Level 6: The real numbers

Numbers · level 6 of 8 · rough age 13-14 (a guide for parents; anyone can start anywhere)

Fractions seem to fill the number line, but they do not: some points, like \(\sqrt2\), can never be written as a fraction. In this level you find out exactly which decimals are fractions, prove (not just check) that \(\sqrt2\) is irrational, and learn to calculate exactly with roots. Then you put every family of numbers into one picture, the real numbers, and finish with the practical side: how to round, how big errors can grow, and a 2000-year-old method that finds square roots astonishingly fast.

Lesson	You will be able to
Repeating decimals to fractions	Turn any repeating decimal into a fraction, and explain why every fraction's decimal must stop or repeat
Rational and irrational numbers	Tell rational from irrational numbers, build new irrationals, and explain why fractions leave holes in the number line
Proof that √2 is irrational	Write the classic proof by contradiction, adapt it to \(\sqrt3\), \(\sqrt5\), \(\sqrt6\), and see why it fails for \(\sqrt4\)
Radicals	Simplify, add, multiply and compare radicals, and rationalise a denominator
The real numbers	Place any number in its families, test closure, and disprove a claim with a counterexample
Estimation and precision	Round or truncate, measure errors with absolute value, and avoid the traps of rounding early
Approximating roots	Use the Babylonian divide-and-average method, explain why it is so fast, and count the steps needed

Previous: Level 5: integers, powers and roots · Next: Level 7: building number systems · Numbers topic page