1. Algebraic Notation
What is algebra, really?
Algebra is the part of maths where we use letters to stand in for numbers we don't know yet, or numbers that could be anything. Instead of writing "a number plus 6 equals 10", we write x + 6 = 10. The letter x is called a variable because its value can vary — it can be different numbers in different situations.
Think of a variable like a box. Instead of saying "the weight of a bag of apples is unknown", you write W = mass of the bag. Now W is a label for that unknown weight. If you're told the total weight is 5kg, W = 5. If it's 3kg, W = 3. Same variable, different values.
Variable: A letter (like x, y, a, or b) that represents a number. We don't know what the number is yet — that's why we use a letter instead.
How do we write calculations in algebra?
Here's where it gets interesting. Algebra uses the same rules as normal arithmetic, but with a few shortcuts to make writing easier:
Add and subtract use + and − (same as numbers)
Multiplication: we write ab instead of a × b
Numbers and letters together: 3ab means 3 × a × b
Division: we use fractions, so a ÷ b becomes a/b
Powers: a² means a × a, just like with numbers
Why no multiplication sign? Because ab is clearer and quicker to write than a × b. Once you're used to it, it's easier to read too. The × symbol can look confusing next to x (the variable), so mathematicians decided to drop it.
Writing algebra
Normal way: 3 × a × b + c ÷ 3
Algebraic way: 3ab + c/3
Why they're the same: Order of operations still applies. We do 3 × a × b and c ÷ 3 first, then add.
Powers and roots in algebra
These work exactly like they do with numbers:
- a² = a × a (a squared)
- 4a² = 4 × a × a (remember: the power applies to a first, then multiply by 4)
- √a = the square root of a
Order of operations matters here. In 4a², we calculate a² first, then multiply by 4. So if a = 3, then 4a² = 4 × (3²) = 4 × 9 = 36, not (4 × 3)² = 144.
Brackets in algebra
Brackets work the same way as in arithmetic. They tell you what to do first.
Brackets in action
3(a + b) means 3 × (a + b). Add a and b first, then multiply by 3.
If a = 2 and b = 5, then 3(a + b) = 3 × (2 + 5) = 3 × 7 = 21.
Without brackets, 3a + b would be (3 × 2) + 5 = 11. Different answer!
Worked example: Points in a game
Raheem has q points. He earns 6 more.
Total points: q + 6
The game then doubles his points.
New total: 2(q + 6) or equivalently 2q + 12
(Both answers are correct — one is factored, one is expanded.)
Write 5 × a × 3 + b ÷ 2 in algebraic notation
Exam tip: Write algebraic notation clearly. Don't confuse × (multiplication) with x (the variable). If you're unsure, use a multiplication dot (·) instead of ×, but algebraic notation is cleaner: just write ab for a × b.