Decimal Expansions of Real Numbers

There are three types of decimal expansions of real numbers.

1.Terminating

2. Non Terminating, Repeating

3. Non Terminating, Non Repeating

1.(Terminating) The remainder becomes zero.

Let us take examples to know it.

Example 1: Expansion of  7/4

In the example of  7/4  

we found that after some steps the remainder becomes zero and the decimal expansion of

                       7/4    = 1.75

The remainder is 0, and the decimal expansion end at 5, 
so it means expansion is Terminating.

Example 2: Expansion of  32/5

Again we found that after some steps the remainder becomes zero and the decimal expansion of

                       32/5    = 6.4

Similarly expansion of 32/5 is Terminating.

Example 3: Expansion of 578/25

decimal expansion of 

                 578/25 = 23.12

expansion of 578/52 is Terminating.

2. The remainder never becomes zero.

There are two cases,

a.   Non Terminating, Repeating.

b.  Non Terminating, Non Repeating

Example 4: Expansion of 10/3

decimal expansion of 

                 10/3 = 3.333333… 
                             _
                       = 3.3

The expansion of 10/3 does not end that is not terminating, and number 3 is repeating, so it is a non 
terminating, Repeating expansion.
b. Non terminating, Non Repeating

Example 5:  Expansion of 1/7

decimal expansion of 

                 1/7 = 1.142857… 

The expansion of 1/7 does not end that is not terminating, and numbers 142857 are repeating, so it is a non terminating, Repeating expansion.

                          ______
              1/7 = 1.142857…