Class 10 / Maths
Class 10 Maths Real Numbers Notes
Revision notes on Euclid's division lemma, HCF, LCM, prime factorization, and irrational numbers.
Topic introduction
Real Numbers builds the foundation for number theory questions in Class 10. Focus on Euclid's division algorithm, prime factorisation, HCF-LCM relation, and irrationality proofs.
Key points
- Euclid's division lemma says that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 <= r < b.
- The HCF of two numbers is the product of the smallest powers of common prime factors.
- For two positive integers a and b, HCF(a, b) x LCM(a, b) = a x b.
- A number is irrational if it cannot be written in the form p/q, where p and q are integers and q is not zero.
Definitions
Euclid's division lemma: For positive integers a and b, there exist unique integers q and r such that a = bq + r and 0 <= r < b.
HCF: Highest common factor of two or more numbers.
LCM: Least common multiple of two or more numbers.
Irrational number: A number that cannot be written as p/q where p and q are integers and q is not zero.
Formulas
- a = bq + r, where 0 <= r < b
- HCF(a, b) x LCM(a, b) = a x b
Examples
- For 225 and 135: 225 = 135 x 1 + 90, 135 = 90 x 1 + 45, 90 = 45 x 2 + 0, so HCF = 45.
- If HCF of two numbers is 6 and product is 180, then LCM = 180/6 = 30.
- sqrt(2) is irrational, so 5 + sqrt(2) is also irrational.
Euclid's division algorithm
Use repeated division to find the HCF of two positive integers. Divide the larger number by the smaller number, then divide the previous divisor by the remainder until the remainder becomes zero.
The last non-zero remainder is the HCF. This method is useful in board questions because it shows clear step-by-step reasoning.
Prime factorization method
Write each number as a product of prime factors. The HCF uses common factors with the smallest powers. The LCM uses all factors with the greatest powers.
This method is useful when a question asks for both HCF and LCM or asks you to verify HCF x LCM = product of numbers.
Irrational number proofs
Many irrationality proofs begin by assuming the number is rational. Then write it as p/q in lowest terms and show that both p and q must have a common factor.
This creates a contradiction, so the original assumption is false. Therefore the number is irrational.
Common mistakes
- Stopping Euclid's algorithm at the first remainder instead of the last non-zero remainder.
- Using greatest powers for HCF instead of smallest common powers.
- Writing an irrational proof without clearly stating the contradiction.
Quick practice
- Find the HCF of 135 and 225 using Euclid's division algorithm.
- Find the LCM and HCF of 84 and 120 by prime factorization.
- Verify HCF x LCM = product of numbers for 36 and 54.
- Explain why 5 + sqrt(2) is irrational.
Related practice
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