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8.3: Polar Form of Complex Numbers

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From previous classes, you may have encountered “imaginary numbers” – the square roots of negative numbers – and, more generally, complex numbers which are the sum of a real number and an imaginary number. While these are useful for expressing the solutions to quadratic equations, they have much richer applications in electrical engineering, signal analysis, and other fields. Most of these more advanced applications rely on properties that arise from looking at complex numbers from the perspective of polar coordinates.

We will begin with a review of the definition of complex numbers.

Definition: IMAGINARY NUMBER clipboard_e75e8b4437bfc1b52f97e72bbc0f110c2.png

The most basic complex number is i, defined to be i=1, commonly called an imaginary number. Any real multiple of i is also an imaginary number.

Example 8.3.1

Simplify 9.

Solution

We can separate 9 as 91. We can take the square root of 9, and write the square root of -1 as i.

9=91=3i

A complex number is the sum of a real number and an imaginary number.

Definition: COMPLEX NUMBER

A complex number is a number z=a+bi, where a and b are real numbers

a is the real part of the complex number
b is the imaginary part of the complex number

i=1

Plotting a complex number

We can plot real numbers on a number line. For example, if we wanted to show the number 3, we plot a point:

A numberline from -5 to 5 with a dot at 3.

To plot a complex number like 34i, we need more than just a number line since there are two components to the number. To pAxes, with the horizontal axis labeled real and the vertical axis labeled imaginary.lot this number, we need two number lines, crossed to form a complex plane.

COMPLEX PLANE

In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.

Example 8.3.2

Plot the number 34i on the complex plane.

Solution

The real part of this number is 3, A coordinate grid with the horizontal axis labeled real and the vertical axis labeled imaginary. A point is drawn at 3 comma negative 4and the imaginary part is - 4. To plot this, we draw a point 3 units to the right of the origin in the horizontal direction and 4 units down in the vertical direction.

Because this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number z=a+bi as if we were plotting the point (a, b) in Cartesian coordinates. Sometimes people write complex numbers as z=x+yi to highlight this relation.

Arithmetic on Complex Numbers

Before we dive into the more complicated uses of complex numbers, let’s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.

Example 8.3.3

Add 34i and 2+5i.

Solution

Adding (34i)+(2+5i), we add the real [arts and the imaginary parts

3+24i+5i

5+i

Exercise 8.3.1

Subtract 2+5i from 34i.

Answer

(34i)(2+5i)=19i

We can also multiply and divide complex numbers.

Example 8.3.4

Multiply: 4(2+5i).

Solution

To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials.

4(2+5i)Distribute
=42+45iSimplify
=8+20i

Example 8.3.5

Multiply: 23i)(1+4i).

Solution

To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL – “first outer inner last”).

(23i)(1+4i)Expand
=2+8i3i12i2 Since i=1, i2=1
=2+8i3i12(1) Simplify
=14+5i

Example 8.3.6

Divide (2+5i)(4i).

Solution

To divide two complex numbers, we have to devise a way to write this as a complex number with a real part and an imaginary part.

We start this process by eliminating the complex number in the denominator. To do this, we multiply the numerator and denominator by a special complex number so that the result in the denominator is a real number. The number we need to multiply by is called the complex conjugate, in which the sign of the imaginary part is changed. Here, 4+i is the complex conjugate of 4i. Of course, obeying our algebraic rules, we must multiply by 4+i on both the top and bottom.

(2+5i)(4i)(4+i)(4+i)

In the numerator,

(2+4i)(4+i)Expand
=8+20i+2i+5i2Since i=1, i2=1
=8+20i+2i+5(1)Simplify
=3+22i

Multiplying the denominator

(4i)(4+i)Expand
(164i+4ii2)Since i=1, i2=1
(16(1))
=17

Combining this we get 3+22i17=317+22i17

Exercise 8.3.2

Multiply 34i and 2+3i.

Answer

(34i)(2+3i)=18+i

With the interpretation of complex numbers as points in a plane, which can be related to the Cartesian coordinate system, you might be starting to guess our next step – to refer to this point not by its horizontal and vertical components, but using its polar location, given by the distance from the origin and an angle.

Polar Form of Complex Numbers

Remember, because the complex plane is analogous to the Cartesian plane that we can think of a complex number z=x+yi as analogous to the Cartesian point (x, y) and recall how we converted from (x, y) to polar (r, θ) coordinates in the last section.

Bringing in all of our old rules we remember the following:Axes are labeled real on the horizontal and imaginary on the vertical. A circle centered on axes with a line labeled r drawn at an angle of theta.  The point where the line meets the circle is labeled x plus y times i.  A vertical line is drawn from that point to the x axis forming a triangle, with the vertical length labeled y.  The horizontal leg of the triangle from the origin to the vertical line is labeled x.

cos(θ)=xrx=rcos(θ)

sin(θ)=yry=rsin(θ)

tan(θ)=yxx2+y2=r2

With this in mind, we can write z=x+yi=rcos(θ)+irsin(θ).

Example 8.3.7

Express the complex number 4i using polar coordinates.

Solution

A coordinate grid with the horizontal axis labeled real and the vertical axis labeled imaginary. A point is drawn at 0 comma 4.  A measurement of length 4 is shown from the origin to the point, and an angle of pi over 2 is shown from the horizontal positive axis to the vertical positive axis on which the point lies.On the complex plane, the number 4i is a distance of 4 from the origin at an angle of π2, so 4i=4cos(pi2)+i4sin(π2)

Note that the real part of this complex number is 0.

In the 18th century, Leonhard Euler demonstrated a relationship between exponential and trigonometric functions that allows the use of complex numbers to greatly simplify some trigonometric calculations. While the proof is beyond the scope of this class, you will likely see it in a later calculus class.

Definition: POLAR FORM OF A COMPLEX NUMBER AND EULER'S FORMULA

The polar form of a complex number is z=rcos(θ)+irsin(θ)

An alternate form, which will be the primary one used, is z=reiθ

Euler's Formula states reiθ=rcos(θ)+irsin(θ)

Similar to plotting a point in the polar coordinate system we need r and θ to find the polar form of a complex number.

Example 8.3.8

Find the polar form of the complex number -8.

Solution

Treating this is a complex number, we can write it as 8+0i.A coordinate grid with the horizontal axis labeled real and the vertical axis labeled imaginary. A point is drawn at negative 8 comma 0.  A measurement of length 8 is shown from the origin to the point, and an angle of pi is shown from the horizontal positive axis to the horizontal negative axis on which the point lies.

Plotted in the complex plane, the number -8 is on the negative horizontal axis, a distance of 8 from the origin at an angle of π from the positive horizontal axis.

The polar form of the number -8 is 8eiπ

Plugging r=8 and θ=π back into Euler's formula, we have:

8eiπ=8cos(π)+8isin(π)=8+0i=8as desired.

Example 8.3.9

Find the polar form of 4+4i.

Solution

On the complex plane, this complex number would correspond to the point (-4, 4) on a Cartesian plane. We can find the distance r and angle θ as we did in the last section.

r2=x2+y2
r2=(4)2+42
r=32=42A coordinate grid with the horizontal axis labeled real and the vertical axis labeled imaginary. A point is drawn at negative 4 comma 4.  A measurement of length 4 square root of 2 is shown from the origin to the point, and an angle of 3 pi over 4 is shown from the horizontal positive axis a line from the origin to the point.

To find θ, we can use cos(θ)=xr

cos(θ)=442=22

This is one of known cosine values, and since the point is in the second quadrant, we can conclude that θ=3π4.

The polar form of this complex number is 42e3π4i

Example 8.3.10

Find the polar form of 35i.

Solution

On the complex plane, this complex number would correspond to the point (-3, -5) on a Cartesian plane. First, we find r.

A coordinate grid with the horizontal axis labeled real and the vertical axis labeled imaginary. A point is drawn at negative 3 comma negative 5.  A measurement of square root of 34 is shown from the origin to the point, and an angle of 4.1720 is shown from the horizontal positive axis a line from the origin to the point.

r2=x2+y2
r2=(3)2+(5)2
r=34

To find θ, we might use tan(θ)=yx

tan(θ)=53
θ=tan1(53)=1.0304

This angle is in the wrong quadrant, so we need to find a second solution. For tangent, we can find that by adding π.

θ=1.0304+π=4.1720

The polar form of this complex number is 34e4.1720i.

Exercise 8.3.3

Write 3+i in polar form.

Answer

3+i would correspond with the point (3,1) in the first quadrant.

r=32+12=4=2
sin(θ)=12, so θ=π6
3+i in polar form is 2eiπ/6

Example 8.3.11

Write 3eπ6i in complex a+bi form.

Solution

3eπ6i=3cos(π6)+i3sin(π6)Evaluate the trig functions
=332+i312Simplify
=332+i32

The polar form of a complex number provides a powerful way to compute powers and roots of complex numbers by using exponent rules you learned in algebra. To compute a power of a complex number, we:

1) Convert to polar form
2) Raise to the power, using exponent rules to simplify
3) Convert back to a+bi form, if needed

Example 8.3.12

Evaluate (4+4i)6.

Solution

While we could multiply this number by itself five times, that would be very tedious. To compute this more efficiently, we can utilize the polar form of the complex number.

In an earlier example, we found that 4+4i=42e3π4i. Using this,

(4+4i)6Writing the complex number in polar form
=(42e3π4i)6Utilize the exponent rule (ab)m=ambm
=(42)6(e3π4i)6On the second factor, use the rule (am)n=amn
=(42)6e3π4i6Simplify
=32768e2π2i

At this point, we have found the power as a complex number in polar form. If we want the answer in standard a+bi form, we can utilize Euler’s formula.

32768e9π2i=32768cos(9π2)+i32768sin(9π2)

Since 9π2 is coterminal with π2, we can use our special angle knowledge to evaluate the sine and cosine.

32768cos(9π2)+i32768sin(9π2)=32768(0)+i32768(1)=32768i

We have found that 4+4i)6=32768i.

The result of the process can be summarized by DeMoivre’s Theorem. This is a shorthand to using exponent rules.

DEMOIVRE'S THEOREM

If z=r(cos(θ)+isin(θ)), then for any integer, zn=rn(cos(nθ)+isin(nθ))

We omit the proof, but note we can easily verify it holds in one case using Example 12:

(4+4i)6=(42)6(cos(63π4+isin(63π4))=32768(cos(9π2)+isin(9π2))=32768i

Example 8.3.13

Evaluate 9i.

Solution

To evaluate the square root of a complex number, we can first note that the square root is the same as having an exponent of 12: 9i=(9i)1/2

To evaluate the power, we first write the complex number in polar form. Since 9i has no real part, we know that this value would be plotted along the vertical axis, a distance of 9 from the origin at an angle of π2. This gives the polar form: 9i=9eπ2i.

9i=(9i)1/2Use the polar form
=(9eπ2i)1/2Use exponent rules to simplify
=91/2(eπ2i)1/2
=91/2eπ2i12Simplify
=3eπ4iRewrite using Euler's formula if desired
=3cos(π4)+i3sin(π4)Evaluate the sine and cosine
=322+i322

Using the polar form, we were able to find a square root of a complex number.

9i=322+322i

Alternatively, using DeMoivre's Theorem we could write

(9eπ2i)1/2=91/2(cos(12π2+isin(12π2))=3(cos(π4)+isin(π4) and simplify

Exercise 8.3.4

Evaluate (3+i)6 using polar form.

Answer

(3+i)6=(2eiπ/6)6=26eiπ=64cos(π)+i64sin(π)=64

You may remember that equations like x2=4 have two solutions, 2 and -2 in this case, though the square root 4 only gives one of those solutions. Likewise, the square root we found in Example 11 is only one of two complex numbers whose square is 9i.

Similarly, the equation z3=8 would have three solutions where only one is given by the cube root. In this case, however, only one of those solutions, z=2, is a real value. To find the others, we can use the fact that complex numbers have multiple representations in polar form.

Example 8.3.14

Find all complex solutions to z3=8.

Solution

Since we are trying to solve z3=8, we can solve for z is z=81/3. Certainly one of these solutions is the basic cube root, giving z=2. To find others, we can turn to the polar representation of 8.

Since 8 is a real number, is would sit in the complex plane on the horizontal axis at an angle of 0, giving the polar form 8e0i. Taking the 1/3 power of this gives the real solution:

(8e0i)1/3=81/3(e0i)1/3=2e0=2cos(0)+i2sin(0)=2

However, since the angle 2π is coterminal with the angle of 0, we could also represent the number 8 as 8e2πi. Taking the 1/3 power of this gives a first complex solution:

(8e2πi)1/3=81/3(e2πi)1/3=2e2π3i=2cos(2π3)+i2sin(2π3)=2(12)+i2(32)=1+3i

A coordinate grid with the horizontal axis labeled real and the vertical axis labeled imaginary.  A dashed circle of radius 2 centered at the origin is shown.  Three points equally spaced around the circle are drawn, at 2 comma 0, negative 1 comma square root of 3, and negative 1 comma negative square root of 3.For the third root, we use the angle of 4π, which is also coterminal with an angle of 0.

(8e4πi)1/3=81/3(e4πi)1/3=2e4π3i=2cos(4π3)+i2sin(4π3)=2(12)+i2(32)=13i

Altogether, we found all three complex solutions to z3=8, z=2, 1+3i, 13i

Graphed, these three numbers would be equally spaced on a circle about the origin at a radius of 2.

Important Topics of This Section

  • Complex numbers
  • Imaginary numbers
  • Plotting points in the complex coordinate system Basic operations with complex numbers
  • Euler’s Formula
  • DeMoivre’s Theorem
  • Finding complex solutions to equations

This page titled 8.3: Polar Form of Complex Numbers is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by David Lippman & Melonie Rasmussen (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.

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