Everything to learn better...

Home

Maths

Types of numbers

Special types of numbers

Tutor: Bilal

# 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 $\underline7$.

## FAQs - Frequently Asked Questions

### What is an integer?

Beta

I'm Vulpy, your AI study buddy! Let's study together.