Adjectives that modify Nouns

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#8

$1.25

$2.50

Introduction to

Decimals

Taking the Fear out of Math

© Math As A Second Language All Rights Reserved

Prelude

When it comes to using the adjective / noun theme for teaching arithmetic, students seem to be most motivated when the nouns are related to money.

For example , even students who are

“afraid” of fractions are comfortable with such quantities as quarters and half-dollars.

So as a prelude to decimal fractions , we can talk about quantities such as $1.25.


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Renaming $1.25

We know that $1.25 is more than $1 but less than $2. Since there are no whole numbers between 1 and 2, it means that the amount is a fractional number of dollars (in fact, in the language of mixed numbers , we can write the amount as $1 25 /

100 or in “reduced” form $1 1 /

4

.

Often when we write a check we express

$1.25 in the form, $1 25 /

100

).


However, if we change the noun from

“dollars” to “cents”, the fractional adjective 1.25 is replaced by the whole number adjective 125. That is…

1.25 dollars = 125 cents

With this in mind, it might seem that the study of decimal fractions 1 could begin immediately after the arithmetic of whole numbers .

note

1

Usually we simply say “ decimal

“ when we mean “ decimal fraction

” and we say

“ fraction

” when we mean “ common fraction

”. For the sake of brevity we will adopt this common usage and say “ fraction ” when we mean “ common fraction ” and “ decimal ” when we mean “ decimal fraction ”..


In fact, when students are taught decimals after they are taught fractions , their usual comment is that it would have been much more logical to have studied decimal arithmetic first because “it is so easy”.

At first glance this comment seems to make good sense. However, what is actually true is that if all we wanted was for students to use rote memory, then it is true that decimal arithmetic should be taught before we teach the arithmetic of common fractions .


However, from a logical point of view, decimals are simply a different way of representing common fractions whose denominators are powers of 10.

In terms of whole number division, decimals result when we divide a number by powers of 10. To see how this causes a problem consider the following situation.

You want 3 people to share a “ corn bread ” equally 2 .

note

2 We enclosed corn bread in quotation marks to remind us that the “ corn bread

” is simply a visual model for representing any quantity.


Finding the Answer

Knowing fractions we would simply divide the corn bread into 3 pieces of equal size and give each person 1 piece. Pictorially...

corn bread

…and in the language of fractions.

1 /

3

1 /

3

1 /

3


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The Fly in the Ointment

Suppose that we had not yet invented fractions , and we wanted to use decimals to solve the problem. Since we can only divide by powers of ten, it will be impossible to divide the “ corn bread ” into a number of equally-sized pieces that is divisible by 3.

In fact every power of 10 leaves a remainder of 1 when it is divided by 3.


Notes

To see why every power of 10 leaves a remainder of 1 when divided by 3, notice that the powers of 10 can be written in the form of a multiple of 9 with a remainder of 1; and 9 is a multiple of 3. Specifically…

10 = 9 + 1

100 = 99 + 1

1,000 = 999 + 1

10,000 = 9,999 + 1

100,000 = 99,999 + 1

1,000,000 = 999,999 + 1


10 ÷ 3 = 3R1

100 ÷ 3 = 33R1

1,000 ÷ 3 = 333R1

10,000 ÷ 3 = 3,333R1

100,000 ÷ 3 = 33,333R1

1,000,000 ÷ 3 = 333,333R1

Applying the Above Discussion to

Our Monetary System

Suppose oranges are being sold at a rate of 3 oranges per $1 (or, in the common vernacular, “3 for a dollar”).


If you buy 3 oranges, each orange costs

1 /

3 of a dollar. However, given our monetary system, if you want to buy

1 orange, the cost would be 33 1 /

3 cents.

There is no coin in our monetary system whose value is exactly of 1 /

3 a cent. In such a case the merchant would most likely charge you 34 cents for the orange.


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Note

If the cost of each orange was rounded off to the nearest cent, each orange would only cost 33 cents (which is why the merchant charges you 34 cents if you only buy 1 orange (if he only charged 33 cents per orange you could buy 3 for 99 cents rather than for $1). However, in most cases paying

34 cents instead of 33 cents for the orange is not much of a problem.


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Note

However, if you were a buyer for a huge chain of stores and you were buying

1 million oranges, the difference between

33 cents per orange and 34 cents per orange is not negligible (unless you consider $10,000 to be negligible).


The above discussion illustrates what problems would have arisen if we had decided to introduce decimals as an extension of place value.

However, once we point out this difficulty, it is worth noting that there is a close resemblance between whole number arithmetic and decimal arithmetic.


A user-friendly way to introduce decimals is by constructing a monetary system that is based on place value .

Notice that in our own English monetary system, which was developed long before place value was invented, we have to memorize many different conversion factors such as…

5 pennies = 1 nickel 2 nickels = 1 dime

5 nickels = 1 quarter 4 quarters = 1 dollar


A similar problem applies to our system of measuring distance.

For example …

12 inches = 1 foot

3 feet = 1 yard

1,760 yards = 5,280 = 1 mile.


However, once place value was invented it became much more convenient to “memorize” that 10 was the only conversion factor we had to use.

So let’s reinvent history a little and see what might have happened if a nation that had been using the English system had decided to change its monetary system in a way that was based on the same principle as place value was. That is, 10 of any denomination was equal to 1 of the next greater denomination.


In any monetary system there has to be a smallest denomination ( sd ) 3 .

So we start with 1 sd and then construct the denominations 10 sd , 100 sd , etc.

Whatever we pick to be our smallest denomination, it is clear that no item can cost less than that amount note

3 In our own monetary system, " sd " would stand for “cents" because that is our least denomination. If we chose “dollars" as our noun , then 1 sd would mean 1 dollar.


Imagine next that technology has advanced to the stage where an item that originally cost 1 sd should only cost 1 /

2 of an sd . In terms of our previous discussion this dilemma can be overcome by selling the item at rate of 2 for 1 sd .

However, a new problem arises when a person wants to buy only 1 of these items.


The only way to resolve this problem is to create a smaller denomination. In order to preserve the place value property, no matter what name we give this new denomination its value must be

1 /

10 of an sd .

4

Continuing in this way, we can construct a monetary system that looks like…

1,000 sd 100 sd 10 sd 1 sd 1 /

10 sd 1 /

100 sd 1 /

1000 sd note

4 In our own monetary system if we use a dollar as being 1 sd , the next smaller denomination is called a dime and a dime is 1 /

10 of a dollar. In fact, the word

“dime” is a “corruption” of the word “ decimal ”.


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1,000 sd 100 sd 10 sd 1 sd 1 /

10 sd 1 /

100 sd 1 /

1000 sd

And to generalize the above chart we can replace the noun “ sd ” by other nouns such as “ meters ”. In this case, using “ m ” as an abbreviation for “ meters ”, the above chart can be rewritten in the form…

1,000 m 100 m 10 m 1 m 1 /

10 m 1 /

100 m 1 /

1000 m

…or usually expressed as… kilometers hectometers decameters meters decimeters centimeters millimeters


To make our discussion “ noun free” we can say that the denominations are… tens ones tenths hundredths thousandths

…and we can extend the chart ad infinitum in both directions, no matter what unit the denominations are modifying.


The problem occurs because we have now introduced denominations that are less than one. Specifically, in whole number place value notation, the digit furthest to the right names the ones place.

Thus, for example , when we write 35, we know that it means 35 ones. And if we had meant 35 tens, we would make sure that the digit furthest to the right (in this case, 5) was entered in the tens place, as shown below.

ones hundreds tens

3 5


35 tens

Notice that as long as the denominations are visible there is no need to annex a zero. However if we omit the denominations then we have to annex a zero to remind us that the 5 is in the tens place.

hundreds

3 tens

5 ones

0 350 ones = 35 tens

So while zeroes help us keep track of the denomination when we are dealing with whole numbers , the problem resurfaces when we are dealing with proper fractions .


For example , as long as the denominations are visible the chart below tells us that we talking about 35 tenths.

hundreds tens ones

3 tenths

5

However, once the denominations are omitted it is impossible to tell what denomination is being modified by 35.

That is, just as it could mean 35 hundreds or 35 tens, it could also mean 35 tenths,

35 hundredths, 35 thousandths, etc.


To see this in the form of a real life situation, we can think in terms of our own monetary system. In fact in addition to dimes and cents, let’s also include an obsolete coin called a mill. The mill was used at a time when a cent had enough purchasing power that prices were thought of in terms of mills. 5 note

5 Even today, in states where the tax rate is based on per hundred rather than on per thousand dollars, to figure the taxes accurately, the tax rate is measured to the nearest mill rather than to the nearest cent.


10 mills were equal to 1 cent, and therefore, 1,000 mills were worth a dollar.

The word “mill” is derived from the Latin word “milla” which means a thousand.

So just as we nowadays write, for example ,

$27.49 to abbreviate…

$10-bills $1-bills

2 7 dimes

4 cents

9

…in the “old” days, people would have written $27.493 as an abbreviation for…


2 7

© Math As A Second Language All Rights Reserved dimes

4 cents

9 mills

3

And if there had been a time when there was a denomination that was less than a mill and we had called it a decimill (where

10 decimills = 1 mill), then we would have written $27.4931 as an abbreviation for…


2 7 dimes

4 cents

9 mills decimills

3 1

There is no need to limit our discussion to the development of a monetary system now; any more than it was necessary when we talked only about whole numbers .



2 7 dimes

4 cents

9 mills decimills

3 1

We can generalize the above chart by writing.

10

2

1

7

1 /

10

4

1 /

100

9

1 /

1000

3

1 /

10,000

1

Notice that as long as the denominations are visible, there is no need to introduce a symbol to separate the whole number portion from the fractional portion of a number that’s written in place value notation.


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Notes

However, when we omit the denominations, the decimal point plays the role of a place holder. Just as we use zeroes to keep track of whole number denominations, the decimal point helps us keep track of the fractional denominations.


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Notes

In terms of mixed number notation, the digits to the left of the decimal point represent the whole number part of the mixed number and the digits to the right of the decimal point represent the fractional part of the number.


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Notes

Just as the two zeroes in 4,500 tell us that the 5 is modifying hundreds , the two digits to the right of the decimal point in 3.56 tell us that

6 is modifying hundredths .


Notes

Be careful with respect to symmetry.

For example , tens and tenths are symmetric with respect to 1; not with respect to the decimal point . That is, the second digit to the left of 1 represents hundreds , and the second digit to the right of 1 represents hundredths .

However, the second digit to the left of the decimal point represents tens.


Notes

A teacher once told me that when she taught decimals she told her students to say “one” when discussing whole numbers but to say “oneths” when discussing fractional parts.

In terms of a diagram… hundred ten one oneths tenths hundredths

In this context “ oneths ” means “1 of what it takes 1 of to make the whole”.


Using Fractional Notation

If we omit the decimal point , what we see is the whole number adjective or the numerator if we are talking about fractions .

The denominator is a 1 followed by as many 0’s as there are digits to the right of the decimal point .


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So, for example , let’s look at the different numbers 3.45, 34.5 and 0.0345.

10

3

1

3

4

1 /

10

4

5

1 /

100

5

1 /

1,000

1 /

10,000

3 4 5

In each case, the whole number adjective is 345, but the noun it modifies is named by the column in which the digit furthest to the right (5) appears.

3.

45 = 345 hundredths

34.

5 = 345 tenths

0.

0345 = 345 ten thousandths


In terms of common fractions , it means that the numerator is 345 .

The denominator will be a 1 followed by as many zeroes as there are digits to the right of the decimal point .

With respect to 3.45 there are two digits

(4 and 5). Hence, the denomination is hundredths ; which in terms of common fractions means that the denominator is 1 00 .

Therefore, written as a common fraction

3.45 is 345 /

1 00 and is read as 345 hundredths .


Notes

345 hundredths does not look like

“3 and 45 hundredths ”, but the two numerals are equivalent.

More specifically, 345 /

100 means

345 ÷ 100, and as a mixed number , this is 3 45 /

100

, which is read as

“3 and 45 hundredths ”.


Notes

34.5 tenths does not look like

“34 and 5 tenths ”, but these two numerals are equivalent.

More specifically, 345 /

10 means

345 ÷ 10, and as a mixed number , this is 34 5 /

10

, which is read as

“34 and 5 tenths ”.


Notes

With respect to 0.0345, there are 4 digits to the right of the decimal point (0, 3, 4, and 5)

Therefore, the denomination is 1 0 , 000ths , which in terms of common fractions means that the denominator is 1 0 , 000 .

Written as a common fraction , 0.0345 is

345 /

1 0 , 000 and is read as 345 ten thousandths .


Using Adjective/Noun Theme

An interesting way to demonstrate the “ adjective / noun ” theme is to think in terms of probability .

For example , to see the difference between,

0.9 and 0.0009, notice that in both

0.9 and 0.0009 the adjective is 9 .

6 note

6 In the decimal .9 it is easy to overlook the decimal point . It is rather small in appearance and it can easily be mistaken for a random blemish on the paper.

So to help make sure that we notice that a decimal point is there, we place a 0 to the left of the decimal point . Rather than write .9 we often choose to write

0.9. There are several other conventions we follow in order to avoid confusion.

For example , we often write 0 as 0 in order not to confuse the digit 0 with the letter O. Mathematicians often put a “bar” through the letter Z in order not to confuse it with the numeral 2. That is they might write 3Z in order not to confuse it with the numeral 32.


Using Adjective/Noun Theme

However, since there is only one digit to the right of the decimal point in 0.9, the

9 modifies tenths . That is, 0.9 = 9 /

10

On the other hand, since there are four digits to the right of the decimal point in

0.0009, and since a 1 followed by four 0’s is 10,000, the 9 modifies ten thousandths .

That is, 0.0009 = 9 /

10,000

.


So suppose there are 10 ping pong balls in a bag and 9 of them are colored red and the other one white .

You make a bet that you will, without looking, randomly select one of the ping pong balls and that the ball will be red .

Clearly, you have 9 chances of picking a winner, and since there are a total of

10 balls in the bag, we say that your chance of winning is 9 tenths , and we write the probability as 0.9 or 9 /

10

.


Now suppose that there are 10,000 ping pong balls in the bag (it’s a really big bag!) and that 9 of them are still colored red but the rest are colored white .

You again bet that you will, without looking, reach into the bag randomly and pick out a red ball. You still have the same 9 chances of winning the bet, but now since there are

10,000 balls in the bag, you have

9,991 chances of losing.


Your probability of winning is now

9 ten thousands which we write as

0.0009 or 9 /

10,000

.

7 note

7 Closely connected with probability is the term “odds”. If you have 9 chances of winning and 1 of losing (with all the outcomes being equally likely), we say that the odds in favor of your winning are 9 to 1.

If instead you have 9,991 chances of losing we say that the odds against you are

9,991 to 1,or, the odds of you winning are 9 to 9,991..


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Decimals , as an

Extension of Percents

This completes our present discussion. In our next presentation we will use decimals as an extension of percents.


Adjectives that modify Nouns

#8

Introduction to

Decimals

Prelude

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Adjectives that modify Nouns

#8

Introduction to

Decimals

Prelude

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