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Lesson 3  Section 3

THE FUNCTION OF 0

COMPARING DECIMALS

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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.

The expanded form of 2001 is

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

The expanded form of 21 is

2 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 expanded form.
 

Consider these 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. Those 0's are called trailing 0's. They are 0's followed by no other digits.

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

65 = 065 = 0000065

Those are called leading 0's.

Example 1.   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. They are neither leading nor trailing.

Comparing decimals

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

Example 2.  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 digit, .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, the first decimal digits are equal.  Therefore we will compare the second digits.  .34 has 4 hundredths, while .306 has 0 hundredths.  4 hundredths are more than 0 hundredths.  Therefore, .34 is the largest number.

Example 3.   Which is the largest number?

.02   .0201   .021

Answer.  .021

.02   .0201   .021

For, the first two decimal digits are equal:  .02.  Therefore we must look at the third digit.  .02 effectively has 0 as the third digit.  The next number .0201 also has 0 as the third digit.  But .021 has 1 as the third digit.  1 is more than 0.  Therefore, .021 is the largest number.


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