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The Complex Plane (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 20 April 2026

Complex numbers & rectangular coordinates

What is a complex number?

A complex number is a number of the form $a + b i$ , where
- $a$ and $b$ are real numbers
- $i$ is the imaginary unit, defined by $i^{2} = - 1$
  - or equivalently, $i = \sqrt{- 1}$
The real number $a$ is called the real part of the complex number
The real number $b$ is called the imaginary part of the complex number
- E.g. the complex number $3 - 5 i$ has real part $3$ and imaginary part $- 5$
Every real number is also a complex number
- A real number $a$ can be written as $a + 0 i$
  - i.e. with real part $a$ and imaginary part $0$

Examiner Tips and Tricks

Familiarity with complex numbers is a prerequisite for the AP® Precalculus course.

How are complex numbers represented using rectangular coordinates?

A complex number can be represented as a point in a plane, called the complex plane
- The horizontal axis is the real axis (Re)
- The vertical axis is the imaginary axis (Im)
The complex number $a + b i$ is represented by the point with rectangular coordinates $(a, b)$
- The real part $a$ gives the horizontal coordinate
- The imaginary part $b$ gives the vertical coordinate
E.g. the complex number $4 + 2 i$ corresponds to the point $(4, 2)$ in the complex plane
- or going the other way, the point with rectangular coordinates $(- 3, - 3)$ in the complex plane corresponds to the complex number $- 3 - 3 i$
Note that a real number like $2 = 2 + 0 i$ is located on the real axis
- and a purely imaginary number like $i = 0 + i$ is located on the imaginary axis

Graph with complex numbers plotted as points: 4+2i at (4,2), -3-3i at (-3,-3), 2 at (2,0), and i at (0,1), with labels for each. — **Representing complex numbers using rectangular coordinates in the complex plane**

Complex numbers & polar coordinates

How are complex numbers represented using polar coordinates?

Since a complex number corresponds to a point in the complex plane, it can also be described using polar coordinates $(r, θ)$
- $r$ is the signed radius value
- $θ$ is the angle in standard position whose terminal ray includes the point
If a complex number has polar coordinates $(r, θ)$
- then its rectangular coordinates are $(r \cos θ, r \sin θ)$
  - This follows directly from the standard conversion formulas
    - $x = r \cos θ$ and $y = r \sin θ$
- Substituting into the $a + b i$ form gives the polar form of the complex number:
  - $(r \cos θ) + i (r \sin θ)$

E.g. the complex number with polar coordinates $(4, \frac{π}{3})$ can be written as $(4 \cos \frac{π}{3}) + i (4 \sin \frac{π}{3})$
- which simplifies to $2 + 2 \sqrt{3} i$ in the $a + b i$ form
  - i.e. because $\cos \frac{π}{3} = \frac{1}{2}$ and $\sin \frac{π}{3} = \frac{\sqrt{3}}{2}$
Going the other way, to convert a complex number $a + b i$ to polar form
- use the same rectangular-to-polar conversion formulas from the polar coordinate system
  - $r = \sqrt{a^{2} + b^{2}}$
  - $θ = \arctan (\frac{b}{a})$ for $a > 0$
  - $θ = \arctan (\frac{b}{a}) + π$ for $a < 0$
E.g. to convert $- \sqrt{3} + i$ to polar form
- $r = \sqrt{(- \sqrt{3})^{2} + 1^{2}} = \sqrt{4} = 2$
- Since $a = - \sqrt{3} < 0$
  - $θ = \arctan (\frac{1}{- \sqrt{3}}) + π = - \frac{π}{6} + π = \frac{5 π}{6}$
- So the polar form is $(2 \cos \frac{5 π}{6}) + i (2 \sin \frac{5 π}{6})$
  - with polar coordinates $(2, \frac{5 π}{6})$

Polar grid with two marked complex points. The first is at radius 2, angle 5π/6; the second at radius 4, angle π/3. Both are annotated with equivalent expressions in (rcosθ)+i(rsinθ) form. — **Representing complex numbers using polar coordinates**

Examiner Tips and Tricks

The polar form of a complex number is written with two factors of $r$ , one multiplying the cosine and one multiplying the sine.

A common error is to leave the $r$ out and write $\cos θ + i \sin θ$ , which only works when $r = 1$

When converting from $a + b i$ to polar form, treat it exactly like converting the rectangular coordinates $(a, b)$ to polar coordinates.

The only difference is in how the final answer is written

Worked Example

A complex number is represented by a point in the complex plane. The complex number has the rectangular coordinates $(- 1, \sqrt{3})$ . Which of the following is one way to express the complex number using its polar coordinates $(r, θ)$ ?

(A) $(2 \cos \frac{2 π}{3}) + i (2 \sin \frac{2 π}{3})$

(B) $(\cos \frac{2 π}{3}) + i (\sin \frac{2 π}{3})$

(D) $(\cos (- \frac{π}{3})) + i (\sin (- \frac{π}{3}))$

Answer:

Start by finding $r$ from the rectangular coordinates

$r = \sqrt{(- 1)^{2} + (\sqrt{3})^{2}} = \sqrt{1 + 3} = \sqrt{4} = 2$

Since the polar form of a complex number is $(r \cos θ) + i (r \sin θ)$ , this rules out options (B) and (C)

The point $(- 1, \sqrt{3})$ lies in Quadrant II (negative real part, positive imaginary part)

so use the $a < 0$ version of the angle formula:

$θ = \arctan (\frac{\sqrt{3}}{- 1}) + π = - \frac{π}{3} + π = \frac{2 π}{3}$

The polar form is therefore

$(2 \cos \frac{2 π}{3}) + i (2 \sin \frac{2 π}{3})$

which is option (A)

Note that option (C) is what you would get if you forgot to use the version of the angle formula for $a < 0$

I.e. if you used $\arctan (\frac{b}{a})$ directly without adjusting by $+ π$ because the real part is negative
$(2 \cos (- \frac{π}{3})) + i (2 \sin (- \frac{π}{3}))$ is a number in Quadrant IV (positive real part, negative imaginary part) instead of Quadrant II

(A) $(2 \cos \frac{2 π}{3}) + i (2 \sin \frac{2 π}{3})$

Examiner Tips and Tricks

Note that the worked example question says "Which of the following is one way to express the complex number...".

That is because, as with regular polar coordinates, there are an infinite number of ways to represent a complex number in polar form, i.e. by adding or subtracting an integer multiple of $2 π$ to the angle $θ$ .

E.g. $\frac{2 π}{3} + 2 π = \frac{8 π}{3}$ , and $\frac{2 π}{3} - 2 π = - \frac{4 π}{3}$
So the complex number $(2 \cos \frac{2 π}{3}) + i (2 \sin \frac{2 π}{3})$ could equivalently be written as $(2 \cos \frac{8 π}{3}) + i (2 \sin \frac{8 π}{3})$ or $(2 \cos (- \frac{4 π}{3})) + i (2 \sin (- \frac{4 π}{3}))$

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The Complex Plane (College Board AP® Precalculus): Study Guide

Complex numbers & rectangular coordinates

What is a complex number?

Examiner Tips and Tricks

How are complex numbers represented using rectangular coordinates?

Complex numbers & polar coordinates

How are complex numbers represented using polar coordinates?

Examiner Tips and Tricks

Worked Example

Examiner Tips and Tricks

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis