Praxis Mathematics (5165)

The real number system forms the foundation of all mathematical computation and analysis. Real numbers include all rational and irrational numbers, representing every point on the continuous number line.

Rational numbers are numbers expressible as the quotient of two integers where the denominator is nonzero. The set is denoted Q and includes integers, fractions, terminating decimals, and repeating decimals. Every rational number has a decimal representation that either terminates or repeats. For example, 3/4 = 0.75 (terminating) and 1/3 = 0.333... (repeating). The density property states that between any two rational numbers, infinitely many other rational numbers exist.

Irrational numbers cannot be expressed as ratios of integers. Their decimal representations neither terminate nor repeat. Common examples include √2, √3, π, and e. The sum or product of a rational and irrational number is irrational. However, the sum or product of two irrational numbers may be rational or irrational. For instance, √2 × √2 = 2 (rational), while √2 + √3 remains irrational.

The real numbers satisfy fundamental algebraic properties under addition and multiplication. The closure property ensures that sums and products of real numbers are real numbers. The commutative property states a + b = b + a and ab = ba. The associative property gives (a + b) + c = a + (b + c) and (ab)c = a(bc). The distributive property connects operations: a(b + c) = ab + ac. Additive identity is 0, multiplicative identity is 1. Every real number a has an additive inverse -a and every nonzero real number has a multiplicative inverse 1/a.

Real numbers possess a total ordering. The trichotomy property states that for any two real numbers a and b, exactly one holds: a < b, a = b, or a > b. Transitivity ensures that if a < b and b < c, then a < c. Adding the same number to both sides preserves inequality direction. Multiplying both sides by a positive number preserves direction; multiplying by a negative number reverses direction.

The absolute value of a real number represents its distance from zero on the number line. For any real number x, |x| = x if x ≥ 0 and |x| = -x if x < 0. Key properties include |ab| = |a||b|, |a/b| = |a|/|b| for b ≠ 0, and the triangle inequality |a + b| ≤ |a| + |b|. Absolute value equations |x| = a have two solutions x = a and x = -a when a > 0, one solution x = 0 when a = 0, and no solution when a < 0.

Complex numbers extend the real number system to include solutions of equations like x² + 1 = 0. A complex number has the form a + bi where a and b are real numbers and i is the imaginary unit satisfying i² = -1.

In the complex number a + bi, the real part is a and the imaginary part is b. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Addition combines like parts: (a + bi) + (c + di) = (a + c) + (b + d)i. Multiplication uses distribution and the relation i² = -1: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

The conjugate of a + bi is a - bi, denoted by a bar over the number. Multiplying a complex number by its conjugate yields a real number: (a + bi)(a - bi) = a² + b². This property facilitates division: to divide by c + di, multiply numerator and denominator by c - di. The quotient becomes [(ac + bd) + (bc - ad)i]/(c² + d²).

The complex plane represents complex numbers geometrically, with the horizontal axis as the real axis and vertical axis as the imaginary axis. The point (a, b) corresponds to a + bi. The modulus (or absolute value) of a + bi equals √(a² + b²), representing distance from the origin. The argument is the angle θ measured counterclockwise from the positive real axis, where tan θ = b/a.

Complex numbers can be written in polar form as r(cos θ + i sin θ) or r·cis θ, where r is the modulus and θ is the argument. Multiplication in polar form multiplies moduli and adds arguments. Division divides moduli and subtracts arguments. De Moivre's Theorem states [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ), enabling efficient computation of powers.

The nth roots of a complex number r·cis θ are given by r^(1/n)·cis[(θ + 2πk)/n] for k = 0, 1, 2, ..., n-1. Every nonzero complex number has exactly n distinct nth roots, equally spaced around a circle of radius r^(1/n) centered at the origin.

Every polynomial equation of degree n with complex coefficients has exactly n roots in the complex numbers, counting multiplicity. This theorem guarantees that polynomial equations always have solutions when complex numbers are permitted. Complex roots of polynomials with real coefficients occur in conjugate pairs.