Types of Numbers

Number Sets

Natural numbers are the counting numbers starting from 1, 2, 3, and so on. They are used for counting objects.

Whole numbers include all natural numbers plus zero. So, 0, 1, 2, 3, ... are whole numbers.

Integers extend whole numbers to include negative numbers as well as positive numbers and zero. For example, -3, -2, -1, 0, 1, 2, 3 are integers. (Note: Negative numbers are covered in other topics.)

Rational numbers are numbers that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. This includes integers, fractions, and decimals that terminate or repeat.

Irrational numbers cannot be written as a simple fraction. Their decimal expansions neither terminate nor repeat. Examples include $\pi$ and $\sqrt{2}$.

For instance, $\frac{3}{4}$ and 0.25 are rational because $\frac{3}{4} = 0.75$ (terminating decimal), but $\pi \approx 3.14159...$ is irrational.

Prime Numbers and Factors

Prime numbers are natural numbers greater than 1 that have exactly two distinct factors: 1 and the number itself. They cannot be divided evenly by any other number.

Composite numbers are natural numbers greater than 1 that have more than two factors. For example, 4 is composite because it has factors 1, 2, and 4.

Factors of a number are integers that divide the number exactly without leaving a remainder.

Multiples of a number are the products of that number and any integer.

Prime factorisation is expressing a number as a product of prime numbers only.

For example, 7 is prime because its only factors are 1 and 7. The number 12 is composite because it has factors 1, 2, 3, 4, 6, and 12.

Prime factorisation of 12 is:

$12 = 2 \times 2 \times 3 = 2^2 \times 3$

Squares, Cubes and Roots

Square numbers are numbers multiplied by themselves. The square of $n$ is written as $n^2$.

Examples: $1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$.

Cube numbers are numbers multiplied by themselves twice more. The cube of $n$ is written as $n^3$.

Examples: $1^3 = 1$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, $5^3 = 125$.

Square roots are the inverse operation of squaring. The square root of $x$ is a number $y$ such that $y^2 = x$. It is written as $\sqrt{x}$.

For example, $\sqrt{25} = 5$ because $5^2 = 25$.

Cube roots are the inverse operation of cubing. The cube root of $x$ is a number $y$ such that $y^3 = x$. It is written as $\sqrt[3]{x}$.

For example, $\sqrt[3]{27} = 3$ because $3^3 = 27$.

Reciprocals

Reciprocal of a number is its multiplicative inverse. When a number is multiplied by its reciprocal, the product is 1.

For any non-zero number $a$, the reciprocal is $\frac{1}{a}$.

For example, the reciprocal of 5 is $\frac{1}{5}$, and the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.

The reciprocal of a decimal can be found by writing it as a fraction and then flipping numerator and denominator.

Learning example: Find the reciprocal of 0.2.

Write 0.2 as $\frac{1}{5}$. The reciprocal is $\frac{5}{1} = 5$.