Types of Numbers

Natural Numbers (ℕ)

Natural numbers are the basic counting numbers starting from 1. These are the numbers we use to count objects in real life.

Examples: \( 1, 2, 3, 4, 5, \ldots \)

Note: In IGCSE, 0 is not considered a natural number.

Integers (ℤ)

Integers include all whole numbers — both positive and negative — along with zero. They do not include decimals or fractions.

Examples: \( \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \)

Prime Numbers

A prime number is a whole number greater than 1 that has exactly two distinct positive divisors: 1 and itself. This means it cannot be made by multiplying two smaller natural numbers.

Examples: \( 2, 3, 5, 7, 11, 13, 17, 19, \ldots \)

Note: 1 is not a prime number because it only has one positive divisor.

Square Numbers

A square number is the result of multiplying a whole number by itself. These are also called “perfect squares.”

Examples: \( 1^2 = 1, \; 2^2 = 4, \; 3^2 = 9, \; 4^2 = 16, \; 5^2 = 25, \ldots \)

The square root of a number \( x \), written as \( \sqrt{x} \), is the positive number that, when squared, gives \( x \). That is, \( \sqrt{x} = n \) if \( n^2 = x \).

Examples:

• \( \sqrt{9} = 3 \), because \( 3^2 = 9 \)
• \( \sqrt{16} = 4 \), because \( 4^2 = 16 \)
• \( \sqrt{25} = 5 \), because \( 5^2 = 25 \)

Cube Numbers

A cube number is the result of multiplying a number by itself twice. These are also known as “perfect cubes.”

Examples: \( 1^3 = 1, \; 2^3 = 8, \; 3^3 = 27, \; 4^3 = 64, \ldots \)

Common Factors

A common factor is a number that divides exactly into two or more numbers. The highest common factor (HCF) is the greatest number that divides two or more numbers without leaving a remainder.

Example: Common factors of 12 and 18 are \( 1, 2, 3, 6 \).
HCF of 12 and 18 is \( 6 \).

Common Multiples

A common multiple is a number that is a multiple of two or more different numbers. The lowest common multiple (LCM) is the smallest such number.

Example: Common multiples of 4 and 6 are \( 12, 24, 36, \ldots \).
LCM of 4 and 6 is \( 12 \).

Rational Numbers (ℚ)

Rational numbers are numbers that can be expressed as a ratio \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \ne 0 \).

This includes:

• All integers (e.g., \( 5 = \frac{5}{1} \))
• Terminating decimals (e.g., \( 0.25 = \frac{1}{4} \))
• Repeating decimals (e.g., \( 0.\overline{3} = \frac{1}{3} \))

Irrational Numbers

Irrational numbers cannot be written in the form \( \frac{a}{b} \). Their decimal expansions are non-terminating and non-repeating.

Examples:

• \( \pi = 3.141592\ldots \)
• \( \sqrt{2} \approx 1.414213\ldots \)
• \( \sqrt{5}, \; e, \; \text{etc.} \)

Reciprocals

The reciprocal of a non-zero number \( x \) is \( \frac{1}{x} \). When you multiply a number by its reciprocal, the result is 1.

Examples:

• Reciprocal of \( 5 \) is \( \frac{1}{5} \)
• Reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \)
• Reciprocal of \( -2 \) is \( -\frac{1}{2} \)