Terminating decimal

A terminating decimal is a decimal that has a finite number of digits. All terminating decimals can be expressed in the form of a fraction, and all of the digits of the terminating decimal can be determined by carrying out the division problem. Technically, an infinite number of zeros can be added to the end of a decimal. However, since the value of the decimal does not change regardless of the number of zeros added, these decimals would still be considered terminating decimals. The following are all terminating decimals.

• ½ expressed as a decimal is 0.5.
• ¼ expressed as a decimal is 0.25.
• "0.250000000000000000..." is still equal to ¼ regardless of how many zeros are added.

Terminating, non-terminating, and repeating decimals

These three types of decimals are often discussed together because they are closely related. As discussed above, a terminating decimal is one that has a finite number of digits. All of the digits in a terminating decimal are known. A non-terminating decimal is a decimal that never ends. It has an infinite number of digits.

There are two types of non-terminating decimals, ones that repeat and ones that do not repeat. Non-terminating decimals that repeat are referred to as repeating decimals. Although they have an infinite number of digits, all of the digits in a repeating decimal are known. Also, to be considered a repeating decimal, the repeating digits cannot all be zero. For non-terminating decimals that do not repeat, not all of the digits are known. No matter how many digits are known, there will always be a digit following it that needs to be determined.

• Terminating decimal: = 0.25
• Non-terminating decimal: = 0.3333333...
• Repeating decimal: = 0.09090909...

Note that ⅓ is both a non-terminating decimal as well as a repeating decimal. Understanding the differences between these types of decimals is important when trying to distinguish rational and irrational numbers. All terminating decimals are rational numbers. The same is true of repeating decimals. Both terminating and repeating decimals can be expresed in the form of a fraction. Together, they make up the rational numbers.

Irrational numbers on the other hand, must be both non-terminating and non-repeating decimals. Examples include π (3.14159...) and the square root of 2 (1.4142135...). Regardless of the number of digits we compute, neither π nor the square root of 2 will ever terminate or repeat.