Real Numbers (Class 10)
About 6 min · Mathematical Reasoning · English
Time worked out from the lesson's 393 words at 150 words a minute, plus a minute for each question.
Real numbers form the foundation of Class 10 mathematics. This topic introduces rigorous number theory concepts that underpin algebra, geometry and higher mathematics. You must master three core pillars: Euclid's division lemma (a tool for finding HCF and proving divisibility), the fundamental theorem of arithmetic (unique prime factorization), and properties of irrational numbers (non-terminating, non-repeating decimals).
Key ideas
- Euclid's Division Lemma: For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. This is the foundation of the division algorithm and HCF computation.
- Euclid's Division Algorithm: To find HCF of two numbers, repeatedly apply the lemma: divide the larger by the smaller, replace the larger with the smaller and the smaller with the remainder, until remainder becomes zero. The last non-zero remainder is the HCF.
- Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes in exactly one way (ignoring order). This uniqueness property is central to solving HCF, LCM and divisibility problems.
- Prime Factorization Method: Express numbers as products of prime powers (e.g. 72 = 2³ × 3²). HCF is the product of lowest powers of common primes; LCM is the product of highest powers of all primes present.
- Rational Numbers: Numbers expressible as p/q where p, q are integers and q ≠ 0. Their decimal expansions either terminate or repeat periodically.
Quick reference
- Euclid's lemma: a = bq + r, 0 ≤ r < b — basis for HCF algorithm
- HCF via Euclid: Repeatedly divide, replace, until remainder = 0; last non-zero remainder is HCF
- Fundamental theorem: Every composite number has unique prime factorization
- Terminating decimal test: Denominator (in lowest terms) must be 2ᵐ5ⁿ only
- √2, √3, √5 (non-perfect squares): All irrational — prove by contradiction
Check yourself: 3 questions
1. Using Euclid's division lemma, find the HCF of 196 and 38220.
2. The prime factorisation of a number is 2^3 × 3^2 × 5^2 × 7. How many consecutive zeros will this number have at the end when written in decimal form?
3. Let p be a prime number and a be a positive integer. If p divides a^2, then which of the following statements is always true according to the Fundamental Theorem of Arithmetic?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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