Problem type

Polar form of a complex number

What it solves: Convert a complex number a + bi to polar form: find its modulus (length) and its principal argument (direction), exactly where the angle is a special one, and check the result by converting it back.
Subject: Complex Numbers
Topic: Length and angle
Also called: modulus and argument, polar form, argument of a complex number, Argand
Free example: Read the complete worked example

Femtobytes

Polar form of -sqrt(3) - i: modulus and argument, step by step

femtobytes.com/solve/complex-polar-form/-1_3_-1_1

The problem

Write this complex number in polar form.

z = -\sqrt{3} - i

The result

z = -\sqrt{3} - i = 2 \left( \cos \left( -\frac{5 \pi}{6} \right) + i \sin \left( -\frac{5 \pi}{6} \right) \right)

The modulus is $r = 2$ and the principal argument is $\theta = -\frac{5 \pi}{6}$ .

checked 9 lines of this solution were evaluated with exact arithmetic before it was shown to you. A solution with a false line is never served.

Picture 1 of 2Step 1: See where the number lives

The number z drawn as an arrow from the origin. Its length is the modulus r and the angle it makes with the positive real axis is the argument, written with the Greek letter theta. Here

r = 2

and

\theta = -\frac{5 \pi}{6}

See where the number lives

Why this step

A complex number is two real numbers travelling together, so it cannot sit on a number line; it needs a plane. Rectangular form $a + bi$ says "go $a$ across and $b$ up". Polar form describes the SAME point as "face this direction, walk this far". The reason to bother: when complex numbers are multiplied, their lengths multiply and their directions add. Rectangular form hides that completely; polar form makes it visible.

Here $z = -\sqrt{3} - i$ , so the real part is $a = -\sqrt{3}$ and the imaginary part is $b = -1$ . The symbol $i$ is the imaginary unit, the number with $i^2 = -1$ . On the plane, $z$ is the point $(a, b)$ : $a$ is read on the horizontal (real) axis and $b$ on the vertical (imaginary) axis.

Both parts are negative, so $z$ lies in the third quadrant: to the left and below. Its principal angle will be between $-\pi$ and $-\frac{\pi}{2}$ .

The number z drawn as an arrow from the origin. Its length is the modulus r and the angle it makes with the positive real axis is the argument, written with the Greek letter theta. Here

r = 2

and

\theta = -\frac{5 \pi}{6}

Find the modulus: how far z is from the origin

Why this step

The arrow to $z$ is the long side of a right-angled triangle whose other sides are $\left| a \right|$ across and $\left| b \right|$ up. Pythagoras gives its length. We square $a$ and $b$ first, which also throws away their signs: a distance cannot care whether we went left or right.

\displaystyle r

=

\displaystyle \sqrt{\left( -\sqrt{3} \right)^{2} + \left( -1 \right)^{2}}

The modulus

r = \left| z \right| = \sqrt{a^2 + b^2}

, by Pythagoras. checked

=

\displaystyle \sqrt{3 + 1}

Square each part. Squaring a square root undoes it, for example

\left( \sqrt{3} \right)^2 = 3

, and the square of a negative number is positive. checked

=

\displaystyle \sqrt{4}

Add. checked

=

\displaystyle 2

4

is a perfect square, so the root comes out exactly. checked

The same arrow, with the right-angled triangle it is the long side of. The side across is the real part and the side up is the imaginary part, so the modulus is the length Pythagoras gives from those two.

Polar form of a complex number

See where the number lives

Find the modulus: how far z is from the origin

Find the argument: which way z points

Write the polar form, and check it by turning it back

Every other angle that names the same direction