Class 10 Maths - CBSE
Real Numbers
The 'Real Numbers' chapter in Class 10 CBSE Mathematics builds a strong foundation in number systems by exploring rational and irrational numbers. You will learn the Fundamental Theorem of Arithmetic, which states that every composite number can be uniquely expressed as a product of primes. This chapter covers essential methods like finding HCF and LCM using prime factorization and proving the irrationality of numbers like root 2 and root 3. This is a high-scoring chapter that typically carries around 2 to 6 marks in the CBSE board exam, featuring straightforward application and proof-based questions.
Start Learning FreeKey Concepts
Euclid's Division Lemma
A proven statement used for proving other statements, stating that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 <= r < b.
The Fundamental Theorem of Arithmetic
Every composite number can be factorized as a product of prime numbers, and this factorization is unique, apart from the order in which the prime factors occur.
HCF and LCM Relationship
For any two positive integers a and b, the product of their HCF and LCM is always equal to the product of the two numbers: HCF(a, b) * LCM(a, b) = a * b.
Irrational Numbers
Numbers that cannot be expressed in the form p/q where p and q are integers and q != 0. Examples include root 2, root 3, and pi.
Important Formulas
Board Exam Info
In the CBSE Class 10 Mathematics board exam, this chapter generally carries about 2 to 6 marks. Common question types include finding HCF and LCM using prime factorization, word problems based on LCM/HCF (like circular tracks or grouping items), and proof questions like proving that root 5 is an irrational number.
Frequently Asked Questions
Can the formula HCF * LCM = a * b be applied to three numbers?
No, this specific formula is strictly valid only for two positive integers. For three numbers, HCF(a,b,c) * LCM(a,b,c) is not necessarily equal to a * b * c.
How do I prove that a number like root 3 is irrational?
You use the method of contradiction. Assume it is rational (equal to p/q where p and q are co-prime), square both sides, show that 3 divides both p and q, and arrive at a contradiction that they have a common factor other than 1.
Is Euclid's Division Lemma the same as Euclid's Division Algorithm?
Euclid's Division Lemma is a proven statement used to solve problems, whereas Euclid's Division Algorithm is a technique based on the Lemma used specifically to compute the HCF of two positive integers.
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