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 $\underline7$ .