Real Numbers
Real Numbers opens the BSEB Matric syllabus and tests divisibility, HCF and LCM, the Fundamental Theorem of Arithmetic, irrational numbers, and terminating decimals. Questions usually ask for a proof or a short computation, so state the lemma or theorem you use.
Core rules
- Euclid’s division lemma: for positive integers a and b, there exist unique integers q and r with a = bq + r and 0 ≤ r < b.
- Fundamental Theorem of Arithmetic: every composite number has a unique prime factorisation apart from order.
- For two positive integers, HCF(a,b) × LCM(a,b) = a × b.
- A rational number p/q in lowest terms has a terminating decimal exactly when q = 2^m5^n.
- √2, √3, √5 are irrational; prove by contradiction using prime factorisation.
Worked example
Find HCF and LCM of 96 and 404. Apply Euclid’s algorithm: 404 = 96 × 4 + 20 96 = 20 × 4 + 16 20 = 16 × 1 + 4 16 = 4 × 4 + 0 The last non-zero remainder is 4, so HCF = 4. Since HCF × LCM = 96 × 404, LCM = (96 × 404)/4 = 9696.
Traps
Do not write r = b or a negative remainder; the remainder must satisfy 0 ≤ r < b. The product relation holds for two numbers, not automatically for three. In irrationality proofs, “assume √2 = p/q with p,q coprime” must be stated and used. For terminating decimals, reduce the fraction first; 3/6 terminates because it is 1/2, although 6 is not of the form 2^m5^n.