Non-Terminating, Non-Repeating Decimal

 

Definition

A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly.

Decimals of this type cannot be represented as fractions, and as a result are irrational numbers.

Example:

Pi is a non-terminating, non-repeating decimal. π = 3.141 592 653 589 793 238 462 643 383 279...

e is a non-terminating, non-repeating decimal. e = 2.718 281 828 459 045 235 360 287 471 352...

Non-terminating, non-repeating decimals can be easily created by using a pattern. Some examples are listed below:

  • 10100100010000100000100000010000000 ...
  • 121122111222111122221111122222111111222222 ...

Non-terminating Decimal:

Definition:

When a fraction is expressed in its decimal form, we perform division and we get some remainder. If the division process does not end, we do not get the remainder equal to zero; then such decimal is known as non-terminating decimal.

Example:

(a) 3.666... is a non-terminating repeating decimal and can be expressed as 3.6.

(b) 0.121212 ... is a non-terminating repeating decimal and can be expressed as 0.12.

Find the decimal representation of 16/45.

Solution:

Using long division method, we get