Types of Number (Edexcel GCSE Maths): Revision Notes

Types of number

Understanding different types of numbers is fundamental to GCSE mathematics. Before diving into calculations, you need to know how numbers are classified and what makes each type special.

Integers

An integer is simply another word for a whole number. This includes positive numbers, negative numbers, and zero. Think of integers as the numbers you can count with, along with their negative counterparts.

Examples of integers: $-365, 0, 1, 17, 989, 1,234,567,890$

Not integers: $0.5, \frac{2}{3}, \sqrt{7}, 13\frac{3}{4}, -1000.1, 66.66, \pi$

Rational and irrational numbers

Every number you'll encounter falls into one of these two categories, and understanding the difference is crucial for your exams.

Rational numbers

Rational numbers are numbers that can be expressed as fractions. Most numbers you work with in everyday maths are rational numbers.

Rational numbers appear in three main forms:

1) Integers - These can be written as fractions by putting them over 1

• For example: $4 = \frac{4}{1}$, $-5 = \frac{-5}{1}$, $-12 = \frac{-12}{1}$

2) Fractions - Where both the top and bottom are integers (and the bottom isn't zero)

• For example: $\frac{1}{4}, \frac{1}{2}, \frac{3}{4}$

3) Terminating or recurring decimals - These are decimals that either stop or repeat in a pattern

• Terminating: $0.125 = \frac{1}{8}$, $0.333333... = \frac{1}{3}$
• Recurring: $0.143143143... = \frac{143}{999}$

Irrational numbers

Irrational numbers are more complex - they cannot be written as simple fractions. These numbers have decimal representations that go on forever without repeating in any pattern.

Common examples include:

• Square roots of non-perfect squares ($\sqrt{2}, \sqrt{3}, \sqrt{7}$ are all irrational, but $\sqrt{4} = 2$ is rational)
• Surds (mathematical expressions containing irrational roots)
• $\pi$ (pi) is also irrational

Squares and cubes

Knowing your squares and cubes by heart is essential for GCSE success, especially in non-calculator papers. These special number patterns come up frequently in exams.

The special squares

You should memorise these perfect squares:

The crucial cubes

Similarly, these cube values are worth memorising:

Prime numbers

Prime numbers are special because they have exactly two factors: 1 and themselves. Understanding primes is important for many areas of mathematics.

Key facts about prime numbers

A prime number cannot be divided evenly by any other number except 1 and itself. The number 1 is specifically not considered prime because it only has one factor.

Prime number sequence

The first several prime numbers are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43...$

Identifying potential primes

When checking if larger numbers might be prime, remember that numbers like $71, 73, 77, 79, 101, 103, 107, 109$ could be prime candidates. However, you need to check carefully that they don't divide by any smaller prime numbers to confirm they're actually prime.