3.1 Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit $i$, where $i^2 = -1$. This lets you take square roots of negative numbers, which is essential for solving polynomial equations that have no real solutions. You'll use complex numbers throughout this unit when analyzing polynomial and rational functions.

Complex Numbers

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Square roots of negative numbers

The imaginary unit $i$ is defined as $i = \sqrt{-1}$, and its most important property is that $i^2 = -1$.

To find the square root of a negative number, separate it into the square root of the positive part times $\sqrt{-1}$:

• $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$
• $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$

A complex number has the general form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. Both $a$ and $b$ are real numbers, and either can be zero. So $0 + 3i$ (usually written $3i$) is purely imaginary, and $2 + 0i$ (just $2$) is a real number. Every real number is technically a complex number with $b = 0$.

Graphical representation on the complex plane

The complex plane is a 2D coordinate system where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. You plot a complex number $a + bi$ at the point $(a, b)$.

• $3 + 2i$ is plotted at $(3, 2)$
• $-1 + 4i$ is plotted at $(-1, 4)$

The modulus (or absolute value) of a complex number is its distance from the origin. For $z = a + bi$:

$|z| = \sqrt{a^2 + b^2}$

This is just the Pythagorean theorem applied to the complex plane. For example, $|3 + 4i| = \sqrt{9 + 16} = 5$.

Complex numbers can also be written in polar form as $r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the angle from the positive real axis. You won't need polar form as much in this section, but it shows up later.

Arithmetic operations with complex numbers

Addition and subtraction: Combine real parts with real parts and imaginary parts with imaginary parts.

• $(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i$
• $(5 - 3i) - (2 + i) = (5 - 2) + (-3 - 1)i = 3 - 4i$

Multiplication: Use the distributive property (FOIL), then replace $i^2$ with $-1$.

• $(3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i^2 = 3 - 10i + 8 = 11 - 10i$
• $(2 - i)(3 + 4i) = 6 + 8i - 3i - 4i^2 = 6 + 5i + 4 = 10 + 5i$

Division: Multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of $a + bi$ is $a - bi$. This eliminates the imaginary part from the denominator.

1. Identify the conjugate of the denominator
2. Multiply top and bottom by that conjugate
3. Simplify, using $i^2 = -1$

$\frac{3 + 2i}{1 - 4i} = \frac{(3 + 2i)(1 + 4i)}{(1 - 4i)(1 + 4i)} = \frac{3 + 12i + 2i + 8i^2}{1 - 16i^2} = \frac{3 + 14i - 8}{1 + 16} = \frac{-5 + 14i}{17} = -\frac{5}{17} + \frac{14}{17}i$

Notice that the denominator becomes a real number because $(a + bi)(a - bi) = a^2 + b^2$. That's the whole point of using the conjugate.

Complex numbers for polynomial equations

When a polynomial equation has no real solutions, the solutions are complex numbers. The discriminant $b^2 - 4ac$ from the quadratic formula tells you what to expect:

• If $b^2 - 4ac > 0$: two distinct real roots
• If $b^2 - 4ac = 0$: one repeated real root
• If $b^2 - 4ac < 0$: two complex conjugate roots

Example 1: Solve $x^2 + 2x + 5 = 0$

1. Identify $a = 1$, $b = 2$, $c = 5$

2. Compute the discriminant: $4 - 20 = -16$

3. Apply the quadratic formula: $x = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$

4. The roots are $-1 + 2i$ and $-1 - 2i$

Notice the roots come in conjugate pairs. Complex roots of polynomials with real coefficients always come in conjugate pairs. This is a fact you'll use repeatedly in this unit.

Example 2: Solve $2x^2 - 3x + 4 = 0$

1. Identify $a = 2$, $b = -3$, $c = 4$

2. Compute the discriminant: $9 - 32 = -23$

3. Apply the quadratic formula: $x = \frac{3 \pm \sqrt{-23}}{4} = \frac{3 \pm i\sqrt{23}}{4}$

4. The roots are $\frac{3 + i\sqrt{23}}{4}$ and $\frac{3 - i\sqrt{23}}{4}$

Advanced Complex Number Concepts

• Euler's formula: $e^{ix} = \cos x + i\sin x$, which connects complex exponentials to trigonometric functions.
• Complex roots of unity are solutions to $z^n = 1$. They form a regular polygon when plotted on the complex plane (for example, the cube roots of unity form an equilateral triangle).
• De Moivre's theorem gives a clean way to raise complex numbers to powers: $(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))$. This is especially useful for finding $n$th roots of complex numbers.