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# 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]\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.

## Contents

## 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]1 \frac{1}{2}[/math], [math]\frac{3}{2}[/math], or [math]3/2[/math].

More examples of fractions that are rational numbers include [math]\frac{1}{7}[/math], [math]\frac{-8}{9}[/math], and [math]\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]\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]\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]\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]\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]\frac{a}{b}[/math] and [math]\frac{c}{d}[/math] are equal if [math]ad = bc[/math].

## Related pages

## References

- ↑ "Compendium of Mathematical Symbols" (in en-US). 2020-03-01. https://mathvault.ca/hub/higher-math/math-symbols/.
- ↑ "Rational number" (in en). https://www.britannica.com/science/rational-number.
- ↑ Weisstein, Eric W.. "Rational Number" (in en). https://mathworld.wolfram.com/RationalNumber.html.