Polar Form, de Moivre, and Roots

Overview

This branch develops polar and exponential form, where multiplication becomes scaling plus rotation. It is the main reference for de Moivre’s theorem, powers, roots, and the equal-spacing structure of roots in the Argand diagram.

Polar and Exponential Form

Complex numbers can be written in different forms depending on what we want to emphasise.

The Cartesian form

$$z=x+iy$$

emphasises horizontal and vertical components in the Argand diagram.

Polar and exponential forms emphasise something different:

• the distance of the point from the origin;
• the direction of the point from the positive real axis.

These forms are especially useful for understanding multiplication, division, powers, and roots geometrically.

From coordinates to size and direction

Let

$$z=x+iy$$

be a non-zero complex number.

In the Argand diagram, $z$ is represented by the point

$$(x,y).$$

Let

$$r=|z|$$

and

$$\theta =\arg z.$$

Here, $r$ is the distance from the origin to the point $(x,y)$, and $\theta$ is the direction of the point measured from the positive real axis.

Caption: Polar form records the same complex number by its distance from the origin and its direction from the positive real axis.

From right-triangle geometry,

$$x=r\cos \theta$$

and

$$y=r\sin \theta .$$

Substituting these into

$$z=x+iy$$

gives

$$z=r\cos \theta +ir\sin \theta .$$

So

$$z=r(\cos \theta +i\sin \theta ).$$

This is the polar form of a complex number.

Polar form

If $z$ has modulus $r$ and argument $\theta$, then

$$z=r(\cos \theta +i\sin \theta ).$$

In this form:

• $r$ gives the size or distance from the origin;
• $\theta$ gives the direction or argument.

So polar form separates a complex number into

$$\text{size}+\text{direction}.$$

For a non-zero complex number,

$$r>0.$$

The argument $\theta$ is not unique. If $\theta$ is an argument of $z$, then

$$\theta +2k\pi$$

is also an argument of $z$ for any integer $k$.

Usually, we use the principal argument, chosen in the range

$$-\pi <\theta \le \pi .$$

The number $0$ has modulus $0$, but its argument is undefined. Therefore, polar form with a specified argument is normally used only for non-zero complex numbers.

Exponential form

Euler’s formula states that

$$e^{i\theta }=\cos \theta +i\sin \theta .$$

Using this formula, the polar form

$$z=r(\cos \theta +i\sin \theta )$$

can be written more compactly as

$$z=r{e}^{i\theta }.$$

This is the exponential form of a complex number.

Exponential form is not a completely separate idea. It is a compact way of writing polar form.

The expression

$$e^{i\theta }$$

represents a direction in the Argand diagram, while the factor $r$ gives the distance from the origin.

So

$$r{e}^{i\theta }$$

means:

$$\text{distance }r\text{ from the origin, in direction }\theta .$$

Example: writing $1+i$ in three forms

Consider

$$z=1+i.$$

This is the Cartesian form.

To write it in polar form, first find the modulus:

$$r=|z|=\sqrt{1^2+1^2}=\sqrt{2}.$$

Next, find the argument. The point $(1,1)$ lies in the first quadrant, so

$$\theta =\frac{\pi }{4}.$$

Therefore,

$$1+i=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right).$$

This is the polar form.

Using Euler’s formula, we can also write

$$1+i=\sqrt{2}{e}^{i\pi /4}.$$

This is the exponential form.

So the same complex number can be written as

$$1+i,$$ $$\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right),$$

or

$$\sqrt{2}{e}^{i\pi /4}.$$

These are three different forms of the same number.

Converting from Cartesian form to polar or exponential form

To convert

$$z=x+iy$$

to polar or exponential form:

1. Find the modulus:

$$r=\sqrt{x^2+y^2}.$$

2. Find the argument $\theta$ using the correct quadrant.

3. Write either

$$z=r(\cos \theta +i\sin \theta )$$

or

$$z=r{e}^{i\theta }.$$

The quadrant matters. The value of

$${\tan }^{-1}\left(\frac{y}{x}\right)$$

alone may give the wrong argument if the point is not in the first quadrant.

Converting from polar or exponential form to Cartesian form

To convert

$$z=r(\cos \theta +i\sin \theta )$$

to Cartesian form, expand:

$$z=r\cos \theta +ir\sin \theta .$$

So the Cartesian form is

$$z=x+iy,$$

where

$$x=r\cos \theta$$

and

$$y=r\sin \theta .$$

For exponential form,

$$z=r{e}^{i\theta },$$

first use Euler’s formula:

$$e^{i\theta }=\cos \theta +i\sin \theta .$$

Then

$$z=r(\cos \theta +i\sin \theta ),$$

and the same expansion gives the Cartesian form.

Why these forms matter

Different forms are useful for different purposes.

Cartesian form

$$x+iy$$

is best when we need to identify real and imaginary parts, compare complex numbers, or perform basic algebra.

Polar form

$$r(\cos \theta +i\sin \theta )$$

is best when we want to think geometrically in terms of size and direction.

Exponential form

$$r{e}^{i\theta }$$

is usually the most efficient form for multiplication, division, powers, and roots.

For example,

$$r_1{e}^{i{\theta }_1}\cdot r_2{e}^{i{\theta }_2}=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}.$$

This shows the geometric meaning of multiplication:

• multiply the moduli;
• add the arguments.

In words, multiplication scales distance from the origin and rotates direction.

This is why polar and exponential forms become powerful when studying geometric transformations, de Moivre’s theorem, and roots of complex numbers.

Multiplication, Division, and Geometric Effects

Polar and exponential forms make multiplication and division much easier to understand geometrically.

Suppose

$$z_1=r_1{e}^{i{\theta }_1}$$

and

$$z_2=r_2{e}^{i{\theta }_2},$$

where $r_1,r_2>0$.

Then

$$z_1{z}_2=r_1{e}^{i{\theta }_1}\cdot r_2{e}^{i{\theta }_2}.$$

Using the laws of indices,

$$e^{i{\theta }_1}{e}^{i{\theta }_2}=e^{i({\theta }_1+{\theta }_2)}.$$

Therefore,

$$z_1{z}_2=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}.$$

So multiplication has a simple geometric meaning:

• the moduli are multiplied;
• the arguments are added.

In other words, multiplying by a complex number changes both the size and the direction.

Division

Similarly,

$$\frac{z_1}{z_2}=\frac{r_1{e}^{i{\theta }_1}}{r_2{e}^{i{\theta }_2}}.$$

Using the laws of indices,

$$\frac{e^{i{\theta }_1}}{e^{i{\theta }_2}}=e^{i({\theta }_1-{\theta }_2)}.$$

So

$$\frac{z_1}{z_2}=\frac{r_1}{r_2}{e}^{i({\theta }_1-{\theta }_2)}.$$

Thus division has the geometric meaning:

• the moduli are divided;
• the arguments are subtracted.

This is why exponential form is usually the most efficient form for multiplication and division.

Multiplication by a positive real number

If $k>0$, then multiplying a complex number $z$ by $k$ gives

$$kz.$$

Geometrically, this scales the distance from the origin by a factor of $k$, but it does not change the direction.

So:

• if $k>1$, the point moves farther from the origin;
• if $0<k<1$, the point moves closer to the origin;
• the argument stays the same.

Thus multiplication by a positive real number is a scaling about the origin.

Multiplication by a unit complex number

A unit complex number has modulus $1$.

For example,

$$e^{i\varphi }$$

has modulus $1$.

If

$$z=r{e}^{i\theta },$$

then

$$z{e}^{i\varphi }=r{e}^{i\theta }{e}^{i\varphi }=r{e}^{i(\theta +\varphi )}.$$

The modulus is still $r$, but the argument has changed from $\theta$ to $\theta +\varphi$.

Therefore, multiplying by

$$e^{i\varphi }$$

rotates the point anticlockwise by angle $\varphi$ about the origin, without changing its distance from the origin.

If $\varphi$ is negative, the rotation is clockwise.

Special case: multiplication by $i$

The number $i$ can be written as

$$i=e^{i\pi /2}.$$

So multiplying by $i$ means multiplying by

$$e^{i\pi /2}.$$

If

$$z=r{e}^{i\theta },$$

then

$$iz=e^{i\pi /2}r{e}^{i\theta }.$$

Hence

$$iz=r{e}^{i(\theta +\pi /2)}.$$

The modulus is unchanged, while the argument increases by

$$\frac{\pi }{2}.$$

Therefore, multiplication by $i$ rotates a point anticlockwise by

$$\frac{\pi }{2}$$

about the origin.

For example, if

$$z=3+2i,$$

then

$$iz=i(3+2i)=3i+2i^2.$$

Since

$$i^2=-1,$$

we get

$$iz=-2+3i.$$

So the point

$$(3,2)$$

is sent to

$$(-2,3),$$

which is a $90^{\circ }$ anticlockwise rotation about the origin.

General geometric effect of multiplication

Multiplication by a general complex number combines two effects.

If

$$w=R{e}^{i\varphi },$$

then multiplying $z$ by $w$ gives

$$wz=R{e}^{i\varphi }\cdot r{e}^{i\theta }.$$

So

$$wz=Rr{e}^{i(\varphi +\theta )}.$$

This means:

• the modulus of $z$ is scaled by the factor $R$;
• the argument of $z$ is increased by $\varphi$.

Therefore, multiplication by a complex number is a combination of scaling and rotation about the origin.

In summary:

• multiplying by a positive real number scales only;
• multiplying by a unit complex number rotates only;
• multiplying by a general non-zero complex number scales and rotates.

This is the main geometric power of polar and exponential forms: multiplication is no longer just algebraic expansion. It becomes a clear transformation in the Argand diagram.

Caption: Multiplication by $i$ rotates a complex number by $90^{\circ }$ anticlockwise about the origin.

De Moivre’s Theorem and Powers

Polar and exponential forms make powers of complex numbers much easier to understand.

The key idea is:

$$\text{multiplication multiplies moduli and adds arguments.}$$

Therefore, repeated multiplication should raise the modulus to a power and multiply the argument by the same power.

Why powers are simple in polar form

Suppose

$$z=r(\cos \theta +i\sin \theta ).$$

Here:

• $r$ is the modulus of $z$;
• $\theta$ is an argument of $z$.

If we multiply $z$ by itself repeatedly, the modulus is multiplied repeatedly, and the argument is added repeatedly.

So for a positive integer $n$,

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

The modulus becomes

$$r^n,$$

and the argument becomes

$$n\theta .$$

Hence,

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

This is the general polar form of De Moivre’s theorem.

De Moivre’s theorem

In the special case where $r=1$, we obtain

$$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta ),$$

for positive integers $n$.

Equivalently, using exponential form,

$$z=r{e}^{i\theta }$$

gives

$$z^n=(r{e}^{i\theta }{)}^n.$$

So

$$(r{e}^{i\theta }{)}^n=r^n{e}^{in\theta }.$$

This is often the most compact form of De Moivre’s theorem.

Conceptual meaning

De Moivre’s theorem says that powers act separately on size and direction:

• the modulus changes from $r$ to $r^n$;
• the argument changes from $\theta$ to $n\theta$.

So powers do not need repeated expansion. They can be handled by changing the modulus and the argument.

Example: finding a power

Find

$$(1+i{)}^6.$$

First write $1+i$ in exponential form.

The modulus is

$$|1+i|=\sqrt{1^2+1^2}=\sqrt{2}.$$

The argument is

$$\frac{\pi }{4}.$$

So

$$1+i=\sqrt{2}{e}^{i\pi /4}.$$

Therefore,

$$(1+i{)}^6={\left(\sqrt{2}{e}^{i\pi /4}\right)}^6.$$

Using De Moivre’s theorem,

$$(1+i{)}^6=(\sqrt{2}{)}^6{e}^{i6\pi /4}.$$

Now

$$(\sqrt{2}{)}^6=8$$

and

$$\frac{6\pi }{4}=\frac{3\pi }{2}.$$

So

$$(1+i{)}^6=8e^{i3\pi /2}.$$

Since

$$e^{i3\pi /2}=\cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2}=-i,$$

we get

$$(1+i{)}^6=-8i.$$

This example shows why exponential form is useful: a high power becomes a simple operation on the modulus and argument.

nth Roots of a Complex Number

Powers and roots are inverse operations.

If powers multiply arguments, then roots should divide arguments.

This is the main idea behind finding nth roots of a complex number.

Roots reverse powers

To solve

$$z^n=p,$$

we want all complex numbers $z$ whose $n$th power is $p$.

Suppose

$$p=r{e}^{i\theta },$$

where

$$r>0.$$

If

$$z=\rho e^{i\varphi },$$

then

$$z^n={\rho }^n{e}^{in\varphi }.$$

For this to equal

$$r{e}^{i\theta },$$

we need

$${\rho }^n=r$$

and the direction

$$n\varphi$$

to match the direction of $p$.

So the modulus of each root should be

$$\rho =r^{1/n}.$$

The argument should come from dividing an argument of $p$ by $n$.

Why multiple roots appear

The important subtlety is that arguments are not unique.

If $p\ne 0$ and

$$p=r{e}^{i\theta },$$

then the same complex number can also be written as

$$p=r{e}^{i(\theta +2k\pi )},$$

where $k$ is any integer.

Before taking roots, we must include these equivalent arguments:

$$\theta +2k\pi .$$

Then, after dividing by $n$, we get different possible arguments for the roots:

$$\frac{\theta +2k\pi }{n}.$$

This is why an nth-root equation has $n$ distinct roots, not just one.

General formula for nth roots

If

$$p=r{e}^{i\theta },$$

then the $n$ roots of

$$z^n=p$$

are

$$z_k=r^{1/n}{e}^{i(\theta +2k\pi )/n},k=0,1,\dots ,n-1.$$

The values

$$k=0,1,\dots ,n-1$$

give all $n$ distinct roots.

Other choices of $n$ consecutive integer values of $k$ also give the same set of roots, possibly written with different arguments.

If $p=0$, the equation $z^n=0$ has only the root $z=0$.

Geometric meaning

The formula

$$z_k=r^{1/n}{e}^{i(\theta +2k\pi )/n}$$

has a clear geometric meaning.

All roots have the same modulus:

$$r^{1/n}.$$

So all roots lie on the circle centered at the origin with radius

$$r^{1/n}.$$

The arguments of neighbouring roots differ by

$$\frac{2\pi }{n}.$$

Therefore, the roots are equally spaced around the circle.

This equal spacing is not an extra rule to memorise. It comes from adding

$$2k\pi$$

to the argument of $p$ and then dividing by $n$.

Workflow for finding nth roots

To solve

$$z^n=p,$$

use the following steps:

1. Write $p$ in exponential form:

$$p=r{e}^{i\theta }.$$

2. Include all equivalent arguments:

$$p=r{e}^{i(\theta +2k\pi )}.$$

3. Take the nth root of the modulus:

$$r^{1/n}.$$

4. Divide the argument by $n$:

$$\frac{\theta +2k\pi }{n}.$$

5. Use $n$ consecutive integer values of $k$, commonly

$$k=0,1,\dots ,n-1.$$

6. Present the roots in exponential, polar, or Cartesian form as required.

Example: cube roots of unity

The cube roots of unity are the solutions of

$$z^3=1.$$

Write

$$1=e^{i0}.$$

But because arguments are not unique,

$$1=e^{i(0+2k\pi )}=e^{i2k\pi }.$$

So

$$z^3=e^{i2k\pi }.$$

Taking cube roots gives

$$z_k=e^{i2k\pi /3}.$$

Use

$$k=0,1,2.$$

For $k=0$,

$$z_0=e^0=1.$$

For $k=1$,

$$z_1=e^{i2\pi /3}.$$

For $k=2$,

$$z_2=e^{i4\pi /3}.$$

Therefore, the cube roots of unity are

$$1,e^{i2\pi /3},e^{i4\pi /3}.$$

In Cartesian form,

$$e^{i2\pi /3}=\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i,$$

and

$$e^{i4\pi /3}=\cos \frac{4\pi }{3}+i\sin \frac{4\pi }{3}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i.$$

So the three cube roots of unity are

$$1,-\frac{1}{2}+\frac{\sqrt{3}}{2}i,-\frac{1}{2}-\frac{\sqrt{3}}{2}i.$$

Geometrically, these three roots lie on the unit circle and are equally spaced by angle

$$\frac{2\pi }{3}.$$

They form the vertices of an equilateral triangle centered at the origin.

Key idea

For powers:

$$\text{modulus is raised to a power, and argument is multiplied.}$$

For roots:

$$\text{modulus is rooted, and argument is divided.}$$

The multiple roots appear because the same complex number has many equivalent arguments differing by multiples of

$$2\pi .$$

Caption: For $p\ne 0$, the solutions of $z^n=p$ lie equally spaced on a circle in the Argand diagram.

Related

• Complex Numbers
• Complex Number Algebra and Roots
• Argand Diagram, Modulus, Argument, and Loci
• Content Map