1. **Problem Statement:** Learn about different types of numbers and their properties as per the GCSE syllabus. 2. **Types of Numbers:** - **Natural numbers:** Counting numbers starting from 1, 2, 3, ... - **Integers:** Whole numbers including positive, zero, and negative numbers, e.g., -3, 0, 4. - **Prime numbers:** Numbers greater than 1 with only two factors: 1 and itself, e.g., 2, 3, 5, 7. - **Square numbers:** Numbers that are squares of integers, e.g., $1^2=1$, $2^2=4$, $3^2=9$. - **Cube numbers:** Numbers that are cubes of integers, e.g., $1^3=1$, $2^3=8$, $3^3=27$. - **Common factors:** Factors shared by two or more numbers. - **Common multiples:** Multiples shared by two or more numbers. - **Rational numbers:** Numbers that can be expressed as a fraction $\frac{a}{b}$ where $a,b$ are integers and $b\neq0$. - **Irrational numbers:** Numbers that cannot be expressed as a simple fraction, e.g., $\pi$, $\sqrt{2}$. - **Reciprocals:** For a number $x$, its reciprocal is $\frac{1}{x}$. 3. **Examples and Important Rules:** - Convert words to numbers: "six billion" is 6000000000. - Convert numbers to words: 10007 is "ten thousand and seven". - Express 72 as product of prime factors: $$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$$ - Find Highest Common Factor (HCF) of 18 and 24: Prime factors of 18: $2 \times 3^2$ Prime factors of 24: $2^3 \times 3$ Common prime factors: $2$ and $3$ HCF = $2^1 \times 3^1 = 6$ - Find Lowest Common Multiple (LCM) of 18 and 24: Take highest powers of prime factors: $2^3$ and $3^2$ LCM = $2^3 \times 3^2 = 72$ 4. **Summary:** - Understand each type of number and their properties. - Use prime factorization to find HCF and LCM. - Convert between words and numbers accurately. This covers the foundational concepts from lowest to higher levels as per your syllabus.