Real Numbers

1. Chapter at a glance

• Every composite number can be expressed as a product of primes and this factorisation is unique apart from the order of the factors (Fundamental Theorem of Arithmetic).
• Prime factorisation of a natural number is unique except for the order of its factors.
• If a prime p divides a², then p divides a (used to prove irrationality).
• HCF of two or more numbers is the product of the smallest powers of the common prime factors; LCM is the product of the greatest powers of all prime factors involved.
• For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
• √2, √3, √5 and, in general, √p (p prime) are irrational; proved by contradiction using the above results.
• Sum or difference of a rational and an irrational is irrational; product or quotient of a non-zero rational and an irrational is irrational.
• Prime factorisation determines whether a rational number has terminating or non-terminating decimal expansion (via denominator).

2. Definitions, theorems and results

Theorem 1.1 (Fundamental Theorem of Arithmetic): Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
(The prime factorisation of a natural number is unique, except for the order of its factors.)

Theorem 1.2: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
(Requires proof in the exam; proof uses FTA.)

Theorem 1.3: √2 is irrational.
(Requires proof in the exam; proof by contradiction using Theorem 1.2.)

Standard results (proved in text): - √3 is irrational (same method as Theorem 1.3). - If a rational number is subtracted from or added to an irrational, the result is irrational (e.g., 5 – √3 is irrational). - Product of a non-zero rational and an irrational is irrational (e.g., 3√2 is irrational).

No definitions of new terms beyond the statements above; all results are derived from FTA.

3. Formula sheet

4. Solved-example patterns

Type 1: Prime factorisation of a number

Method: Repeatedly divide by smallest prime factors until the quotient is 1; write as product of powers of primes in ascending order.

Type 2: Find HCF and LCM of two or three numbers by prime factorisation
Method: Factorise each number; HCF = product of lowest powers of common primes; LCM = product of highest powers of all primes appearing. Verify HCF × LCM = product only when exactly two numbers are involved.

Type 3: Use HCF to find LCM (or vice versa)
Method: Compute one using prime factorisation; apply relation LCM = (a × b) / HCF for two numbers.

Type 4: Check whether a number of the form aⁿ ends with digit 0
Method: Assume it ends with 0 ⇒ divisible by 5; check whether 5 appears in prime factorisation of aⁿ; use uniqueness from FTA to reach contradiction if it does not.

Type 5: Prove √p (p prime) is irrational
Method: Assume √p = a/b in lowest terms (a, b coprime, b ≠ 0); square to get p b² = a²; apply Theorem 1.2 to show p divides both a and b, contradicting coprimeness; conclude irrational.

Type 6: Prove an expression involving rational and irrational is irrational
Method: Assume the expression is rational; rearrange algebraically to isolate the irrational part; show this implies the irrational is rational, leading to contradiction.

5. Common mistakes and exam pitfalls

• Extending HCF × LCM = product to three or more numbers (explicitly false in text).
• Forgetting that FTA uniqueness holds only up to order; writing factors in non-ascending order without combining powers.
• In irrationality proofs: assuming a and b are coprime initially but failing to reach the final contradiction clearly; missing the step that applies Theorem 1.2 twice.
• Stating HCF or LCM without writing the prime factorisation first.
• Claiming a number ends with zero without checking presence of both 2 and 5 in factorisation.
• Omitting “apart from order” when stating FTA.