Real Numbers | SFALXO

Core Concepts

1. Euclid’s Division Lemma

Given positive integers a and b, there exist unique integers q and r satisfying:

a = bq + r, where 0 ≤ r < b

This is extensively used to find the Highest Common Factor (HCF) of two positive integers.

2. Fundamental Theorem of Arithmetic

Every composite number can be expressed (factored) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

3. Relation Between HCF and LCM

For any two positive integers a and b:

HCF(a, b) × LCM(a, b) = a × b

4. Rational Numbers and Their Decimal Expansions

A rational number in the form ^p/_q has a terminating decimal expansion if, in its lowest form, the prime factorisation of the denominator q is of the form:

q = 2^m5ⁿ

where m and n are non-negative integers. Otherwise, its decimal expansion is non-terminating recurring.

Points to Remember

Crucial Rule: Always reduce the fraction ^p/_q to its lowest form by cancelling out common factors before checking the denominator's prime factorisation.

Irrational Numbers: Numbers that cannot be written in the form ^p/_q have decimal expansions that are non-terminating and non-recurring (e.g., √2, √3, √5, π).

Important Practice Questions

Use Euclid’s division algorithm to find the HCF of 135 and 225.
Prove that √5 is an irrational number.
Without actual long division, determine whether the rational number ¹³/₃₁₂₅ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

Quick Revision

Euclid's Division Lemma

a = bq + r (0 ≤ r < b). Used for finding HCF step-by-step.

Fundamental Theorem

Composite numbers have a unique prime factorisation.

HCF & LCM Property

HCF × LCM = Product of two numbers.

Terminating Decimals

Denominator prime factors are exclusively in powers of 2 and/or 5.

Non-Terminating Decimals

Denominator contains prime factors other than 2 or 5.

Irrationality Proofs

Proven using proof by contradiction technique.