Non Terminating Decimal

Non-terminating decimals are decimal numbers that do not have an end term. We get such decimals when we divide a dividend by a divisor. However, the remainder is never zero, and the division continues forever. The digits are infinite after the decimal point in such decimals.

An example is shown below with the expansion:

We can further divide non-terminating decimals into two categories:

Non-Terminating Repeating Decimals

The decimal numbers that continue forever with a digit (or group of digits) repeating periodically. Such decimals are also called recurring or repeating decimals. These are always rational numbers.  E.g., 15/7 = 2.142857142857…

Converting to Rational Numbers

Suppose we are given a repeating decimal, i.e., 0.5555…

We can write it as ${ 0.\overline{5}}$

Let us convert the decimal to fraction.

Let, x = 0.5555… ——- (1)

Multiplying 10 on both sides,

10x = 5.5555… ——- (2) (this is done to eliminate the digits after the decimal point)

Subtracting (1) from (2)

10x – x = 5.5555… – 0.5555…

9x = 5

x = 5/9

∴${ 0.\overline{5}}$ = ${\dfrac{5}{9}}$

Let us take another example of a mixed repeating decimal (where a group of digits is repeated periodically).

Given a decimal 0.7575…

We can write it as ${ 0.\overline{75}}$

Now, let, x = 0.7575… ——- (1)

Multiplying 100 on both sides,

100x = 75.7575… ——- (2)

Subtracting (1) from (2)

100x – x = 75.7575 – 0.7575

99x = 75

x = 75/99

Non-Terminating Non-Repeating Decimals

The decimal numbers that continue forever with no digit (or group of digits) repeating. Such decimals are also called non-recurring or non-repeating decimals. These are always irrational numbers. E.g., π = 3.141592653…, √2 = 1.4142135623…, 737.537269541…, or 1.41421356237309504….

Solved Examples