1. **Natural Numbers**: These are the counting numbers starting from 1, 2, 3, and so on. They do not include zero or negative numbers. 2. **Integers**: This set includes all whole numbers, both positive and negative, including zero. For example, ..., -3, -2, -1, 0, 1, 2, 3, ... 3. **Prime Numbers**: These are natural numbers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, 7, 11. 4. **Square Numbers**: Numbers that are the square of an integer. For example, $1^2=1$, $2^2=4$, $3^2=9$, $4^2=16$. 5. **Cube Numbers**: Numbers that are the cube of an integer. For example, $1^3=1$, $2^3=8$, $3^3=27$, $4^3=64$. 6. **Common Factors**: Factors that two or more numbers share. For example, common factors of 12 and 18 are 1, 2, 3, and 6. 7. **Common Multiples**: Multiples that two or more numbers share. For example, common multiples of 3 and 4 include 12, 24, 36, etc. 8. **Rational Numbers**: Numbers that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. Examples: $\frac{1}{2}$, 0.75, -3. 9. **Irrational Numbers**: Numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Examples include $\pi$, $\sqrt{2}$. 10. **Reciprocals**: The reciprocal of a number $x$ (where $x \neq 0$) is $\frac{1}{x}$. For example, the reciprocal of 5 is $\frac{1}{5}$, and the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. These concepts form the foundation of number theory and are essential for IGCSE mathematics.