The four families of numbers
Every number you meet on a 0580 paper sits in one of these nested families. Knowing which family it belongs to tells you what you can do with it.
Numbers in the IGCSE syllabus are organised into four nested families. Each family is a SUBSET of the next.
Natural numbers () are the positive whole numbers used for counting: . Some conventions include ; Cambridge usually doesn't.
Integers () are all whole numbers — positive, negative and zero: . Every natural number is an integer.
Rational numbers () are numbers that can be written as where and are integers and . Examples:
- is rational ().
- is rational ().
- is rational ().
- is rational ().
A decimal is rational when it either terminates (e.g. , ) or recurs (e.g. , ).
Real numbers () are every point on the number line. Real numbers split into rationals and irrationals.
Irrational numbers are real numbers that are NOT rational — their decimal expansion goes on forever WITHOUT settling into a repeating pattern. The classics:
- (square root of any non-perfect square is irrational).
- (you'll meet this in A-level, not IGCSE).
- ⊂ ⊂ ⊂ — each family contains the previous one.
- Rational = fraction of two integers. Includes terminating AND recurring decimals.
- Irrational = neither terminates nor recurs. and are the textbook examples.
- Real numbers cover every point on the number line.