The word **decimal** technically refers to the base-ten number system that we use every day. When most people talk about decimals, however, they are referring to *decimal numerals *such as 5.24, so on the rest of this page, the word decimal will be used to mean *decimal numeral*.

A decimal contains the decimal point. Any number on the number line – whether, positive, negative or zero – can be expressed as a decimal. The whole numbers (integers) can be written with zeroes in place value columns after the decimal point. For example, 7 can be written as 7.0, 7.00, 7.000 etc. and -18 can be written as -18.0, -18.00, -18.000 etc. (Note that in the context of rounding, these may mean different things, despite all appearing to refer to the same integer.)

Decimals are most commonly used when dealing with numbers that are not whole numbers. Any fraction is a rational number, and can be written as either a **terminating** or **recurring** decimal:

\(\dfrac{3}{4}=0.75\) is a terminating decimal as its decimal expansion comes to an end.

\(2 \frac{5}{8}=2.625\) is a terminating decimal as its decimal expansion comes to an end.

\(\dfrac{1}{3}=0.33333… = 0.\dot{3}\) is a recurrring decimal as its decimal expansion goes on forever but has a clearly repeating pattern.

\(\dfrac{124}{990}=0.0124124124… = 0.0\dot{1}2\dot{4}\) is a recurring decimal as its decimal expansion goes on forever but has a clearly repeating pattern.

Irrational numbers – such as surds and \(\pi\) – can’t be expressed exactly as fractions (with integer numerator and denominators). These numbers have decimal expansions which go on forever i.e. they do not terminate and they don’t even have a recurring pattern.

Decimals can also be expressed as percentages.

**Relevant lessons:**

- N1b – Ordering decimals
- N1e – Ordering integers, decimals and fractions and using the symbols =, ≠, <, >, ≤, ≥
- N2d – Applying the four operations to decimals
- N10a – Converting terminating decimals into fractions and vice versa
- N10b – Converting recurring decimals into fractions and vice versa
- N15a – Rounding numbers and measures to an appropriate degree of accuracy
- N16a – Applying and interpreting limits of accuracy, including upper and lower bounds
- R9a – Converting between fractions, decimals and percentages
- R9c – Finding percentages and percentage changes multiplicatively using decimals