Number and Quantity (Subtest I)
Number Systems and Their Structure
The Number and Quantity domain tests how fluently you move among N ⊂ Z ⊂ Q ⊂ R ⊂ C and how well you justify properties rather than merely compute. Expect questions on field axioms (closure, commutativity, associativity, identities, inverses, distributivity), order properties of the reals, absolute value and distance, and the density of the rationals. Know why √2 is irrational and be able to reproduce a proof by contradiction: if √2 = a/b in lowest terms, then a² = 2b² forces a and b both even, contradicting lowest terms.
Number Theory Tools
Divisibility, prime factorization, gcd/lcm, and modular arithmetic appear regularly. The Euclidean algorithm computes gcd(a,b) by repeated remainder: gcd(252, 105) = gcd(105, 42) = gcd(42, 21) = 21. Use the prime factorization rule gcd = product of shared primes to the lower power and lcm = product of all primes to the higher power, with gcd(a,b)·lcm(a,b) = ab.
Complex Numbers, Vectors, and Matrices
Complex arithmetic follows i² = −1; division uses the conjugate: (a+bi)/(c+di) = (a+bi)(c−di)/(c²+d²). Modulus |a+bi| = √(a²+b²) and argument give polar form r(cos θ + i sin θ), where multiplication multiplies moduli and adds arguments. Vectors add componentwise; the dot product measures angle via u·v = |u||v|cos θ. Matrix multiplication is associative but not commutative, and AB is defined only when columns of A equal rows of B.
Worked Example
Find the last digit of 3^2024. Last digits of powers of 3 cycle with period 4: 3, 9, 7, 1. Since 2024 = 4·506, the exponent is a multiple of 4, so 3^2024 ≡ 3^4 ≡ 1 (mod 10). The last digit is 1. The reasoning depends on recognizing the cycle modulo 10, not on computing the huge power.
Traps That Cost Marks
- Declaring a number irrational because its decimal looks long; only non-repeating, non-terminating decimals are irrational.
- Forgetting that √(a+b) ≠ √a + √b.
- Using gcd and lcm interchangeably, or taking the higher power for gcd.
- Dividing complex numbers without multiplying by the conjugate.
- Assuming AB = BA for matrices, or reversing order in a composition.