TExES Mathematics 7-12 (235)

This chapter covers Competency 002, the complex number system: why complex numbers are needed, their properties (conjugate, modulus, multiplicative inverse), their structure as a field, the rectangular, vector, polar and exponential forms, and the geometric meaning of addition, multiplication and roots.

The equation x^2 + 1 = 0 has no real solution, because the square of a real number is never negative. Defining the imaginary unit i with i^2 = -1 gives the equation two solutions, i and -i. A complex number is a number of the form a + bi, where a and b are real; a is the real part and b is the imaginary part. Every real number is complex (b = 0), and numbers with a = 0 are called pure imaginary.

With complex numbers, every quadratic equation has solutions. The Fundamental Theorem of Algebra goes further: every polynomial of degree n ≥ 1 with complex coefficients has exactly n complex roots, counted with multiplicity.

Worked Example: Solve x^2 + 2x + 5 = 0.

The discriminant is b^2: 4ac = 4 - 20 = -16, so there are no real roots.

Simplify: x = -1 ± 2i.

Result: x = -1 + 2i or x = -1 - 2i, a pair of complex conjugates.

When a polynomial has real coefficients, its non-real roots come in conjugate pairs. So a cubic with real coefficients has either three real roots or one real root and a conjugate pair.

Add and subtract complex numbers by combining real parts and imaginary parts. Multiply with the distributive property and replace i^2 by -1. Powers of i repeat in a cycle of four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, so i^n depends only on the remainder of n divided by 4. For example, i^47 = i^3 = -i because 47 = 4(11) + 3.

Worked Example: Let z = 2 + 3i and w = 1 - 2i. Find zw and z/w in a + bi form.

For the quotient, multiply numerator and denominator by the conjugate of w: z/w = (2 + 3i)(1 + 2i)/((1 - 2i)(1 + 2i)).

Numerator: (2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i^2 = -4 + 7i. Denominator: 1 + 4 = 5. So z/w = (-4 + 7i)/5 = -4/5 + (7/5)i.

Result: zw = 8 - i and z/w = -0.8 + 1.4i.

Common Trap: sqrt(-4) · sqrt(-9) is not sqrt(36) = 6. Write each as an imaginary number first: (2i)(3i) = 6i^2 = -6. The rule sqrt(a)·sqrt(b) = sqrt(ab) requires a and b to be nonnegative.

The modulus (absolute value) of z = a + bi is |z| = sqrt(a^2 + b^2), the distance from z to the origin in the complex plane. Useful properties:

z·z̄ = |z|^2.

|zw| = |z|·|w| and |z/w| = |z|/|w|.

The conjugate of a sum is the sum of the conjugates; the conjugate of a product is the product of the conjugates.

z + z̄ = 2a (real) and z - z̄ = 2bi (pure imaginary).

The multiplicative inverse of a nonzero z is 1/z = z̄/|z|^2.

|z - w| is the distance between the points z and w.

Worked Example: Let z = 3 - 4i. Find |z|, the conjugate of z, and 1/z. Describe where 1/z lies compared with z. |z| = sqrt(9 + 16) = 5, and z̄ = 3 + 4i. 1/z = z̄/|z|^2 = (3 + 4i)/25 = 0.12 + 0.16i.

|1/z| = 1/|z| = 1/5, so 1/z lies at distance 1/5 from the origin, while z lies at distance 5.

The complex numbers satisfy every field axiom: closure, associativity and commutativity of both operations, identities 0 and 1, additive inverses -z, multiplicative inverses for every nonzero z, and the distributive property. Unlike the real numbers, the complex numbers cannot be ordered in a way that is compatible with multiplication. If i > 0, then i·i > 0, so -1 > 0, which is false; if i < 0, then -i > 0, and (-i)(-i) = -1 > 0, again false. So statements such as "3 + 2i > 1 + 5i" have no meaning.

The real numbers form a subfield of C, and the set of numbers a + bi with a and b rational is another subfield. The pure imaginary numbers are not closed under multiplication (i · i = -1, which is real).

The complex number a + bi corresponds to the point (a, b) in the complex plane (real axis horizontal, imaginary axis vertical) and to the vector <a, b> from the origin. Adding complex numbers is vector addition: the sum is the fourth vertex of the parallelogram formed by z, w and 0.

In polar form, z = r(cos θ + i sin θ), often written r cis θ, where r = |z| and θ is an argument of z, the angle from the positive real axis. Euler's formula, e^(iθ) = cos θ + i sin θ, gives the exponential form z = re^(iθ). Setting θ = pi gives e^(i·pi) = -1.

Converting Between Forms

Common Trap: The calculator's arctan(b/a) returns an angle between -90° and 90°. For -2 + 2i, arctan(2/(-2)) = -45°, but the point is in the second quadrant, so θ = 135° (or 3pi/4). Always place the point before naming the angle. So -2 + 2i = 2·sqrt(2) cis 135° = 2·sqrt(2)e^(i·3pi/4).

Polar form makes multiplication geometric. If z = r cis α and w = s cis β, then zw = rs cis (α + β): multiply the moduli and add the arguments. Division divides the moduli and subtracts the arguments. Multiplying by i (which is 1 cis 90°) rotates a point 90° counterclockwise about the origin without changing its distance from the origin. Multiplying by 2i rotates by 90° and doubles the distance.

Result: (1 + i·sqrt(3))^6 = 64.

Expanding the sixth power with the binomial theorem would take far longer, which is exactly why the test includes questions like this.

Every nonzero complex number has exactly n distinct n-th roots. If z = r cis θ, its n-th roots are the n-th root of r times cis ((θ + 360°k)/n) for k = 0, 1, ..., n - 1. Geometrically, the roots lie on a circle of radius (n-th root of r) centered at the origin, equally spaced 360°/n apart, so they form the vertices of a regular n-gon.

Worked Example: Find the three cube roots of 8i.

The cube root of the modulus is 2. The arguments are (90° + 360°k)/3 for k = 0, 1, 2, which gives 30°, 150° and 270°.

Result: the cube roots of 8i are sqrt(3) + i, -sqrt(3) + i and -2i, the vertices of an equilateral triangle inscribed in the circle of radius 2.

Memory Aid: "Multiply lengths, add angles." For powers, raise the length and multiply the angle. For roots, take the root of the length and divide the angle, then spread the roots evenly around the circle.

Because complex arithmetic has geometric meaning, many plane-geometry questions can be phrased with complex numbers, and the test uses this connection.

Complex Operations as Transformations of the Plane

Equations in z describe sets of points. The equation |z| = 3 is the circle of radius 3 centered at the origin. The equation |z - (2 + i)| = 1 is the circle of radius 1 centered at (2, 1). The equation |z - 1| = |z + 1| describes the points equidistant from 1 and -1, which is the imaginary axis (the perpendicular bisector of the segment joining them).

Complex numbers also model quantities that combine magnitude and phase. In alternating-current circuits, impedance is written as a complex number, and adding impedances in series is complex addition; this is one of the connections to science that Competency 019 mentions.

Complex numbers a + bi solve equations such as x^2 + 1 = 0; by the Fundamental Theorem of Algebra a degree-n polynomial has n complex roots, and non-real roots of real polynomials come in conjugate pairs.

Divide by multiplying by the conjugate; 1/z = z̄/|z|^2, and on the unit circle 1/z = z̄.

Polar and exponential forms (r cis θ = re^(iθ)) turn multiplication into "multiply moduli, add arguments"; place the point in its quadrant before naming θ.

De Moivre's Theorem gives powers quickly, and the n-th roots of a number are equally spaced on a circle.