九至十二年级Number计划中

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

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前置知识

Rational numbers Proof: direct, contrapositive, contradiction and induction

12 项必需的前置知识（一直追溯到底）

你将学会

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.

定义

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.

定理

√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|.

常见误解

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.

通向

Countability and Cantor's diagonal argument Complex numbers Quadratic equations and functions Constructing ℤ, ℚ and ℝ Linear algebra Real analysis

可选发现

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

学习路径

Accelerated secondary progression