# Rational number

In mathematics, a **rational number** is a number that can be written as a fraction. The set of rational number is often represented by the symbol [math]\displaystyle{ \mathbb{Q} }[/math], standing for "quotient" in English.^{[1]} ^{[2]}

Rational numbers are all real numbers, and can be positive or negative. A number that is not rational is called irrational.^{[3]}

Most of the numbers that people use in everyday life are rational. These include fractions, integers and numbers with finite decimal digits. In general, a number that can be written as a fraction while it is in its own form is rational.

## Writing rational numbers

## Fraction form

All rational numbers can be written as a fraction. Take 1.5 as an example, this can be written as [math]\displaystyle{ 1 \frac{1}{2} }[/math], [math]\displaystyle{ \frac{3}{2} }[/math], or [math]\displaystyle{ 3/2 }[/math].

More examples of fractions that are rational numbers include [math]\displaystyle{ \frac{1}{7} }[/math], [math]\displaystyle{ \frac{-8}{9} }[/math], and [math]\displaystyle{ \frac{2}{5} }[/math].

## Terminating decimals

A terminating decimal is a decimal with a certain number of digits to the right of the decimal point. Examples include 3.2, 4.075, and -300.12002. All of these are rational. Another good example would be 0.9582938472938498234.

## Repeating decimals

A repeating decimal is a decimal where there are infinitely many digits to the right of the decimal point, but which follow a repeating pattern.

An example of this is [math]\displaystyle{ \frac{1}{3} }[/math]. As a decimal, it is written as 0.3333333333... The dots indicate that the digit **3** repeats forever.

Sometimes, a group of digits repeats. An example is [math]\displaystyle{ \frac{1}{11} }[/math]. As a decimal, it is written as 0.09090909... In this example, the group of digits **09** repeats.

Also, sometimes the digits repeat *after* another group of digits. An example is [math]\displaystyle{ \frac{1}{6} }[/math]. It is written as 0.16666666... In this example, the digit **6** repeats, following the digit 1.

If you try this on your calculator, sometimes it may make a rounding error at the end. For instance, your calculator may say that [math]\displaystyle{ \frac{2}{3} = 0.6666667 }[/math], even though there is no 7. It rounds the 6 at the end up to 7.

## Irrational numbers

The digits after the decimal point in an irrational number do not repeat in an infinite pattern. For instance, the first several digits of π (Pi) are 3.1415926535... A few of the digits repeat, but they never start repeating in an infinite pattern, no matter how far you go to the right of the decimal point.

## Arithmetic

- Whenever you add or subtract two rational numbers, you always get another rational number.

- Whenever you multiply two rational numbers, you always get another rational number.

- Whenever you divide two rational numbers, you always get another rational number (as long as you do not divide by zero).

- Two rational numbers [math]\displaystyle{ \frac{a}{b} }[/math] and [math]\displaystyle{ \frac{c}{d} }[/math] are equal if [math]\displaystyle{ ad = bc }[/math].

## Related pages

## References

- ↑ "Compendium of Mathematical Symbols".
*Math Vault*. 2020-03-01. Retrieved 2020-08-11. - ↑ "Rational number".
*Encyclopedia Britannica*. Retrieved 2020-08-11. - ↑
## Rational Number Media

The rational numbers Template:Tmath are included in the real numbers Template:Tmath, while themselves including the integers Template:Tmath, which in turn include the natural numbers Template:Tmath.

Weisstein, Eric W. "Rational Number".

*mathworld.wolfram.com*. Retrieved 2020-08-11.