Non-terminating decimal

A non-terminating decimal is a decimal that goes on without end. Said differently, when a fraction is expressed in decimal form but always has a remainder regardless how far the long division process is carried through, the resultant decimal is a non-terminating decimal.

Non-terminating, terminating, and repeating decimals

These three types of decimals are often discussed together because they are closely related. As mentioned, 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.

A terminating decimal is one that has a finite number of digits. All of the digits in a terminating decimal are known.

Note that ⅓ is also a repeating decimal.

Rational and irrational numbers

Non-terminating decimals are one of the ways that rational numbers and irrational numbers are distinguished. A rational number is either a terminating decimal (ends after a certain number of decimal places), or a repeating decimal (a decimal number in which a set of digits repeat endlessly). All rational numbers can be written in the form of a fraction. It is also worth noting that a fraction involving integers in the numerator and denominator can always be expanded as a terminating decimal or a repeating decimal.

Irrational numbers are numbers that cannot be expressed as a fraction. An irrational number does not terminate and does not repeat.


Determine whether the following numbers are rational or irrational numbers.

1. :

Dividing 1 by 3 results in the decimal 0.3. The bar indicates that 3 continues to repeat itself indefinitely. Thus, 0.3 or 0.333... is a rational number because it repeats. It is also a non-terminating decimal.

2. :

Dividing 3 by 11 results in the decimal 0.27. In this number, the "27" repeats. Thus, 0.27, or 0.2727..., is a rational number because it repeats. It is also a non-terminating decimal.

3. π:

We cannot express π in the form of a fraction (though 22/7 is used as an approximation), so π is an irrational number. If we were to try calculate π we could continue finding more and more decimal places, but we would never find a pattern, and there will always be another decimal place after the one we find. π is one of the most well-known irrational numbers, and is a non-terminating decimal that does not repeat. Its first 10 digits are: 3.14159265...