Number and Quantity
Number and Quantity
This domain covers the real and complex number systems, number-theoretic ideas (divisibility, primes, modular reasoning), vectors and matrices as quantities, and proportional reasoning. Questions test structure-why a number is irrational, how operations close within a system-as much as computation.
Core rules and concepts
- Hierarchy: naturals ⊂ integers ⊂ rationals ⊂ reals ⊂ complex. Irrational numbers have non-repeating, non-terminating decimals.
- Complex arithmetic: i² = −1; to divide, multiply numerator and denominator by the conjugate.
- Vectors have magnitude and direction; add componentwise, and scale by a scalar.
- Matrices: multiplication is not commutative; AB is defined when columns of A equal rows of B.
- Proportional reasoning: set up ratios with matching units and solve by cross-multiplication or unit rate.
Worked example
Simplify (3 + 2i)/(1 − i). Multiply by the conjugate (1 + i)/(1 + i): (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 3 + 5i − 2 = 1 + 5i. Denominator: (1 − i)(1 + i) = 1 − i² = 1 + 1 = 2. Result: (1 + 5i)/2 = 1/2 + (5/2)i. Reasoning: the conjugate removes i from the denominator because (a − bi)(a + bi) = a² + b².
Traps that cost marks
- Treating i² as +1 instead of −1.
- Conjugating only the numerator, or forgetting to multiply the denominator too.
- Confusing vector magnitude with a component.
- Assuming √(a + b) = √a + √b, or that all roots are rational.