# 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. Below are a few non-terminating decimal examples:

- 0.10101010101010...
- 0.1
- 0.753838383838...
- 1.28

Notice that there are two different ways that non-terminating decimals are expressed above; the first uses a "..." after showing the pattern of repeating digits; the second uses a bar over the digits to indicate which digits repeat.

## 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.

### Repeating and non-repeating decimals

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. Below is a non-terminating and repeating decimal example.

- Non-terminating decimal: = 0.3333333...
- Repeating decimal: = 0.09090909...

Generally, we refer to decimals that do not terminate and do not repeat as "non-terminating decimals" and refer to those that do repeat as "repeating decimals." Technically, "non-terminating decimal" includes repeating decimals, so when it is relevant, we should be specific about how we refer to the decimals.

### Terminating decimals

A terminating decimal is one that has a finite number of digits. All of the digits in a terminating decimal are known. For example, the fraction ¼ can be written as a terminating decimal:

= 0.25

## Convert repeating decimals to fractions

There are two types of repeating decimals we can convert using formulas. The first type is repeating decimals in which only one of the digits after the decimal point repeats, such as 0.3. The second type is ones where only some of the digits after the decimal point repeat, such as 0.312. Depending on which type we have, the formula for converting the repeating decimal to a fraction varies.

### Repeating decimals where all digits repeat

To convert repeating decimals in which all the digits repeat, use the following formula:

For example, in the repeating decimal 0.3, the only repeated digit is the 3. Using the formula, the repeated term is "3" and since the term only contains one number, we divide 3 by a single 9:

Thus, the fraction form of 0.3 is ⅓. As another example, consider the repeating decimal 0.32789. In this example, there are 5 repeated digits. These 5 repeated digits make up the numerator and we divide the numerator by 5 9's since there are 5 digits in the repeated term:

The above fraction is already in simplified form and if we were to carry out the division, we would find it equal to the repeating decimal 0.32789.

### Repeating decimals where some digits repeat

To convert repeating decimals in which not all of the digits repeat, use the following formula:

For example, consider the repeating decimal 0.1712. In this decimal, the "12" repeats while the "17" does not. To convert the decimal to a fraction, we use the formula as follows:

Dividing 113 by 660 confirms the repeating decimal 0.1712. As another example, consider the repeating decimal 0.00602; there are 3 digits that repeat and 2 digits that do not, so:

## 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.

Examples

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...