कक्षा 9–12Numberनियोजित
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
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
12 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- √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|.
आम भ्रांतियाँ
- 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.
आगे ले जाता है
वैकल्पिक खोजें
- 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
अधिगम पथ
आगे पढ़ने के लिए
- Ivan Niven, Numbers: Rational and Irrational (1961)
- Richard Courant and Herbert Robbins, What Is Mathematics? (1941)