Complex Numbers — AP® Precalculus Notes

A quadratic whose graph misses the $x$ -axis has no real zeros, but the quadratic formula still produces answers: they just involve the square root of a negative number. Topic 1.5 counts every zero of a polynomial function, real or not, so this page covers what those answers mean and how to compute with them.

The imaginary unit

No real number squares to $-1$ . The imaginary unit $i$ is defined by $i^2 = -1$ . Then every negative number has square roots: $\sqrt{-16} = \sqrt{16} \cdot i = 4i$ , since $(4i)^2 = 16i^2 = -16$ .

A complex number has the form $a + bi$ with $a$ and $b$ real. $a$ is the real part and $b$ the imaginary part. Real numbers are the complex numbers with $b = 0$ .

Arithmetic

Add and subtract real parts and imaginary parts separately: $(4 - 3i) - (1 - 5i)$ $= 3 + 2i$ .

Multiply like binomials, then replace $i^2$ with $-1$ .

Worked example 1. Multiply $(3 + 2i)(1 - 4i)$ .

Distribute: $3 - 12i + 2i - 8i^2$ .

Replace $i^2$ : $-8i^2 = -8(-1) = 8$ . Combine: $(3 + 8) + (-12 + 2)i$ $= 11 - 10i$ .

Powers of $i$ repeat in a cycle of four:

$i^1$	$i^2$	$i^3$	$i^4$
$i$	$-1$	$-i$	$1$

Complex conjugates

The complex conjugate of $a + bi$ is $a - bi$ . A number times its conjugate is always real:

$(a + bi)(a - bi) = a^2 - b^2 i^2 = a^2 + b^2.$

For example, $(5 - 2i)(5 + 2i)$ $= 25 + 4$ $= 29$ .

Quadratics with no real solutions

Worked example 2. Solve $x^2 + 4x + 13 = 0$ .

Discriminant: $16 - 4(1)(13) = -36$ . Negative, so no real solutions.

Quadratic formula: $x$ $= \frac{-4 \pm \sqrt{-36}}{2}$ $= \frac{-4 \pm 6i}{2}$ $= -2 \pm 3i$ .

The parabola y = x² + 4x + 13 = (x + 2)² + 9 has its vertex at (−2, 9) and never reaches the x-axis. The equation x² + 4x + 13 = 0 has no real solutions; its solutions are the complex numbers −2 + 3i and −2 − 3i. — **Figure 1.** The parabola has no $x$ -intercepts; its two solutions are complex. LevlPrep original diagram.

The two solutions are complex conjugates of each other. That is no accident: the $\pm$ in the formula produces $a + bi$ and $a - bi$ together whenever the coefficients are real and the discriminant is negative. Topic 1.5 extends this pairing to every polynomial function with real coefficients.