Numbers and their operations

Welcome to the World of Numbers!

Welcome! This chapter is the foundation of everything you will do in O-Level Mathematics. We are looking at Numbers and their operations. Think of numbers as the "alphabet" of math—once you master how they work, you can start "writing" more complex things like algebra and geometry. Don't worry if some of this feels like a lot to take in; we will break it down step-by-step!

1. Primes and Prime Factorisation

Every whole number (greater than 1) is either a Prime Number or a Composite Number.

Prime Numbers: These are numbers that have exactly two factors: 1 and themselves. Examples include 2, 3, 5, 7, 11, and 13.
Did you know? The number 2 is the only even prime number! The number 1 is not a prime number because it only has one factor.

Prime Factorisation

Every composite number can be broken down into a product of its prime factors. It's like finding the "DNA" of a number. You can use a Factor Tree or Repeated Division.

Example: Prime factorise 60.
60 = 2 × 30
60 = 2 × 2 × 15
60 = 2 × 2 × 3 × 5
In index notation: \( 60 = 2^2 \times 3 \times 5 \)

Key Takeaway:

Prime Factorisation is just breaking a big number down into a team of prime numbers that multiply together to make that original number.

2. HCF, LCM, and Roots

Once you know prime factorisation, you can find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) easily.

HCF and LCM

HCF: The largest number that can divide into two or more numbers. Think: "What is the biggest shared piece?"
LCM: The smallest number that is a multiple of two or more numbers. Think: "Where is the first place these numbers meet on a timeline?"

The Prime Factor Trick:
To find HCF: Take the lowest power of each common prime factor.
To find LCM: Take the highest power of every prime factor present.

Squares, Cubes, and Roots

We can use prime factorisation to find square roots (\( \sqrt{} \)) and cube roots (\( \sqrt[3]{} \)).
- For a Perfect Square, all the powers in its prime factorisation must be even.
- For a Perfect Cube, all the powers must be multiples of 3.

Quick Review:

If \( 36 = 2^2 \times 3^2 \), then \( \sqrt{36} = 2^{2\div2} \times 3^{2\div2} = 2^1 \times 3^1 = 6 \). Easy!

3. The Number Family

Numbers come in different types. Understanding these helps you know what tools to use!

Integers: Whole numbers (positive, negative, or zero). \( \{..., -2, -1, 0, 1, 2, ...\} \)
Rational Numbers: Numbers that can be written as a fraction \( \frac{a}{b} \). This includes decimals that end or repeat.
Real Numbers: All the numbers on the number line, including "messy" decimals like \( \pi \) or \( \sqrt{2} \).

Number Lines and Symbols

We represent these numbers on a Number Line. We use symbols to compare them:
\( < \) : is less than
\( > \) : is greater than
\( \le \) : is less than or equal to
\( \ge \) : is greater than or equal to

Analogy: Think of the symbol like an alligator's mouth—it always wants to eat the bigger number!

4. Approximation and Estimation

In the real world, we don't always need an exact number. Sometimes "close enough" is better.

Rounding Off

Decimal Places (d.p.): Count digits to the right of the decimal point.
Significant Figures (s.f.): Count from the first non-zero digit. This is very important in Science and Math!

Common Mistake to Avoid:

When counting significant figures, zeros at the start of a number (like 0.005) don't count, but zeros in the middle (505) or at the end after a decimal point (5.0) do count!

Estimation

Estimation is a "sanity check" for your calculations. Before using your calculator, round the numbers to 1 significant figure to guess the answer. If your calculator gives you something totally different, you might have pressed the wrong button!

5. Standard Form

Standard Form is a way to write very big numbers (like the distance to the sun) or very tiny numbers (like the size of a cell) easily.

It always looks like: \( A \times 10^n \)
- \( A \) must be between 1 and 10 (it can be 1, but must be less than 10).
- \( n \) is an integer (positive for big numbers, negative for tiny decimals).

Example: 5,200 becomes \( 5.2 \times 10^3 \). 0.0052 becomes \( 5.2 \times 10^{-3} \).

6. Indices (Powers)

Indices are shorthand for repeated multiplication. Instead of \( 2 \times 2 \times 2 \), we write \( 2^3 \). There are rules (Laws of Indices) that make working with them much faster.

The Laws of Indices

1. Multiplication: \( a^m \times a^n = a^{m+n} \) (Add the powers)
2. Division: \( a^m \div a^n = a^{m-n} \) (Subtract the powers)
3. Power of a Power: \( (a^m)^n = a^{m \times n} \) (Multiply the powers)
4. Zero Index: \( a^0 = 1 \) (Anything to the power of zero is 1!)
5. Negative Index: \( a^{-n} = \frac{1}{a^n} \) (A negative power means "flip it")
6. Fractional Index: \( a^{\frac{1}{n}} = \sqrt[n]{a} \) (The bottom of the fraction is the root)

Memory Aid:

When multiplying numbers, we add powers. When dividing numbers, we subtract powers. Think of indices as being "one step simpler" than the operation being done to the base numbers.

Final Wrap-Up

Mastering these basics will make the rest of your O-Level journey much smoother. Remember:
- Use Prime Factorisation for HCF, LCM, and roots.
- Watch your Significant Figures when rounding.
- Practice the Laws of Indices until they become second nature.
- Don't be afraid to use your calculator to check your work, but always try to understand the logic behind it first!