a three-digit period in the base four non-decimal representation of
1/7. Now let us write 1/7 with 63 as denominator: 1/7 = 9/63. The numerator 9
represents the decimal value of the period, and we have to express it in base four.
Since 9 = 21four, we conclude that 1/7 = 0.021four. The result is easy to check:
0.021 four = 2/4 2 + 1 /4 3 + 2/4 5 + 1 /4 6 + 2/4 8 + 1 / 4 9 . . .
82
R. Zazkis and D. Whitkanack
This sum can be seen as the sum of two separate infinite decreasing geometric
progressions:
Taking the sum of each progression we obtain:
= 8/63 + 1/63 = 9/63 = 1/7
7. More examples
We invite the reader to work out in detail the following examples.
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(a) Express 1/4 in base six.
Since 4 divides 62, we expect a finite non-decimal with two digits after the separating
point.
1/4 = 9/36; 9ten = 13six.
Therefore
l/4 = 0.13six
(b) Express 1/3 in base five.
Since 3 and 5 are relatively prime and 3 divides (52 —1) or 24, we expect a repeating
non-decimal with a two-digit period.
1/3 = 8/24; 8ten = 13five. Conclusion: 1/3 = O.l3five
(c) Express 1/5 in base seven.
Since 5 and 7 are relatively prime and 5 divides (74 — 1) or 2400, we expect a repeating
non-decimal with a four-digit period.
1/5=480/2400; 480ten = 1254s<:ven. Conclusion: l/5=0.1254seven
(d) Express 1/10 in base four.
Since 1/10 = 1/2 — 2/5, we start by expressing 1/2 and 2/5 in base four.
l/2 = 0.2four
2/5 = 6/15
2
Since 1 5 = 4 - 1 and 6ten = 12four, we find that 2/5=0.Ti four . Therefore
1/10 = 1/2-2/5 = 0.2four-0.!Yfour = 0.0l2four
Warning: the above subtraction is in base four.
8. Some interesting facts and relationships
It is easily shown that l/9 = 0.1 ten . This equality can be proved directly by
representation of the repeating decimal as a sum of a geometric sequence, or
indirectly by using the fact that 1 = 0.9ten.
Students can discover the following pattern:
l/2 = 0.Tthree
l/3 = 0.Tfour
l/4=0.T fIW
Non-decimals: fractions in bases other than ten
83
and in general
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Another fascinating pattern occur when we express 1/2 in different bases:
l/2 = 0.1 two
l/2 = O.l thr «
l/2=0.2 f o u r
l/2 = 0.2five
l/2 = 0.3 six
l/2 = 0.3 seven
l/2 = 0.4eight
l/2 = 0.4 nine
l/2 = 0.5 ten
l/2 = 0.5 eleven
You may ask your students to discover the decimal representation of the nondecimal 0.2 or 0.3 in various non-decimal bases and to explain the pattern. Other
interesting activities are expressing 1 /3 in various bases and discovering a pattern for
in base b.
9. Non-decimali in the mathematics classroom
The above approach suggests a variety of activities at different levels: junior high
school through college.
In the beginning it is possible just to introduce the non-decimal representation
and 'make sense' of it by practising conversion from non-decimals to decimals or
common fractions. Comparison and ordering tasks are also useful and interesting.
For example, which is bigger: 0.2five or 0.2six? Another activity is to perform basic
operations, especially addition and subtraction, with non-decimals. As mentioned
before, these experiences increase a student's understanding of place value,
computations with decimal fractions and appreciation of our standard number
system.
Drawing an analogy with decimals, it is possible to discuss with students the
conditions needed for a finite non-decimal to be converted to either a finite or
repeating decimal. A more advanced activity is to perform the conversion in the
opposite direction, i.e. from fractions and decimals to non-decimals. We may start
with easy tasks, such as expressing 1/2 in base four or six, or 1/4 in base two. Then it
is natural to try to express 1/2 in other bases. In this case, if the base is not even, we
end up with a repeating non-decimal. A general conclusion may be drawn about the
conditions needed to convert a given fraction into either finite or repeating nondecimal. By summarizing the examples students may develop a formal algorithm and
prove or program it into the computer. The approach might be extended to bases
greater than ten as well.
It is fascinating to observe how many tools and topics the seemingly innocent
subject of non-decimals brings together. Therefore for each level an appropriate
issue and set of activities can be chosen for the joy of learning mathematics.
[1]
Bibliography
G. L., and BURGER, W. F., 1991, Mathematics for Elementary Teachers: A
Contemporary Approach, 2nd edition (New York: Macmillan), pp. 58-61.
MUSSER,
[2] NIVEN, I., and ZUCKERMAN, H. S., 1980, An Introduction to the Theory of Numbers, 4th
edition (New York: Wiley).
[3] ORE, O., 1967, Invitation to Number Theory (Washington: Mathematical Association of
America).