Introduction to Algebra

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Cambridge IGCSE · International Maths (Extended)

Introduction to Algebra

The Big Idea: Algebra is just arithmetic with a placeholder — instead of always working with known numbers, we use letters to stand in for numbers we don't know yet (or that can be anything), and all the same rules of add, subtract, multiply, divide, powers and roots still apply.

Summary — What This Chapter Covers

Algebraic notation — the shorthand rules for writing letters, multiplication, division, powers and brackets.
Algebraic vocabulary — the language you need: term, factor, coefficient, expression, equation, formula.
Substitution — replacing letters in a formula with actual numbers to calculate a value.
Collecting like terms — simplifying expressions by combining terms that "match".

Everything in this chapter is — it's the grammar of algebra. Get this rock-solid now, because every later topic (equations, expanding brackets, factorising, graphs) assumes you can do this fluently and without thinking twice.

1. Algebraic Notation

What is algebra, really?

Algebra uses letters (called variables) to represent numbers that are either unknown (we don't yet know their value, e.g. "what number plus 5 equals 12?") or general (they can represent any number, e.g. "for any two numbers x and y..."). Think of a letter like an empty box waiting to be filled with a number — we just don't know the number yet, or we're keeping it flexible on purpose.

Analogy

Imagine a vending machine slot. It's built to accept coin of the right size — you don't know in advance which exact coin will go in. The letter x is that slot: a fixed "shape" in the calculation that can hold different numbers depending on the situation.

The shorthand rules of algebraic notation

Algebra has its own writing conventions — they exist purely to save space and avoid clutter. You must know these cold, because exam questions assume you can read and write them instantly.

Operation	How it's written	What it means
Add / Subtract	a + b, c + d − e	Same as normal — + and − are written out
Multiply	ab, 3ab	No × symbol! ab means a × b; 3ab means 3 × a × b
Divide	a/b (as a fraction)	Means a ÷ b — division is always shown as a fraction, never with a ÷ sign
Powers	a², 4a²	a² means a × a; in 4a², only a is squared, multiplied by 4
Roots	√a	The square root of a — same idea as with plain numbers
Brackets	3(a + b)	Means 3 × (a + b) — work out what's inside the brackets conceptually first

Key Rule

Order of operations still applies in algebra

In ab + c/3, you're meant to work out a × b and c ÷ 3 as separate "chunks" before adding them — exactly like BIDMAS/PEMDAS with ordinary numbers. Algebra doesn't invent new rules; it just hides the symbols.

Watch Out

4a² is not the same as (4a)². In 4a², the power only applies to the a — square a first, then multiply the result by 4. If you wanted to square the whole thing including the 4, it would need to be written (4a)². This tiny detail trips up a huge number of students.

Worked Example — Building Expressions From a Story

(a) Write down the total number of points Raheem has.

He started with q and gained 6 more, so we just add: Raheem has q + 6 points.

(b) Write down his new number of points.

(q + 6) Raheem has 2(q + 6) points 2q + 12

2q + 6 q

Practice Question 1

Practice Question 2

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Also in the full note

2. Algebraic Vocabulary
3. Substitution
4. Collecting Like Terms
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes

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Introduction to Algebra