Handbook · Chapter 7

Complex Numbers

The algebra of i, conjugates, the complex plane, polar form and De Moivre, roots of unity, and complex numbers as a tool for trigonometry.

Book pages 127–132 · 36 entries

7.1 Basic Definitions

Definition 7.1.1

A complex number is a number that can be written as $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the “imaginary unit”. $a$ is the real part of the number and $bi$ is the imaginary part. Complex numbers are often named with the letter $z$ .

Note: this topic is mainly relevant for the AMC 12.

Learn it:Definition 44.1.1 Section 44.1a The Algebra of i

Book p. 127

Definition 7.1.2

i = \sqrt{-1},\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1

Learn it:Definition 44.1.2

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Remark 7.1.3

Powers of $i$ cycle every 4 terms, so $i^{4n} = i^4 = 1$ , $i^{4n+1} = i = \sqrt{-1}$ , $i^{4n+2} = i^2 = -1$ , $i^{4n+3} = i^3 = -i$ .

Learn it:Remark 44.1.3

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Theorem 7.1.4 · Adding Complex Numbers

(a + bi) + (c + di) = (a + c) + (b + d)i

Learn it:Theorem 44.1.4 · Adding Complex Numbers

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Theorem 7.1.5 · Subtracting Complex Numbers

(a + bi) - (c + di) = (a - c) + (b - d)i

Learn it:Theorem 44.1.5 · Subtracting Complex Numbers

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Theorem 7.1.6 · Multiplying Complex Numbers

(a + bi)\cdot(c + di) = (ac - bd) + (bc + ad)i

Learn it:Theorem 44.1.6 · Multiplying Complex Numbers

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Definition 7.1.7 · Real and Imaginary Parts

The imaginary part of a complex number $a + bi$ is $b$ , and the real part is $a$ .

Learn it:Definition 44.1.7 · Real and Imaginary Parts

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Remark 7.1.8

The imaginary part does not include a factor of $i$ : the imaginary part of $3 - 5i$ is $-5$ , not $-5i$ .

Learn it:Remark 44.1.8

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7.2 Conjugates

Definition 7.2.1

The complex conjugate of a complex number is found by flipping the sign of its imaginary part. The conjugate of $z$ is written $\bar z$ .

Learn it:Definition 44.1.9 Section 44.1a The Algebra of i

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Theorem 7.2.2 · Finding Conjugates

\bar z = \overline{a + bi} = a - bi

Learn it:Theorem 44.1.10 · Finding Conjugates

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Theorem 7.2.3 · Multiplying Complex Numbers with their Conjugates

(a + bi)(a - bi) = a^2 + b^2

Learn it:Theorem 44.1.14 · Multiplying Complex Numbers with their Conjugates Theorem 44.1.15

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7.3 Complex Roots

Definition 7.3.1

A polynomial of degree $n$ has $n$ roots (counted with multiplicity), and these roots may be complex. For quadratics, if the discriminant is negative, the roots are complex.

Correction

The book says “For binomials, if our discriminant is negative we have complex roots”; the discriminant test is for quadratics, so the app says “quadratics”.

Learn it:Definition 44.2.1 Section 44.2 Complex Numbers in Polynomials

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7.4 Complex Plane

Definition 7.4.1

A complex number can also be drawn geometrically: write $a + bi$ as the point $(a, b)$ in the complex plane. The $x$ -axis is the real axis and the $y$ -axis is the imaginary axis.

Learn it:Definition 44.1.11 · Complex Plane Section 44.1b Polar Form, De Moivre and Roots of Unity

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Definition 7.4.2

The magnitude of a complex number, written $|z|$ , is the distance of the point $(a, b)$ from the origin.

Learn it:Definition 44.1.12

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Theorem 7.4.3 · Magnitude of a Complex Number

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

Learn it:Theorem 44.1.13 · Magnitude of a Complex Number

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7.5 Polar Form

Definition 7.5.1

The angle that the positive real axis makes with the ray from the origin to a complex number is called the argument of that complex number. It is written $\theta$ .

Learn it:Definition 44.1.17 Section 44.1b Polar Form, De Moivre and Roots of Unity

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Theorem 7.5.2 · Argument of a Complex Number

The argument $\theta$ of a complex number $a + bi$ satisfies

\tan\theta = \frac{b}{a}.

$\tan\theta = \frac ba$ does not pin down the quadrant: $-1 + i$ and $1 - i$ both have $\frac ba = -1$ , but their arguments are $135^\circ$ and $-45^\circ$ . Check which quadrant $(a, b)$ is in.

Learn it:Theorem 44.1.18 · Argument of a Complex Number

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Definition 7.5.3

The distance between $0$ and a complex number is also called the modulus of that complex number, written $r$ .

Learn it:Definition 44.1.19

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Theorem 7.5.4 · Modulus of a Complex Number

The modulus $r$ of a complex number $a + bi$ is

r = |a + bi| = \sqrt{a^2 + b^2}.

Learn it:Theorem 44.1.20 · Modulus of a Complex Number

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Definition 7.5.5

Polar form is another way to write a complex number, using its modulus $r$ and argument $\theta$ .

Learn it:Definition 44.1.21

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Theorem 7.5.6 · Polar Form

z = a + bi = r(\cos\theta + i\sin\theta) = r\operatorname{cis}\theta

Learn it:Theorem 44.1.22 · Polar Form Section 44.1b Polar Form, De Moivre and Roots of Unity

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Remark 7.5.7

$\operatorname{cis}\theta$ is just short for $\cos\theta + i\sin\theta$ .

Learn it:Remark 44.1.23

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Remark 7.5.8

Trigonometric ratios tell us that $\cos\theta = \frac ar$ and $\sin\theta = \frac br$ , which we can rearrange to $r\cos\theta = a$ and $r\sin\theta = b$ . Plugging these in gives the polar form.

Learn it:Remark 44.1.24

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Remark 7.5.9

$\cos\theta + i\sin\theta$ can also be written as $\operatorname{cis}\theta$ .

Learn it:Remark 44.1.25

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Theorem 7.5.10 · Euler’s Formula

Euler's Formula says

\cos\theta + i\sin\theta = e^{i\theta},

z = a + bi = r(\cos\theta + i\sin\theta) = re^{i\theta}.

(Here $\theta$ is in radians.)

Learn it:Theorem 44.1.26 · Euler's Formula

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Remark 7.5.11

Euler's Identity is a special case of Euler's Formula ( $\theta = \pi$ ):

e^{\pi i} = -1.

Learn it:Remark 44.1.27

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Theorem 7.5.12 · De Moivre’s Theorem

For a complex number $z = re^{i\theta}$ and an integer $n$ ,

z^n = (re^{i\theta})^n = r^n\left[\cos(n\theta) + i\sin(n\theta)\right].

Correction

The book says “a real number $n$ ”. For a non-integer $n$ , $z^n$ has several values (for $n = \frac12$ , $4 = 4e^{0i} = 4e^{2\pi i}$ gives both $2$ and $-2$ ), so the app states the theorem for integer $n$ .

Learn it:Theorem 44.1.31 · De Moivre's Theorem

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Remark 7.5.13

We can use this to evaluate expressions like

(3 + 2i)^8

much more easily: convert to polar form and apply De Moivre's Theorem.

Learn it:Remark 44.1.32

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Remark 7.5.14

De Moivre's Theorem is very useful when dealing with complex numbers and exponents.

Learn it:Remark 44.1.33

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Theorem 7.5.15 · Rotating a Point

To rotate a point $\theta$ radians counterclockwise about the origin, convert it to its complex number and multiply by $e^{i\theta}$ . Converting back to an ordered pair gives the rotated point.

Learn it:Theorem 44.1.30 · Rotating a Point Section 44.1c Geometry with Complex Numbers

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Concept 7.5.16

Complex numbers and their link to circles make many geometry problems easy, especially ones with polygons such as equilateral triangles or squares.

How to solve geometry problems with complex numbers:

Assign a complex number to one or more of the points.
To find the complex numbers for other points, multiply or divide by $e^{i\theta}$ .
Use the information you have to solve for what the problem asks.

Learn it:Concept 44.1.34 · Geometry with Complex Numbers Section 44.1c Geometry with Complex Numbers

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Remark 7.5.17

We can also view algebraic complex-number problems geometrically.

Learn it:Remark 44.1.35

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7.6 Roots of Unity

Definition 7.6.1

The roots of unity are the complex solutions of $x^n = 1$ for a positive integer $n$ . There are always $n$ solutions of $x^n = 1$ .

Learn it:Definition 44.1.28

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Theorem 7.6.2 · Roots of Unity

The $n$ th roots of unity are

e^{2k\pi i/n}\quad\text{for } k \in \{1, 2, \dots, n\}.

( $k = n$ gives $1$ ; the same set as $k = 0, 1, \dots, n - 1$ .)

Learn it:Theorem 44.1.29 · Roots of Unity

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7.7 Complex Numbers for Trigonometry

Theorem 7.7.1 · Sin and Cos values in terms of complex numbers

\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}

\phantom{\sin\theta} = \frac{e^{i\theta} + e^{i(\pi-\theta)}}{2i}

\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}

\tan\theta = -i\cdot\frac{e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}}

Correction

The book writes $e^{i(180-\theta)}$ , putting degrees into a radian exponent; it means $e^{i(\pi-\theta)} = -e^{-i\theta}$ . Its tangent line, $\frac{e^{i\theta} + e^{i(180-\theta)}}{e^{i\theta} + e^{i(-\theta)}}\,i$ , has the wrong sign (it equals $-\tan\theta$ ): the factor is $-i = \frac1i$ , not $i$ .

Learn it:Theorem 44.3.1 · Sin and Cos values in terms of complex numbers Section 44.3 Complex Numbers in Trigonometry

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Remark 7.7.2

By using these substitutions, we can bash out the value of trigonometric expressions without the clever manipulation of trigonometric identities that would otherwise be needed.

Learn it:Remark 44.3.2

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