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THE MEANING OF DECIMALS

Lesson 2  Section 3


In the Prologue, Elementary Addition, we introduced the number 0. We continue with the following question:
 7.   In our system of numbering, what is the function of 0?
 
  0 signifies that the unit at that place is absent.
0 holds the place.

This number

2001

is obviously very different from this number:

21.

Those 0's signify No tens and No hundreds.  Those units are absent from the expanded sum.

2001 = 2 Thousands + 0 Hundreds + 0 Tens + 1 One.

 8.   Why do 0's written on the right of a decimal not change its value?
 
.2400000 = .24
 
 
  Those 0's add nothing to the sum, and they do not affect the place values of the first digits.

Consider the expanded forms:

.24  =  2 tenths + 4 hundredths
 
.2400000  =  2 tenths + 4 hundredths + 0 thousandths + 0 ten-thousandths + etc., etc., etc.

The 0's add nothing to the value.

Therefore a whole number, such as 8, could be written as

8.0  or  8.00  or  8.0000000

0's on the extreme right of a decimal
do not change its value.

Similarly, 0's on the left of a whole number do not change its value.

65 = 065 = 0000065

Example.   Rewrite each number and eliminate the unnecessary 0's.

a)  5.6000 = 5.6

b)  08 = 8

c)  .08  This cannot be changed.  8 is in the hundredths place.

d)  204.006  This cannot be changed either. We may not eliminate
d)  204.006  internal 0's.

Comparing decimals
 9.   How do we compare decimals?
 
  Compare them digit by digit, starting with the first decimal place.

Example 1.   Which of these numbers is smallest, and which is largest?

.34   .306   .2986

Answer.  .2986 is the smallest number, and .34 is the largest.  For, consider each digit in turn.

.34   .306   .2986

With respect to the first decimal place, .2986 has only 2 tenths, while the others have 3. (Question 5.) 2 tenths is less than 3 tenths (and adding hundredths to it will not make it more).  Therefore .2986 is the smallest number.

As for

.34   versus   . 306

again, they are equal in the first decimal place.  Therefore we will compare them in the second place.  .34 has 4 hundredths, while .306 has 0 hundredths.  4 hundredths are more than 0 hundredths.  Therefore, .34 is the largest number.

Example 2.   Which is the largest number?

.02   .0201   .021

Answer.  .021

.02   .0201   .021

For, these numbers are equal in the first two decimal places; each has .02.  Therefore we must look at the third place.  .02 effectively has a 0 in the third place.  The next number .0201 also has 0 in the third place.  But .021 has 1 in the third place.  1 is more than 0.  Therefore, .021 is the largest number.


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