Real numbers and irrationality

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 12

ستتعلّم

• 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.

تعريف

• 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.

مبرهنة

• √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|.

مفاهيم خاطئة شائعة

• 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.

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اكتشافات اختيارية

• 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

مسارات التعلّم

• Accelerated secondary progression

قراءات إضافية

• Ivan Niven, Numbers: Rational and Irrational (1961)
• Richard Courant and Herbert Robbins, What Is Mathematics? (1941)