Grades 9–12NumberPlanned
Real numbers and irrationality
The real number line: proofs that √2 and other numbers are irrational, decimal expansions, intervals and absolute value, and an informal statement of completeness.
math.number.real-numbers
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 12
You will learn to
- Prove that √2 is irrational, and generalise to √n for every positive integer n that is not a perfect square.
- Describe real numbers by decimal expansions and order them; work with intervals and absolute value.
- State informally that every non-empty set of reals that is bounded above has a least upper bound, and give a set of rationals with no least upper bound in ℚ.
Definition
- A real number is irrational if it is not rational.
- |x| = x if x ≥ 0 and |x| = −x if x < 0; |x − y| is the distance between x and y.
- An upper bound of a set S is a number b with s ≤ b for all s ∈ S; the least upper bound (supremum) is the smallest such b.
Theorem
- √2 is irrational.
- The sum of a rational number and an irrational number is irrational.
- Completeness: every non-empty set of real numbers that is bounded above has a least upper bound in ℝ; ℚ does not have this property.
- Triangle inequality: |x + y| ≤ |x| + |y|.
Common misconceptions
- Thinking π equals 22/7 or 3.14 exactly.
- Believing an infinite decimal that never repeats can be written as a fraction.
- Thinking the product of two irrational numbers must be irrational; √2 · √2 = 2.
Leads to
Optional discoveries
- Sequences and series A decimal expansion is an infinite series; 0.999… = 1 because its partial sums converge to 1.
- ← Fractions and decimals
- ← The Pythagorean theorem
Learning pathways
Further reading
- Ivan Niven, Numbers: Rational and Irrational (1961)
- Richard Courant and Herbert Robbins, What Is Mathematics? (1941)