Class 10 Maths - BIHAR
Real Numbers
The 'Real Numbers' chapter in Class 10 Mathematics is foundational for the Bihar School Examination Board (BSEB) curriculum. It delves deep into the properties of integers and real numbers, focusing heavily on the Fundamental Theorem of Arithmetic and Euclid's Division Lemma. Students learn how to find HCF and LCM using prime factorization and explore proofs for the irrationality of numbers like root 2 and root 5. Scoring well in this chapter is crucial as it sets the mathematical tone for the entire board exam and consistently features in both objective and short-answer questions.
Start Learning FreeKey Concepts
Euclid's Division Lemma
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 is less than or equal to r which is less than b.
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes, 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).
Rational and Irrational Numbers
Rational numbers can be expressed in the form p/q with terminating or repeating decimal expansions, whereas irrational numbers like root 2 cannot be expressed as simple fractions and have non-terminating, non-recurring decimals.
Important Formulas
Board Exam Info
In the Bihar (BSEB) Class 10 Mathematics board exam, the Real Numbers chapter typically carries around 5 to 8 marks. Questions usually include 1-2 objective (multiple choice) questions, one short-answer question on finding HCF/LCM or proving irrationality, and occasionally a conceptual problem based on Euclid's division algorithm.
Frequently Asked Questions
How do I prove that a number like root 3 is irrational?
You use the proof by contradiction method, assuming it is rational, expressing it as a coprime fraction, and showing that it leads to a contradiction regarding common factors.
Is Euclid's Division Lemma the same as Euclid's Division Algorithm?
No, the Lemma is a proven statement used to prove other statements (like a = bq + r), while the Algorithm is a step-by-step procedure based on the lemma used to calculate the HCF of two numbers.
Can the LCM of two numbers be less than their HCF?
No, the LCM of two positive integers is always greater than or equal to their HCF, because HCF is a factor while LCM is a multiple.
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