Letter-numbers

This chapter is about a short, neat way of writing rules and patterns in maths, one that helps us think about them and even explain why they work.

Shabnam is 3 years older than Aftab. When Aftab is 10, Shabnam is 13. When Aftab is 18, Shabnam is 21. The rule is always the same: add 3. So: Shabnam's age = Aftab's age + 3.

Instead of writing whole phrases, we use letters. Let a be Aftab's age and s be Shabnam's age. Shabnam's age is a + 3, and the whole rule is s = a + 3.

Why a letter, not a box? With one unknown, a box works: □ − 3 = 5, so □ = 8. But with two unknowns, two boxes would get mixed up. A letter gives each unknown its own name. And any letter will do; x is just a habit.

Now the rule works for every age. If a = 23, put 23 in place of a: s = 23 + 3 = 26.

Letters that stand for numbers are called letter-numbers. Expressions with letter-numbers, like a + 3, are algebraic expressions. (An expression has no = sign; s = a + 3, with an = sign, is a relation between two things.)

You can turn the rule round: Aftab is 3 years younger, so a = s − 3. If Shabnam is 20, Aftab is 20 − 3 = 17.