Bihar Board Class 10 · Maths · Matric exam 2027
Chapter 1: Real Numbers — notes
The chapter’s 5 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.
1. Euclid's Division Lemma
Okay, so you already know long division.
- • a = bq + r, 0 ≤ r < b
- • q = quotient, r = remainder
- • 17 = 6 × 2 + 5
- • 20 = 4 × 5 + 0
- • Even integer: 2q, Odd integer: 2q + 1
2. Euclid's Division Algorithm to find HCF
Okay, so, Euclid's division algorithm comes straight from the division lemma.
- • HCF = largest number dividing both a and b
- • 455 = 42 × 10 + 35
- • 42 = 35 × 1 + 7
- • 35 = 7 × 5 + 0
- • HCF(455, 42) = 7
3. The Fundamental Theorem of Arithmetic
Okay, so let's talk about this theorem.
- • Composite number = product of primes, uniquely
- • 32760 = 2³ × 3² × 5 × 7 × 13
- • 6 = 2¹ × 3¹, 20 = 2² × 5¹
- • HCF(6, 20) = 2¹ = 2
- • LCM(6, 20) = 2² × 3¹ × 5¹ = 60
- • HCF(a, b) × LCM(a, b) = a × b
4. Proving √2 is Irrational
Okay, so let's prove that √2 is irrational.
- • Irrational: cannot be written as p/q, q ≠ 0
- • If p divides a², then p divides a
- • Assume √2 = a/b, a and b coprime
- • 2b² = a², so 2 divides a; a = 2c
- • 2b² = 4c², so b² = 2c², 2 divides b
- • Contradiction, so √2 is irrational
5. Decimal Expansions of Rational Numbers
Okay, so in Class IX you learned something about decimals.
- • Terminating: q = 2ⁿ5ᵐ
- • 0.375 = 375/1000 = 3/8, 8 = 2³
- • Non-terminating repeating: q not of form 2ⁿ5ᵐ
- • 1/7: remainders 3, 2, 6, 4, 5, 1, ...
- • 1/7 = 0.142857142857...
Summary
Okay, so let's look back at everything we covered. We started with Euclid's division lemma. It says, for positive integers a and b, we can write a = bq + r, with 0 ≤ r < b. That's just ordinary long division. From this came Euclid's division algorithm — you keep applying the lemma, again and again, until the remainder becomes zero. The divisor at that last step, that's your HCF. Then we moved to the Fundamental Theorem of Arithmetic. It tells us every composite number can be broken into primes, and that way is unique, apart from the order. This same idea gives us HCF and LCM by prime factorisation. Using this theorem, and a method called proof by contradiction, you proved that √2 and √3 are irrational. And finally, you learned something about decimals. A rational number p/q has a terminating decimal expansion when q's prime factorisation looks like 2ⁿ5ᵐ. Otherwise, it's non-terminating and repeating. Now tell me — out of all these, which one did you find the toughest to follow?
AskMedha · Medha Ma’am — Bihar Board Class 10 Maths, Chapter 1. Written from the textbook, checked twice.
Questions parents and students ask
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