Number Systems and Quantity
The Number Systems and Quantity domain covers the structure and properties of number systems, number-theoretic reasoning, and quantitative reasoning with units and precision.
Hierarchy and properties
The counting numbers sit inside the integers, which sit inside the rationals, then the reals, then the complex numbers. The rationals form a field: closed under addition, subtraction, multiplication, and division by a nonzero rational, with commutativity, associativity, distributivity, identities 0 and 1, and additive and multiplicative inverses. Irrational numbers have nonterminating, nonrepeating decimal expansions.
Irrationality argument
Suppose √2 = p/q in lowest terms. Then p² = 2q², so p² is even, hence p is even; write p = 2r, giving 4r² = 2q², so q² = 2r² and q is even. This contradicts lowest terms, so √2 is irrational.
Modular arithmetic
a ≡ b (mod n) when n divides a − b. Congruence respects addition and multiplication, which lets you reduce large powers. Since 3⁶ = 729 ≡ 1 (mod 7) and 100 = 6·16 + 4, we get 3¹⁰⁰ ≡ 3⁴ = 81 ≡ 4 (mod 7).
Quantity and precision
Use dimensional analysis so unit labels cancel. Absolute error is the measured minus the true value; relative error divides that by the true value and is often given as a percent. Scientific notation a × 10ⁿ with 1 ≤ |a| < 10 keeps significant figures visible.
Worked example
Convert 60 mi/h to meters per second: 60 mi/h × 1609.344 m/mi ÷ 3600 s/h ≈ 26.8 m/s. The unit labels cancel explicitly, and the result carries three significant figures.
Traps
Treating a rounded decimal as exact; applying a divisibility rule under the wrong modulus; dropping units mid-conversion; confusing relative and absolute error when the true value is small.