# Special types of numbers

## In a nutshell

All the numbers that you use can be categorised into different groups.

## Integers

**Integers** are whole numbers - both positive and negative. Zero is also considered an integer.

## Odd and even numbers

**Odd numbers** are integers that end in $1,3,5,7,9$. They all leave a remainder of $1$ when divided by $2$.

**Even numbers** are integers that end in $0,2,4,6,8$. They are all divisible by $2$.

## Rational and irrational numbers

**Rational numbers** are numbers that can be written as fractions where the numerator and denominator are both integers. Examples of rational numbers include:

$-\frac{1}{6},\frac{500}{49},12\,(=\frac{12}{1})$

**Irrational numbers** are numbers that *can't *be written as fractions where the numerator and denominator are both integers. Examples of irrational numbers include $\pi$ and any surd.

## Square and cube numbers

**Square numbers** are integers that can be written as the square of another integer.

The first ten square numbers are:

$x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |

$x^2=x\times x$ | $\underline1$ | $\underline4$ | $\underline9$ | $\underline{16}$ | $\underline{25}$ | $\underline{36}$ | $\underline{49}$ | $\underline{64}$ | $\underline{81}$ | $\underline{100}$ |

**Cube numbers** are integers that can be written as the cube of another integer.

The first five cube numbers are:

$x$ | $1$ | $2$ | $3$ | $4$ | $5$ |

$x^3=x\times x\times x$ | $\underline1$ | $\underline8$ | $\underline{27}$ | $\underline{64}$ | $\underline{125}$ |

## Prime numbers

**A prime number **is a positive integer that is only divisible by $1$ and itself.

### Key facts about prime numbers

- $1$ is
*not *a prime number - a prime number has to have two different factors (numbers it is divisible by). - $2$ is the only
*even prime* - all the other even numbers are divisible by $2$ and hence not prime. - The single digit prime numbers are $2,3,5,7$.

- All prime numbers greater than $10$ end in $1,3,7,9$.

- Just because a number ends in $1,3,7,9$, doesn't necessarily mean it is prime. For example, $33$ isn't prime as it is divisible by $3$.
- There are an infinite number of primes.

### Finding two digit prime numbers

It is very difficult to find large prime numbers, as larger numbers have a higher chance of having additional factors. However, for two digit numbers, there is a procedure.

#### procedure

1. | Check if the number ends in $1,3,7,9$. If it doesn't end with those numbers, it's not prime. |

2. | Check if the number is divisible by $3$ and $7$. If it is divisible by either of those numbers, it's not prime. |

3. | If the number ends in $1,3,7,9$ and it is *not *divisible by $3$ and $7$, then it's a prime. |

**Note: **This procedure only works for two digit numbers.

##### Example

*Verify that $71$ is prime.*

*It ends in $1$. So, check if it is divisible by $3$ and $7$*:

$71\div3=23$ *remainder $2$.*

*$71\div7=10$ remainder $1$.*

*Therefore, *$\underline{71}$* is prime because it ends in $\underline1$ and is NOT divisible by $\underline{3}$ and $\underline7$.** *