Mathematical Processes and Number Concepts
Core number systems
Candidates must move fluently among natural, integer, rational, real, and complex numbers and justify properties such as closure, commutativity, associativity, distributivity, identities, and inverses. Know divisibility tests, prime factorization, greatest common divisor and least common multiple, and the Euclidean algorithm. Distinguish rational numbers (terminating or repeating decimals) from irrational numbers, and recognize that irrationality often requires proof by contradiction. Proportional reasoning underlies percent, ratio, rate, and unit conversion, while scientific notation and orders of magnitude support estimation. Place-value reasoning in bases other than ten and modular arithmetic (remainders) also support divisibility arguments.
Traps
- Percent changes compose multiplicatively: a 20% rise followed by a 20% fall gives 0.96 of the original, not 100%.
- An "average of averages" is invalid unless group sizes are equal; weight each group by its size.
- Squaring both sides or multiplying by a variable expression can introduce extraneous solutions.
- Zero is neither positive nor negative; division by zero is undefined.
- Absolute value |a − b| represents distance, not simply "make positive."
Worked example
Find gcd(84, 132): 132 = 1·84 + 48; 84 = 1·48 + 36; 48 = 1·36 + 12; 36 = 3·12 + 0. Thus gcd = 12. Then lcm = (84·132)/12 = 924. Rule of thumb: for positive integers, gcd·lcm equals the product.