Advanced Topics · Chapter 44

Complex Numbers

Put a second axis through the number line and arithmetic turns into geometry: multiplying rotates, De Moivre powers spiral, and roots of unity make regular polygons.

What's this chapter about?

44.1a The Algebra of i
44.1b Polar Form, De Moivre and Roots of Unity
44.1c Geometry with Complex Numbers
44.2 Complex Numbers in Polynomials
44.3 Complex Numbers in Trigonometry
44.4 Practice Problems

You'll need

open on YouTube

44.1a0%

The Algebra of i

In this section you will be able to add, multiply and divide complex numbers, use powers of i, and work with conjugates and moduli.

What number squares to −1? No real number does. So invent one, call it i, and put it on a second axis. Now arithmetic becomes geometry.

A complex number $a + bi$ is just the point $(a, b)$ . You add complex numbers like vectors. You multiply them like binomials, then replace $i^2$ with $-1$ .

Look: the conjugate $\bar z$ is the mirror image of $z$ in the real axis. And $z \bar z = |z|^2$ is always real. That one fact clears $i$ out of any denominator.

Exploreplay first, formulas later

Try this: Step n from 1 to 8 and watch i, i², i³, … go round. Every 4 steps you're back at 1. So what is i to the power 2025?
Try this: Press Conjugate and drag z. Where is z̄? Now press z · z̄: why does the product always land on the real axis?
Try this: Press z · w and drag w around the unit circle. What happens to the length of z · w? What happens to its angle?

Name it

Definition 44.1.1

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 $z$ .

Definition 44.1.2

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

Theorem 44.1.4 · Adding Complex Numbers

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

Theorem 44.1.5 · Subtracting Complex Numbers

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

Theorem 44.1.6 · Multiplying Complex Numbers

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

Definition 44.1.7 · Real and Imaginary Parts

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

Definition 44.1.9

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

Theorem 44.1.10 · Finding Conjugates

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

Definition 44.1.11 · Complex Plane

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

Definition 44.1.12

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

Theorem 44.1.13 · Magnitude of a Complex Number

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

Theorem 44.1.14 · Multiplying Complex Numbers with their Conjugates

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

Theorem 44.1.15

For a complex number $z$ ,

z\cdot\bar z = |z|^2.

Watch itpredict, then reveal

Example 44.2

AIME

The complex numbers $z$ and $w$ satisfy the system

z + \frac{20i}{w} = 5 + i,\qquad w + \frac{12i}{z} = -4 + 10i.

Find the smallest possible value of $|zw|^2$ .

1
We want $zw$ , so multiply the two equations together. What does the left side $\left(z + \frac{20i}{w}\right)\left(w + \frac{12i}{z}\right)$ expand to?

open on YouTube

Example 44.3

AIME

There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that

\frac{z}{z + n} = 4i.

Find $n$ .

1
Clear the fraction. What linear equation does $z$ satisfy?

open on YouTube

Try it

Problem 44.4.10(additional)

AMC 12

For complex numbers $u = a + bi$ and $v = c + di$ (where $i = \sqrt{-1}$ ), define the binary operation

u \otimes v = ac + bdi.

Suppose $z$ is a complex number such that $z \otimes z = z^2 + 40$ . What is $|z|$ ?

Problem 44.4.13(additional)

AIME

Find $c$ if $a$ , $b$ , and $c$ are positive integers which satisfy $c = (a + bi)^3 - 107i$ , where $i^2 = -1$ .

Problem 44.4.14(additional)

AMC 12

For what value of $n$ is $i + 2i^2 + 3i^3 + \dots + ni^n = 48 + 49i$ ? Note: here $i = \sqrt{-1}$ .

Problem 44.4.19(additional)

AIME

The complex number $z$ is equal to $9 + bi$ , where $b$ is a positive real number and $i^2 = -1$ . Given that the imaginary parts of $z^2$ and $z^3$ are the same, what is $b$ equal to?

Contest cornerreal contest problems

Problem 44.4.4

AMC 12

Let $z$ be a complex number satisfying $12|z|^2 = 2|z + 2|^2 + |z^2 + 1|^2 + 31$ . What is the value of $z + \frac6z$ ?

Problem 44.4.18(additional)

AMC 12

A function $f$ is defined by $f(z) = (4 + i)z^2 + \alpha z + \gamma$ for all complex numbers $z$ , where $\alpha$ and $\gamma$ are complex numbers and $i^2 = -1$ . Suppose that $f(1)$ and $f(i)$ are both real. What is the smallest possible value of $|\alpha| + |\gamma|$ ?

Checkpoint for section 44.1a

Get 2 of 3 right to master this section. One try each, no hints.

44.1b0%

Polar Form, De Moivre and Roots of Unity

In this section you will be able to write complex numbers in polar form, raise them to powers with De Moivre, and use the roots of unity.

What is $(\sqrt3 + i)^8$ ? Multiplying it out eight times is painful. In polar form it takes one line.

Every complex number is a length $r$ and an angle $\theta$ : $z = r$ cis $\theta$ = re^(iθ). Multiplying two complex numbers multiplies the lengths and adds the angles. So $z^n$ has length $r^n$ and angle $n\theta$ . That is De Moivre’s Theorem.

Look: the solutions of $z^n = 1$ all have length 1, and their angles are multiples of $360^\circ / n$ . They are the corners of a regular polygon. Ex 44.1 is printed earlier in the book, but it needs polar form, so it lives here.

Exploreplay first, formulas later

Try this: Drag z into each quadrant. Does tan θ = b/a tell you the angle? Compare −2 + 2i with 2 − 2i.
Try this: Press Powers and drag z to 1 + i. Multiply by it eight times: the lengths double, the angles add 45° each time.
Try this: Press Rotate. Multiplying by cis 60° turns every point 60° about the center. Change the angle and predict where w lands.

Name it

Definition 44.1.17

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 $\theta$ .

Theorem 44.1.18 · Argument of a Complex Number

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

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

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

Definition 44.1.19

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

Theorem 44.1.20 · Modulus of a Complex Number

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

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

Definition 44.1.21

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

Theorem 44.1.22 · Polar Form

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

Remark 44.1.23

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

Remark 44.1.25

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

Theorem 44.1.26 · 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.)

Definition 44.1.28

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

Theorem 44.1.29 · Roots of Unity

The $n$ th roots of unity are

e^{2k\pi i/n}\quad\text{for } k \in \{0, 1, 2, \dots, n - 1\}.

They are the corners of a regular $n$ -gon inscribed in the unit circle, with one corner at $1$ .

Theorem 44.1.30 · 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.

Theorem 44.1.31 · 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].

(The book says “a real number $n$ ”; for non-integer powers, $z^n$ has several values, so the formula needs integer $n$ .)

Remark 44.1.33

De Moivre's Theorem is very useful whenever complex numbers meet exponents.

Watch itpredict, then reveal

Example 44.1

AIME

Let $N$ be the number of complex numbers $z$ with the properties that $|z| = 1$ and $z^{6!} - z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ .

1
With $|z| = 1$ , write $z = e^{i\theta}$ . Since $6! = 720$ and $5! = 120$ , what are $z^{6!}$ and $z^{5!}$ ?

open on YouTube

Example 44.4

AMC 12

How many integers $n \ge 2$ are there such that whenever $z_1, z_2, \dots, z_n$ are complex numbers such that

|z_1| = |z_2| = \dots = |z_n| = 1 \quad\text{and}\quad z_1 + z_2 + \dots + z_n = 0,

then the numbers $z_1, z_2, \dots, z_n$ are equally spaced on the unit circle in the complex plane?

1
Start with $n = 2$ . If $z_1 + z_2 = 0$ , where is $z_2$ ?

open on YouTube

Example 44.5

AIME

The complex numbers $z$ and $w$ satisfy $z^{13} = w$ , $w^{11} = z$ , and the imaginary part of $z$ is $\sin\frac{m\pi}{n}$ , for relatively prime positive integers $m$ and $n$ with $m < n$ . Find $n$ .

1
Substitute $w = z^{13}$ into $w^{11} = z$ . What equation does $z$ satisfy?

open on YouTube

Example 44.6

AMC 12

Let

z = \frac{1 + i}{\sqrt2}.

What is

\left(z^{1^2} + z^{2^2} + z^{3^2} + \dots + z^{12^2}\right)\cdot\left(\frac{1}{z^{1^2}} + \frac{1}{z^{2^2}} + \frac{1}{z^{3^2}} + \dots + \frac{1}{z^{12^2}}\right)?

1
Write $z$ in polar form. What is its argument, in degrees?

Predict first:
Your prediction

open on YouTube

Try it

Problem 44.4.8(additional)

AMC 12

There are $24$ different complex numbers $z$ such that $z^{24} = 1$ . For how many of these is $z^6$ a real number?

Problem 44.4.9(additional)

AMC 12

What is the sum of the roots of $z^{12} = 64$ that have a positive real part?

Contest cornerreal contest problems

Problem 44.4.1

AMC 12

Of the following complex numbers $z$ , which one has the property that $z^5$ has the greatest real part?

Problem 44.4.2

AMC 12

How many complex numbers satisfy the equation $z^5 = \bar z$ , where $\bar z$ is the conjugate of the complex number $z$ ?

Problem 44.4.24(additional)

AIME

For how many positive integers $n$ less than or equal to $1000$ is

(\sin t + i\cos t)^n = \sin nt + i\cos nt

true for all real $t$ ?

Problem 44.4.25(additional)

AMC 12

A sequence $(a_1, b_1)$ , $(a_2, b_2)$ , $(a_3, b_3)$ , $\dots$ of points in the coordinate plane satisfies

(a_{n+1}, b_{n+1}) = (\sqrt3 a_n - b_n,\ \sqrt3 b_n + a_n)\quad\text{for } n = 1, 2, 3, \dots.

Suppose that $(a_{100}, b_{100}) = (2, 4)$ . What is $a_1 + b_1$ ?

Problem 44.4.30(additional)

AIME

There are $2n$ complex numbers that satisfy both $z^{28} - z^8 - 1 = 0$ and $|z| = 1$ . These numbers have the form $z_m = \cos\theta_m + i\sin\theta_m$ , where $0 \le \theta_1 < \theta_2 < \dots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_2 + \theta_4 + \dots + \theta_{2n}$ .

Checkpoint for section 44.1b

Get 2 of 3 right to master this section. One try each, no hints.

44.1c0%

Geometry with Complex Numbers

In this section you will be able to solve geometry problems by rotating points with multiplication by cis θ.

The points $(0, 0)$ , $(a, 11)$ and $(b, 37)$ make an equilateral triangle. Coordinates alone give messy equations. One rotation by $60^\circ$ gives the answer.

To rotate a point $z$ by $\theta$ about a center $c$ : shift so $c$ is at $0$ , multiply by cis $\theta$ , then shift back. Squares use $\theta = 90^\circ$ (multiply by $i$ ). Equilateral triangles use $\theta = 60^\circ$ .

Look: a right angle at a point $P$ means the quotient of the two side vectors from $P$ is purely imaginary.

Exploreplay first, formulas later

Try this: Rotate z = 3 + i by 90° about the point 1. Predict the answer first: shift, multiply by i, shift back.
Try this: Set the angle to 60°. Where must w be so that the center, z and w form an equilateral triangle?
Try this: In Example 44.7, drag a + 11i along its line until the rotated point lands on the dashed line Im = 37.

Name it

Concept 44.1.34 · Geometry with Complex Numbers

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 $e^{i\theta}$ (a rotation).
Use the information you have to solve for what the problem asks.

Remark 44.1.35

We can also view algebraic complex-number problems geometrically: $|z - w|$ is a distance, multiplying by $e^{i\theta}$ is a rotation, and $|z| = r$ is a circle.

Watch itpredict, then reveal

Example 44.7

AIME

The points $(0, 0)$ , $(a, 11)$ , and $(b, 37)$ are the vertices of an equilateral triangle. Find the value of $ab$ .

1
Think of the points as $0$ , $a + 11i$ and $b + 37i$ . How do you get from $a + 11i$ to $b + 37i$ ?

open on YouTube

Example 44.8

AIME

Given $f(z) = z^2 - 19z$ , there are complex numbers $z$ with the property that $z$ , $f(z)$ , and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$ . There are positive integers $m$ and $n$ such that one such value of $z$ is $m + \sqrt n + 11i$ . Find $m + n$ .

1
The right angle is at $f(z)$ . What must be true of $\dfrac{f(f(z)) - f(z)}{z - f(z)}$ ?

Predict first:
Click a choice or type its letter (A–E).

open on YouTube

Contest cornerreal contest problems

Problem 44.4.6

AMC 12

What is the greatest possible area of the triangle in the complex plane with vertices $2z$ , $(1 + i)z$ , and $(1 - i)z$ , where $z$ is a complex number satisfying $|4z - 2| = 1$ ?

Problem 44.4.7

AIME

Find the greatest possible real part of

(75 + 117i)z + \frac{96 + 144i}{z},

where $z$ is a complex number with $|z| = 4$ . Here $i = \sqrt{-1}$ .

Problem 44.4.28(additional)

AMC 12

The solutions to the equation $(z + 6)^8 = 81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A$ , $B$ , and $C$ . What is the least possible area of $\triangle ABC$ ?

Problem 44.4.29(additional)

AMC 12

Let $\mathcal R$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$ , where $z_1$ lies on the segment with endpoints $3$ and $4i$ , and $z_2$ has magnitude at most $1$ . What integer is closest to the area of $\mathcal R$ ?

Checkpoint for section 44.1c

Get 2 of 3 right to master this section. One try each, no hints.

44.20%

Complex Numbers in Polynomials

In this section you will be able to use complex roots of polynomials, conjugate pairs and roots of unity to evaluate polynomials.

A polynomial of degree 2018 looks hopeless. But plug in a sixth root of unity and every power collapses to one of six values.

Every polynomial of degree $n$ has exactly $n$ complex roots. When the coefficients are real, the non-real roots come in conjugate pairs $z$ and $\bar z$ , mirror images in the real axis.

Look: to evaluate $f(\omega)$ at a root of unity, reduce every exponent mod the order of $\omega$ first.

Exploreplay first, formulas later

Try this: Type z^2 + 1 and press Plot roots. Why are the roots mirror images in the real axis?
Try this: Now type z^2 + i. The coefficients are not all real: are the roots still mirror pairs?
Try this: Type x^10 + (13x - 1)^10 (Example 44.10). Ten roots, five mirror pairs. Why can you pair each r with r̄?

Name it

Definition 44.2.1

A polynomial of degree $n$ has $n$ roots (counted with multiplicity), and these roots may be complex. For quadratics, a negative discriminant means two complex roots. (The book says “binomials”; it means quadratics.) If the coefficients are real, non-real roots come in conjugate pairs $z, \bar z$ .

Watch itpredict, then reveal

Example 44.9

AIME

The polynomial $f(z) = az^{2018} + bz^{2017} + cz^{2016}$ has real coefficients not exceeding $2019$ , and $f\left(\frac{1 + \sqrt3 i}{2}\right) = 2015 + 2019\sqrt3 i$ . Find the remainder when $f(1)$ is divided by $1000$ .

1
Let $\omega = \frac{1 + \sqrt3 i}{2}$ . Write it in polar form. Which small power of $\omega$ is $1$ ?

Predict first:
Your prediction

open on YouTube

Example 44.10

AMC 12

The equation

x^{10} + (13x - 1)^{10} = 0

has $10$ complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5}$ , where the bar denotes complex conjugation. Find the value of

\frac{1}{r_1\overline{r_1}} + \frac{1}{r_2\overline{r_2}} + \frac{1}{r_3\overline{r_3}} + \frac{1}{r_4\overline{r_4}} + \frac{1}{r_5\overline{r_5}}.

0.08 + 0.005i; 0.073 + 0.004i; 0.072 − 0.002i; 0.076 − 0.006i; 0.083 − 0.002i; 0.083 + 0.002i; 0.076 + 0.006i; 0.072 + 0.002i; 0.073 − 0.004i; 0.08 − 0.005i

Real coefficients: the roots come in mirror pairs (z and z̄).

1
$\frac{1}{r\bar r} = \frac{1}{|r|^2} = \left|\frac1r\right|^2$ . So switch to $y = \frac1x$ : divide the equation by $x^{10}$ . What do you get?

open on YouTube

Try it

Problem 44.4.11(additional)

AMC 12

The polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0$ . What is $a + b + c + d$ ?

Contest cornerreal contest problems

Problem 44.4.3

AMC 12

Let $Q(z)$ and $R(z)$ be the unique polynomials such that

z^{2021} + 1 = (z^2 + z + 1)Q(z) + R(z)

and the degree of $R$ is less than $2$ . What is $R(z)$ ?

Problem 44.4.5

AMC 12

The polynomial

(z + i)(z + 2i)(z + 3i) + 10

has three roots in the complex plane, where $i = \sqrt{-1}$ . What is the area of the triangle formed by these three roots?

Problem 44.4.15(additional)

AIME

There are real numbers $a$ , $b$ , $c$ , and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d$ . These two polynomials share a complex root $m + \sqrt n\cdot i$ , where $m$ and $n$ are positive integers and $i = \sqrt{-1}$ . Find $m + n$ .

Problem 44.4.32(additional)

AIME

Let $P$ be the product of the roots of $z^6 + z^4 + z^3 + z^2 + 1 = 0$ that have a positive imaginary part, and suppose that $P = r(\cos\theta^\circ + i\sin\theta^\circ)$ , where $0 < r$ and $0 \le \theta < 360$ . Find $\theta$ .

Checkpoint for section 44.2

Get 2 of 3 right to master this section. One try each, no hints.

44.30%

Complex Numbers in Trigonometry

In this section you will be able to turn sines and cosines into exponentials and use roots of unity to evaluate trig sums and products.

What is cos(2π/7) + cos(4π/7) + cos(6π/7)? No identity on your sheet gives it directly. The 7th roots of unity do.

Add e^(iθ) and its mirror image e^(−iθ): the imaginary parts cancel and you get $2\cos\theta$ . Subtract them and you get $2i\sin\theta$ . So every trig expression can become an algebra problem about points on the unit circle.

Look: the book prints these formulas with a degree slip and a sign error in the tangent; the card below has the fixed version.

Exploreplay first, formulas later

Try this: Change θ. The sum of the two unit vectors always lies on the real axis. Why is it 2cos θ?
Try this: The difference always lies on the imaginary axis. Why is it 2i sin θ?
Try this: Example 44.11 connects cos(2π/7) with the 7th roots of unity. Their real parts add up to 0: check it on the heptagon below.

Name it

Theorem 44.3.1 · Sin and Cos values in terms of complex numbers

\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i},\qquad \cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2},\qquad \tan\theta = -i\cdot\frac{e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}}.

Fixed from the book. The book writes $e^{i(180-\theta)}$ , mixing degrees into a radian exponent; it means $e^{i(\pi-\theta)} = -e^{-i\theta}$ . Its tangent formula also has the wrong sign: it gives $-\tan\theta$ .

Watch itpredict, then reveal

Example 44.11

AMC 12

Suppose that the roots of the polynomial $P(x) = x^3 + ax^2 + bx + c$ are $\cos\frac{2\pi}{7}$ , $\cos\frac{4\pi}{7}$ , and $\cos\frac{6\pi}{7}$ , where angles are in radians. What is $abc$ ?

1
Let $\omega = e^{2\pi i/7}$ . What is $1 + \omega + \omega^2 + \dots + \omega^6$ ?

Predict first:
Your prediction

open on YouTube

Example 44.12

PUMaC

The value of $\sin(2^\circ)\sin(4^\circ)\cdots\sin(90^\circ)$ is $\frac{p\sqrt5}{2^{50}}$ . Find $p$ .

1
Let $\omega = e^{2\pi i/90}$ . The numbers $\omega, \omega^2, \dots, \omega^{89}$ are the roots of $1 + x + \dots + x^{89}$ . Put $x = 1$ : what is $\prod_{k=1}^{89}(1 - \omega^k)$ ?

Predict first:
Your prediction

open on YouTube

Contest cornerreal contest problems

Problem 44.4.22(additional)

AIME

Given that $z$ is a complex number such that $z + \frac1z = 2\cos3^\circ$ , find the least integer that is greater than $z^{2000} + \frac{1}{z^{2000}}$ .

Problem 44.4.33(additional)

AMC 12

There is a unique sequence of integers $a_1, a_2, \dots, a_{2023}$ such that

\tan 2023x = \frac{a_1\tan x + a_3\tan^3 x + a_5\tan^5 x + \dots + a_{2023}\tan^{2023}x}{1 + a_2\tan^2 x + a_4\tan^4 x + \dots + a_{2022}\tan^{2022}x}

whenever $\tan 2023x$ is defined. What is $a_{2023}$ ?

Checkpoint for section 44.3

Get 2 of 3 right to master this section. One try each, no hints.

If $z = e^{i\theta}$ , which expression equals $z + \frac1z$ ?

Given $z + \frac1z = 2\cos20^\circ$ , find $z^9 + \frac{1}{z^9}$ .

Use the 5th roots of unity to find $\cos72^\circ + \cos144^\circ$ .

44.4

Practice Problems

In this section you will be able to mix every tool of the chapter on contest problems.

The rest of the Additional Problems, from AMC 12 warm-ups to an AIME 15. Some you have seen before in this chapter: try them again without the walkthrough.

Contest cornerreal contest problems

Problem 44.4.12(additional)

AMC 12

A function $f$ is defined by $f(z) = i\bar z$ , where $i = \sqrt{-1}$ and $\bar z$ is the complex conjugate of $z$ . How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$ ?

Problem 44.4.16(additional)

AMC 12

This problem also appears in the book as 44.4.9.

What is the sum of the roots of $z^{12} = 64$ that have a positive real part?

Problem 44.4.17(additional)

AIME

This problem also appears in the book as 44.5.

Problem 44.4.20(additional)

AMC 12

Let $F(z) = \frac{z + i}{z - i}$ for all complex numbers $z \ne i$ , and let $z_n = F(z_{n-1})$ for all positive integers $n$ . Given that $z_0 = \frac{1}{137} + i$ and $z_{2002} = a + bi$ , where $a$ and $b$ are real numbers, find $a + b$ .

Problem 44.4.21(additional)

AMC 12A

Consider the set of complex numbers $z$ satisfying $|1 + z + z^2| = 4$ . The maximum value of the imaginary part of $z$ can be written in the form $\frac{\sqrt m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?

Problem 44.4.23(additional)

AMC 12

A sequence of complex numbers $z_0, z_1, z_2, \dots$ is defined by the rule

z_{n+1} = \frac{iz_n}{\overline{z_n}},

where $\overline{z_n}$ is the complex conjugate of $z_n$ and $i^2 = -1$ . Suppose that $|z_0| = 1$ and $z_{2005} = 1$ . How many possible values are there for $z_0$ ?

Problem 44.4.26(additional)

AIME

This problem also appears in the book as 44.1.

Let $N$ be the number of complex numbers $z$ with the properties that $|z| = 1$ and $z^{6!} - z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ .

Problem 44.4.27(additional)

AIME

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$ . Let $\theta = \arg\left(\frac{w - z}{z}\right)$ . The maximum possible value of $\tan^2\theta$ can be written as $\frac pq$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ . (Note that $\arg(w)$ , for $w \ne 0$ , denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.)

Problem 44.4.31(additional)

AMC 12

Let $c$ be a real number, and let $z_1$ and $z_2$ be the two complex numbers satisfying the equation $z^2 - cz + 10 = 0$ . Points $z_1$ , $z_2$ , $\frac{1}{z_1}$ , and $\frac{1}{z_2}$ are the vertices of (convex) quadrilateral $\mathcal Q$ in the complex plane. When the area of $\mathcal Q$ obtains its maximum possible value, $c$ is closest to which of the following?

Problem 44.4.34(additional)

AMC 12Challenge

The solutions to the equations $z^2 = 4 + 4\sqrt{15}i$ and $z^2 = 2 + 2\sqrt3 i$ , where $i = \sqrt{-1}$ , form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q - r\sqrt s$ , where $p$ , $q$ , $r$ , and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p + q + r + s$ ?