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 with the repetition of the decimal values in a specific pattern.

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., $3.1415926535897....$