Chapter 1: Real Numbers
AI-guided lessons for CBSE Class 10
Every integer greater than 1 can be expressed as a unique product of prime numbers — this is the Fundamental Theorem of Arithmetic.
Chapter 1: Real Numbers
1.1 Euclid's Division Lemma
For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This is the foundation of the Euclidean algorithm for finding the HCF of two numbers.
Key Points
- Euclid's Division Lemma applies to positive integers only.
- The remainder r is always less than the divisor b.
- Used to find HCF (Highest Common Factor) of two numbers.
- The algorithm terminates when the remainder becomes 0.
1.2 The Fundamental Theorem of Arithmetic
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
1.3 Irrational Numbers
A number is irrational if it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. Numbers like √2, √3, √5, and π are irrational.
Proof: √2 is Irrational
- Assume √2 = p/q where p, q are co-prime integers.
- Then 2 = p²/q², so p² = 2q² — meaning p² is even.
- If p² is even, then p must be even. Let p = 2m.
- Substituting: 4m² = 2q², so q² = 2m² — q is also even.
- But p and q can't both be even if they're co-prime. Contradiction! ∴ √2 is irrational.
1.4 Decimal Expansions
The decimal expansion of a rational number is either terminating or non-terminating repeating. The decimal expansion of an irrational number is non-terminating and non-repeating.
Chapter 1 Quiz — Real Numbers
5 questions · Earn up to 50 XP
Q1. According to Euclid's Division Lemma, for integers a and b, which equation is correct?
Q2. The HCF of 96 and 404 is:
Q3. Which of the following is an irrational number?
Q4. The decimal expansion of 17/8 is:
Q5. If HCF(a, b) = 12 and a × b = 1800, then LCM(a, b) is:
Click any option to answer. Correct answers earn +10 XP each!
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