Number Systems and Real Numbers
Number Systems and Real Numbers tests how firmly you can move between integers, rationals, irrationals and the real line, and how you justify claims rather than merely compute. Expect Euclid’s division lemma, the fundamental theorem of arithmetic, HCF and LCM, proofs of irrationality, and decimal expansion classification.
Core rules
- Euclid’s lemma: for integers a and b>0, a=bq+r with 0≤r<b; the HCF is the last non-zero remainder.
- Fundamental theorem: every integer >1 has a unique prime factorisation up to order.
- HCF(a,b)×LCM(a,b)=a×b for positive a,b.
- A rational number in lowest terms has a terminating decimal exactly when its denominator has no prime factors except 2 and 5; otherwise the decimal repeats.
- To prove √p irrational for prime p, assume √p=a/b in lowest terms, square, and derive p divides both a and b.
Worked example
Find HCF(252,105). 252=105×2+42; 105=42×2+21; 42=21×2+0. Last non-zero remainder is 21, so HCF=21. Then LCM=252×105/21=1260.
Traps
Do not write a repeating decimal as terminating. In irrationality proofs, the contradiction must come from the lowest-terms assumption, not from the decimal looking non-terminating. For HCF and LCM word problems, identify whether the question asks for grouping (HCF) or simultaneous repetition (LCM). Keep the remainder condition strict: 0≤r<b, not 0<r<b.