comparing decimals © math as a second language all rights reserved next #8 taking the fear out of...

ComparingDecimals

ComparingDecimals

© Math As A Second Language All Rights Reserved

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#8

Taking the Fearout of Math

1.25

1.50

This is relatively short presentation designed to explain the potential pitfalls in

dealing with the size of a decimal.


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Reading Decimals

Too often people tend to read the number 3.17 as “3 point 17” and the number 3.5 as “3 point 5”. The problem that arises when numbers are read this way, people tend to

believe that since 17 is greater than 5, 3.17 is greater than 3.5.


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This is equivalent to a person saying something along the lines of “Since

17 is greater than 5, 17 pennies is more money than 5 nickels”.

The important point is that when we compare two quantities the only time

we can be sure that the greater adjective names the greater quantity is when both

units are the same.

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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 fractions1 could begin

immediately after the arithmetic ofwhole numbers.

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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”.

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It is correct to say…

17 pennies is a greater amount of money than 5 pennies.

17 pounds is a greater weight than 5 pounds.

17 tenths is more than 5 tenths.

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On the other hand, it is not correct to say…

17 pennies is a greater amount of money than 5 dimes.

17 ounces is a greater weight than 5 pounds.

17 hundredths is more than 5 tenths.

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Thus…24 hundreds is written as 2,400

while5 thousands is written as 5,000.


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In place value notation, when we compare the sizes of 24 hundreds and 5 thousands we usually make sure that we write both numbers in the form in which the digit

furthest to the right is in the ones place.

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We may then conclude that 5 thousands is greater than 24 hundreds because 5,000 ones is greater than 2,400 ones.

We use the decimal point to help us determine the unit we want the

adjectives to modify. The approach is based on the fact that the digit furthest

to the right of the decimal point determines the unit (denomination) that

is being modified.


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For example, we can see that 0.132 is greater than 0.099 because they have the same number of digits to the right of the decimal point in both numbers. In other

words, 0.132 represents 132 thousandths while 0.099 represents 99 thousandths.

Since both adjectives (132 and 99) aremodifying the same unit (thousandths), we may conclude that 132 thousandths (0.132)

is greater than 99 thousandths (0.099).


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Problem arise when two decimals do not have the same number of digits to the right

of the decimal point.

In this case, we have to realize that there are many forms for representing the same

decimal fraction.

For example, notice that when we talk about ten dollars, it makes no difference

whether we write $10 or $10.00.

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In either case, the 1 modifies the number of tens.1

A good way to remember whether or not we can add a zero is to notice whether inserting the zero changes the noun

modified by the other digits.

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1 We could also have represented ten dollars as $10.0 or $10.000 etc. However, when it comes to writing dollar amounts in decimal form it is conventional to write to the nearest cent. that is, we usually write $3.70 when it would have

been equally correct to have written $3.7.


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For example, 21.0000 means the same thing as 21 because in either representation

the 2 and 1 modify ten and one respectively2.

However, 0.021 and 0.21 represent different numbers because in the first

case 2 is modifying hundredths while in the second case it’s modifying tenths.

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2 Scientists and engineers talk about significant figures. In mathematics, we can say that a piece of string is exactly 6 inches long, but in the “real world”

measurements are not exact. Thus, the scientist will write 6.0 inches to let us know that the measurement is guaranteed to be accurate to the nearest tenth of an Inch, but she'll write 6.00 inches if he wants to tell us that the measurement is

guaranteed to be accurate to the nearest hundredth of an inch.

Adding zeroes (or, equivalently, changing the noun the decimal fraction modifies)

simply for effect can make an interestingpsychological impact.


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Aside

For example, in the metric system a gram is the basic

unit of weight.

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In this context, the weights 0.1 grams,0.01 grams and 0.001 grams are given the

names 1 decigram, 1 centigram, and 1 milligram, respectively.3


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A gram is a very small amount of weight. In fact it takes 454 grams to equal 1 pound,

and since a milligram is 0.001 grams, it requires 454,000 milligrams to equal

1 pound.note

3 0.1 and .1 mean the same thing. However, by writing the 0 to the left of the 1, it emphasizes the existence of the decimal point. In other words, when one sees .6 one can wonder whether the “dot" is just a random mark on the paper or a decimal point. However, when we write 0.6, it's clear that the

“dot” is a decimal point.

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Because many drugs are very toxic chemicals, small amounts can be very

dangerous. For that reason, we measure drugs in milligrams (mgs).


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Thus, rather than talk about 0.1 grams of a particular drug,

we would refer to it as 100 mgs.4

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4 0.1 grams is the same as 0.100 grams and since the 0 furthest to the left is in the thousandths place, 0.100 is the same as 100 mgs

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Similarly, we would refer to a dosage of 0.2 grams as 200 mgs. While the difference

between 0.1 and 0.2 grams seems negligible, the difference between 100 mgs and 200 mgs

seems non negligible.


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In this context, the casual user might think that there is little difference between taking

0.1 grams or 0.2 grams, but they might think twice about taking 200 mgs rather than

100 mgs.5note

5 The same logic can be applied in reverse. If, for example, the government talks about a deficit of $1,200,000,000,000, the average “concerned citizen” might be

quite alarmed. However, if the government decides to write the deficit in the form $1.2 trillion, the adjective 1.2 doesn’t seem to be very large and the word “trillion” masks the denomination quite successfully. That is, to most people

“trillion” and “billion are “just words”!

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An important thing to notice is that our adjective/noun theme is not only

consistent with the traditional method of comparing decimals, but it also helps to

explain why the rote method works.


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By way of illustration, let’s compare the size of the two decimals,

0.120999 and 0.121100.

The traditional way is to look at the two decimals denomination by denomination until we come to the first place where the

digits are different; in which case the decimal with the greater digit is greater.


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Thus, in the present example…

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0. 1 2 0 9 9 9

0. 1 2 1 1 9 9

And since 0 < 1; 0.120999 is less than 0.121100.

The adjective/noun theme works nicely here and eliminates our having to think

in terms of decimals. Without the decimal points the numbers we would see are 120,999 and 121,100 and sincethere are six digits to the right of the

decimal point in each number,the two decimal fractions modify

“millionths”.


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Notes

In other words…


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Notes

0.120999 = 120,999 millionths

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And whenever 120,999 and 121,100 modify the same noun, 120,999 < 121,100.

0.121100 = 121,100 millionths

In general, comparing the size of two decimal fractions can always be replaced

by comparing an equivalent pair of whole numbers.


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In particular, if both decimal fractions have the same number of digits to the right of

the decimal point they modify thesame noun, then we can compare their

adjectives as if they were whole numbers.

So, for example, to compare 0.01 with 0.000998, notice that 0.01 has only 2 digits to the right of the decimal pointwhile 0.000998 has 6 digits to the right of the decimal point. Therefore, we rewrite

0.01 in the equivalent form 0.010000.


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In that case, 0.01 represents 10,000 millionths and 0.00998 represents

998 millionths. Since 10,000 is greater than 998, then 10,000 millionths would be greater than 998 millionths. In other

words, 0.01 is greater than 0.000998.


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This explains the danger in reading numbers such as 0.491 as “point 491”.

which leads to believing that 0.491 is greater than 0.53 because 491 is greater than 53”.

Specifically, 0.53 means 53 hundredths and 0.491 means 491 thousandths.

We cannot compare adjectives unless they are modifying the same noun.

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To this end, we may rewrite 0.53 as 0.530.

Then… 0.53 = 530 thousandths

0.491 = 491 thousandths

And 530 thousandthsis great than

491 thousandths

So, as we have mentioned many times before, we have to keep track of what 53

and 491 modify in order to determine which quantity is greater.

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This completes our discussion concerning

comparing the sizeof one decimal amount

to another decimal amount. In our next

presentation, we will add and subtract decimals.


Adding and Subtracting Decimals

comparing decimals © math as a second language all rights reserved next #8 taking the fear out of...

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