Real Numbers Class 10 Mathematics — Revision Notes
This chapter explores the properties of real numbers including rational and irrational numbers, Euclids algorithm, and the Fundamental Theorem of Arithmeti
TL;DR: This chapter explores the properties of real numbers including rational and irrational numbers, Euclids algorithm, and the Fundamental Theorem of Arit…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
This chapter explores the properties of real numbers including rational and irrational numbers, Euclids algorithm, and the Fundamental Theorem of Arithmeti
Rational and Irrational Numbers
- Rational numbers can be expressed as p/q where p and q are integers and q is not zero
- Irrational numbers cannot be expressed as p/q (e.g., sqrt(2), pi, e)
- Decimal expansion of rational is terminating or non-terminating repeating
- Decimal expansion of irrational is non-terminating non-repeating
Euclids Division Algorithm
- For positive integers a and b: a = bq + r where 0 <= r < b
- Used to find HCF (Highest Common Factor) of two numbers
- Also called division algorithm
- Example: HCF(4052, 12576) can be found by successive division
Fundamental Theorem of Arithmetic
- Every integer greater than 1 is either prime or product of primes
- Prime factorization of a number is unique
- Example: 60 = 2 x 2 x 3 x 5
- Used to find HCF and LCM of numbers
HCF and LCM Relations
- HCF (a, b) x LCM (a, b) = a x b
- HCF obtained from lowest powers of common prime factors
- LCM obtained from highest powers of all prime factors
- Example: 12 = 2^2 x 3, 18 = 2 x 3^2, HCF = 6, LCM = 36
Key Terms
- Prime Number: Natural number greater than 1 having exactly two factors: 1 and itself
- HCF: Highest Common Factor - greatest number dividing given numbers
- LCM: Least Common Multiple - smallest number divisible by given numbers
Frequently Asked Questions
How do you find HCF using Euclids algorithm?
Apply division algorithm repeatedly: a = bq + r, then b = rq1 + r1, and so on until remainder is 0. The last non-zero remainder is the HCF.
Why is sqrt(2) irrational?
Assume sqrt(2) = p/q in lowest terms. Then 2q^2 = p^2, so p^2 is even, p is even. Let p = 2k, then q^2 = 2k^2, so q is also even. Contradiction, so sqrt(2) is irrational.
More Class 10 Mathematics Revision Notes
- Polynomials
- Pair of Linear Equations in Two Variables
- Quadratic Equations
- Arithmetic Progressions
- Triangles
- Coordinate Geometry
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