# Rational number

The rational numbers (ℚ) are included in the real numbers (ℝ), and in turn include the integers (ℤ), which include the natural numbers (ℕ)

A rational number is any number that can be expressed as the quotient or fraction ${\displaystyle {\frac {p}{q}}}$ of two integers, a numerator p and a non-zero denominator q.[1]

## Examples

1. ${\displaystyle {\frac {1}{2}}}$
2. 5
3. 0.2

Notice the number 5 in second example! It is because all numbers are divisible by 1 and at such it is actually ${\displaystyle {\frac {5}{1}}}$ but it is more convenient to write it as 5. Note Though all numbers are divisible by 1 some numbers are considered irrational ie they cannot be represented in the form ${\displaystyle {\frac {a}{b}}}$ also note that it impossible to have a number with 0 as the denominator (b must not be equal to 0 in ${\displaystyle {\frac {a}{b}}}$).

## Operations Involving Rational Numbers

${\displaystyle {\frac {a}{b}}}$ + ${\displaystyle {\frac {c}{d}}}$ = ${\displaystyle {\frac {ad+bc}{cd}}}$

### Subtraction

${\displaystyle {\frac {a}{b}}}$ - ${\displaystyle {\frac {c}{d}}}$ = ${\displaystyle {\frac {ad-bc}{cd}}}$

### Multiplication

${\displaystyle {\frac {a}{b}}}$${\displaystyle {\frac {c}{d}}}$ = ${\displaystyle {\frac {ac}{bd}}}$

### Division

${\displaystyle {\frac {a}{b}}}$ ÷ ${\displaystyle {\frac {c}{d}}}$ = ${\displaystyle {\frac {a}{b}}}$${\displaystyle {\frac {d}{c}}}$ = ${\displaystyle {\frac {ad}{bc}}}$.