Real Numbers — Chapter Summary
Maharashtra Board · Class 9 · Mathematics
Summary of Real Numbers for Maharashtra Board Class 9 Mathematics. Part of the Maharashtra Board Class 9 Mathematics syllabus.
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Overview
Real numbers form the foundation of mathematics and include all rational and irrational numbers. This chapter explores the properties of rational numbers, introduces irrational numbers like surds, and teaches us how to perform operations on them. Understanding real numbers is essential for solving e
Key Concepts
Rational numbers are numbers that can
Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples include 2/3, -5/7, 4, and 0. The de
Irrational numbers are real numbers
Irrational numbers are real numbers that cannot be expressed as p/q (where p and q are integers with q ≠ 0). Examples include √2, √3, π, and (2 + √5).
We prove √2 is irrational using
We prove √2 is irrational using contradiction: Assume √2 = p/q (in reduced form). Then 2q² = p², meaning p² is even, so p is even. Let p = 2t, then 2q
A surd is an irrational root
A surd is an irrational root of a positive rational number. For example, √7, ∛5, and ⁴√8 are surds because their values are irrational. However, √4 =
A surd is in simplest form
A surd is in simplest form when the radicand has no perfect square factors. For example: √48 = √(16 × 3) = √16 × √3 = 4√3. Similarly, √98 = √(49 × 2)
Learning Objectives
- Understand the properties and decimal representations of rational numbers
- Recognize and prove that certain numbers like √2 and √3 are irrational
- Comprehend the concept of irrational numbers and their properties
- Learn about surds and their simplification in simplest form
- Compare and perform operations on quadratic surds
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Sources & Official References
Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.
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