Non-terminating recurring decimal expansions

Non-terminating decimals are decimals that have never-ending decimal digits and continue forever.

Non-Terminating Decimal Definition

Non-terminating decimals are defined as those decimal numbers that do not have an endpoint in their decimal digits and keep continuing forever. This happens when a dividend is divided by a divisor but the remainder is never $0$ and hence the process keeps repeating and the non-terminating decimal is obtained in the quotient where the decimal digits keep occurring and never come to an end. A non-terminating decimal has an infinite number of decimal places and it is named as non-terminating because the decimal will never terminate. For example, $1.333333....$,$4.65675747775.......$ etc.

Types Non-Terminating Decimal Expansion

A non-terminating decimal expansion has an infinite number of places and its expansion continues forever. We have two types of non-terminating decimal expansions and they are as follows:

1. Non terminating recurring decimal expansion
2. Non terminating non-recurring decimal expansion

Non-Terminating Recurring Decimal Expansion

Non-terminating recurring decimal is also known by the name non terminating repeating decimal. In this expansion, the decimal places will continue forever and never come to an end but since the name says repeating or recurring, it signifies that the repetition of the decimal values forms a specific pattern that can be easily identified.

For example, ${\frac {2}{9}}=0.2222222.....$ is a non-terminating recurring decimal expansion. The repetition of the decimal can also be indicated by showing a bar on top of the numbers that are repeating i.e., $0.2222222.....$ can also be represented as $0.{\bar {2}}$. Similarly, ${\frac {1}{7}}=0.142857142857....$ which can also be written as $0.{\bar {1}}{\bar {4}}{\bar {2}}{\bar {8}}{\bar {5}}{\bar {7}}$ is also a non-terminating repeating decimal expansion as the block of decimals 142857 is repeating after every 6 digits. Non-terminating recurring decimals can always be converted to a rational number.

Non-Terminating Non-Recurring Decimal Expansion

Non-terminating non-recurring decimal is also known by the name non terminating non-repeating decimal as the values after decimal do not repeat or terminate. For example, $1.4142135...$ , $2.35638745...$ Unlike non-terminating recurring decimal, the decimal places do not form any pattern. A non-terminating non-recurring decimal cannot be converted to a rational number. Hence, non terminating non-recurring decimals are also known as irrational numbers. Note that pi ($\pi$) is an irrational number as its expansion is non-terminating non-recurring i.e., $3.1415926535897...$