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# Rational number

In mathematics, a rational number is a number that can be written as a fraction. Rational numbers are all real numbers, and can be positive or negative. A number that is not rational is called irrational.

Most of the numbers that people use in everyday life are rational. These include fractions and integers.

## 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 $1 \frac{1}{2}$, $\frac{3}{2}$, or $3/2$.

More examples of fractions that are rational numbers include $\frac{1}{7}$, $\frac{-8}{9}$, and $\frac{2}{5}$.

### 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 they follow a repeating pattern.

An example of this is $\frac{1}{3}$. As a decimal, it is written as 0.3333333333... The dots tell you that the number 3 repeats forever.

Sometimes, a group of digits repeats. An example is $\frac{1}{11}$. 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 $\frac{1}{6}$. 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 $\frac{2}{3} = 0.6666667$, 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 $\frac{a}{b}$ and $\frac{c}{d}$ are equal if $ad = bc$.