Class 10 Maths - GUJARAT
Real Numbers
The 'Real Numbers' chapter in Class 10 Mathematics for GSEB students builds a strong foundation in number theory 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 prime numbers. A major focus of this chapter is using prime factorization to find the Highest Common Factor (HCF) and Least Common Multiple (LCM), and proving the irrationality of numbers like square root of 2 and square root of 3. This chapter is vital for scoring well in the Gujarat Board exams, consistently carrying about 4 to 6 marks with straightforward numerical problems.
Start Learning FreeKey Concepts
Euclid's Division Lemma
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 factorized 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).
Irrational Numbers
A number is irrational if it cannot be written in the form p/q, where p and q are integers and q is not equal to zero. Examples include square root of 2 and pi.
Important Formulas
Board Exam Info
In the Gujarat Board (GSEB) Class 10 Mathematics exam, the 'Real Numbers' chapter usually carries around 4 to 6 marks. Common question types include finding HCF and LCM using prime factorization, verifying the relationship between HCF and LCM, proving that a given number like square root of 5 is irrational, and short objective questions based on decimal expansions.
Frequently Asked Questions
What is the difference between rational and irrational numbers?
Rational numbers can be expressed as a fraction p/q with non-zero q and have terminating or repeating decimal expansions. Irrational numbers cannot be expressed as simple fractions and have non-terminating, non-repeating decimal expansions.
How do I prove a number is irrational in the exam?
You use the method of contradiction. Assume the number is rational, write it as p/q in simplest form, square both sides to find common factors, and show that this contradicts the Fundamental Theorem of Arithmetic because p and q share a common factor.
Can the formula HCF × LCM = a × b be applied to three numbers?
No, this specific product formula is only valid for two positive integers. For three numbers, HCF(a,b,c) × LCM(a,b,c) is generally not equal to the product of all three numbers.
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