Number Systems and Operations
What the Domain Covers
This domain tests the structure of the real number system, fluent computation with rational and irrational quantities, number-theoretic tools, exponent and radical laws, complex arithmetic, and proportional reasoning. Expect items that ask not only for an answer but for the property that justifies it.
Real Number Structure
The sets nest: counting numbers ⊂ integers ⊂ rationals ⊂ reals. A number is rational exactly when it can be written a/b with integers a, b and b ≠ 0; equivalently, its decimal terminates or repeats. Irrationals such as √2, π, and e have non-terminating, non-repeating decimals. The rationals are closed under addition, subtraction, multiplication, and division by a nonzero rational; the irrationals are not closed under any of these.
Number Theory Tools
Prime factorization drives GCF and LCM: use the smallest power of each shared prime for the GCF and the largest power of every prime appearing for the LCM. For any positive integers, GCF(a, b) · LCM(a, b) = ab. Divisibility shortcuts: 3 and 9 by digit sum, 4 by the last two digits, 8 by the last three, 11 by the alternating digit sum.
Exponents, Radicals, Complex Numbers
a^m · a^n = a^(m+n); a^m / a^n = a^(m−n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = the n-th root of a^m. For complex numbers, i = √(−1) and i² = −1; multiply by distributing and replace i² with −1. Multiply by the conjugate a − bi to rationalize a denominator.
Worked Example
Simplify (3 + 2i)(1 − 4i). Distribute: 3(1) = 3; 3(−4i) = −12i; 2i(1) = 2i; 2i(−4i) = −8i². Combine like terms: 3 − 10i − 8(−1) = 3 − 10i + 8 = 11 − 10i. Check by moduli: |3 + 2i|² = 13 and |1 − 4i|² = 17, so the product's squared modulus must be 221. Indeed 11² + 10² = 121 + 100 = 221.
Traps That Cost Marks
- Assuming (a + b)² = a² + b² or that √(a + b) = √a + √b.
- Applying a negative exponent term-by-term to a sum: (a + b)^(−1) ≠ a^(−1) + b^(−1).
- Swapping GCF and LCM in word problems: "greatest number of identical groups" is GCF; "when do two cycles next coincide" is LCM.
- A 20% increase followed by a 20% decrease is a 4% net decrease, not zero.