Class 10 Maths - ODISHA
Real Numbers
The 'Real Numbers' chapter in Class 10 Mathematics for the Odisha Board (BSE) introduces students to the fundamental properties of numbers. It builds upon previous knowledge by deeply exploring Euclid's Division Lemma and the Fundamental Theorem of Arithmetic. Students will learn how to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using prime factorization, and explore the relationships between them. A major focus is also placed on proving the irrationality of numbers like root 2 and understanding decimal expansions of rational numbers. Mastering this chapter is essential as it carries significant weight in board exams.
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 is less than or equal to r which is strictly less than b.
Fundamental Theorem of Arithmetic
Every composite number can be expressed 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 a and b is always equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).
Irrational Numbers
Numbers that cannot be expressed in the form p/q where p and q are integers and q is not equal to 0. Examples include square root of 2, square root of 3, and pi.
Decimal Expansion of Rational Numbers
Rational numbers with a denominator of the form 2 to the power n into 5 to the power m have terminating decimal expansions, while others are non-terminating repeating.
Important Formulas
Board Exam Info
In the Odisha BSE Class 10 Mathematics board exam, the 'Real Numbers' chapter typically carries around 4 to 6 marks. Common question types include finding HCF and LCM using prime factorization, proving that a given number like root 5 is irrational, and applying Euclid's Division Lemma.
Frequently Asked Questions
How do I prove a number like root 3 is irrational in the exam?
Use the method of contradiction. Assume it is rational (equal to p/q), simplify, and show that p and q share a common factor other than 1, which contradicts the definition of coprime integers.
Is Euclid's Division Lemma the same as Euclid's Division Algorithm?
Lemma is a proven statement used to prove other results (a = bq + r), while an Algorithm is a step-by-step procedure based on the lemma used to calculate the HCF of two numbers.
Can the formula HCF × LCM = a × b be applied to three numbers?
No, the formula a × b = HCF(a, b) × LCM(a, b) is strictly valid only for two positive integers.
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