Real Numbers (Class 10 Chapter 1) introduces Euclid’s division lemma, the Fundamental Theorem of Arithmetic, and irrational numbers. Use Euclid’s algorithm to find HCF and prime factorization to find LCM — the two most-tested concepts in CBSE board exams.
Key takeaways
- 1Euclid’s division lemma: a = bq + r, with 0 ≤ r < b. Use it iteratively to compute HCF.
- 2Fundamental Theorem of Arithmetic: every composite number has a unique prime factorization (up to order).
- 3A rational p/q has a terminating decimal if and only if q = 2^m × 5^n for non-negative integers m, n.
- 4√2, √3, √5 and the square root of any prime are irrational — prove using contradiction.
- 5For HCF × LCM = product of two numbers (works only for 2 numbers).
Introduction to Real Numbers
Real numbers form the foundation of higher mathematics. In this chapter, we explore the properties of real numbers, beginning with a deep look at the Euclidean algorithm and the Fundamental Theorem of Arithmetic. These two concepts have far-reaching applications in number theory and computing.
Real numbers consist of both rational and irrational numbers. Understanding their properties helps us in solving complex mathematical problems efficiently.
Euclid's Division Lemma
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 ≤ r < b.
This is the basis of the Euclidean algorithm, which is used to find the HCF (Highest Common Factor) of two positive integers. The algorithm is elegant, simple, and remarkably efficient.
- 1Find the HCF of 4052 and 12576 using Euclidean algorithm.
- 2Step 1: 12576 = 4052 × 3 + 420
- 3Step 2: 4052 = 420 × 9 + 272
- 4Step 3: 420 = 272 × 1 + 148
- 5Step 4: 272 = 148 × 1 + 124
- 6Step 5: 148 = 124 × 1 + 24
- 7Step 6: 124 = 24 × 5 + 4
- 8Step 7: 24 = 4 × 6 + 0
- 9HCF (4052, 12576) = 4
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of prime numbers, and this factorization is unique, apart from the order in which the prime factors occur. This theorem is fundamental to number theory.
When finding HCF and LCM using prime factorization, always write the factorization in ascending order of primes for a cleaner solution.
Revisiting Irrational Numbers
A number is called irrational if it cannot be written in the form p/q where p and q are integers and q ≠ 0. The square root of any prime number is irrational. We prove this using the method of contradiction.
Decimal Expansion of Rational Numbers
A rational number p/q (q ≠ 0) has a terminating decimal expansion if and only if q can be expressed in the form 2^m × 5^n where m and n are non-negative integers. Otherwise, the decimal expansion is non-terminating but recurring.
Exercise Solutions
1Q1. Use Euclid’s division algorithm to find HCF of 135 and 225.
2Q2. Show that any positive odd integer is of the form 6q+1, 6q+3 or 6q+5.
3Q3. Prove that √2 is irrational.
Chapter Summary
- Euclid’s division lemma helps find the HCF of two integers.
- The Fundamental Theorem of Arithmetic guarantees unique prime factorization.
- Square roots of primes are irrational.
- A rational number p/q has a terminating decimal iff q = 2^m × 5^n.
Written by Dr. Rajiv Kumar
Verified EducatorPhD in Mathematics, IIT Delhi · 12 years of CBSE board coaching · Reviewed by the Shiksha editorial team.