# Terminating and recurring decimals

**Terminating decimals**

A decimal with a fixed number of digits after the decimal point is called a **terminating decimal**.

For example,

0.61, 0.185 and 3.5326

It is also possible to write terminating decimals as vulgar fractions by writing them over the appropriate power of 10 and canceling if possible.

For example,

**Recurring decimals**

**Recurring decimals** is a number with decimals which repeat infinitely. For example:

A calculator displays this as:

0.72727273

This decimal has been rounded to eight decimal places.

The last digit is a 3 because it has been rounded up.

Points to note:

- Some recurring decimals maybe difficult to spot because repeating cycles are very long

**Recurring decimals and fractions**

Points to note:

- A
*ll*fractions convert to either terminating or recurring decimals. - When the denominator of the fraction divides into a power of 10 then the fraction will convert into a terminating decimal.
- If the denominator of the fraction does not divide into a power of 10, then it will convert into a recurring decimal
- Also, all terminating and recurring decimals can be written as fractions in the form where
*a*and*b*are integers and*b*≠ 0.

**Converting recurring decimals to fractions**

To convert terminating decimals into fractions, the place value method is used.

Converting recurring decimals into fractions is more difficult.

**Rational numbers**

Most numbers that can be written in the form (where *a* and *b* are integers and b ≠ 0) is called a **rational number**.

All of the following are rational:

We have seen that all terminating and recurring decimals can be written as fractions in the form . This means that they are also rational.

Therefore any number that can be written exactly as a whole number is a positive or negative form whether number, fraction or decimal is rational.

**Irrational numbers**

**Irrational numbers** cannot be written in the form .

If it was attempted to write an an irrational number then it would be represented by an infinite non-repeating string of digits. Therefore, irrational numbers can only be written as approximations to a given number of decimal places.

Examples of irrational numbers include: