Lesson 5 — Rational, Irrational & Real Numbers

Hands-on · about 10 minutes.

Try these first. They are about the last two number families — and about what a function really is.

Rational numbers: anything you can write as a fraction

A rational number is any number you can write as a fraction of two integers — a ratio. That is a bigger family than it first sounds:

• Every integer is rational: 5 = 5/1.
• Every terminating decimal is rational: 0.25 = 1/4.
• Every repeating decimal is rational: 0.333… = 1/3.

The signature of a rational number's decimal is that it either stops or repeats a pattern forever.

Irrational numbers: the ones no fraction can name

An irrational number cannot be written as a fraction of two integers. Its decimal never stops and never settles into a repeating pattern. The famous ones:

Put the rationals and irrationals together and you get the real numbers — every number that sits somewhere on the number line. Each family nests inside the next: counting → whole → integer → rational → real, with the irrationals filling the gaps the fractions leave behind.

Functions: a machine with one output per input

A function is a rule that takes an input and gives back exactly one output. Think of a machine: you drop a number in, the rule acts on it, one number comes out.

The one unbreakable rule: each input has exactly one output. A machine that sometimes gave 7 and sometimes gave 8 for the same input would not be a function. This is the same "find the rule" habit from Lesson 1 — a function is just that rule, written so it works for any input.

True or false? You decide

Decide whether each statement is true or false. Instant feedback either way.

Quick check

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