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The Polar Form Of A Complex Number

Q: Can there be more than one angle (θ) for the same number?

Yes. Since angles on a circle repeat every $360^\circ$ (or $2\pi$ radians), you can add or subtract any multiple of $360^\circ$ to an angle and it will point to the same location. We usually use the 'principal value', which is the angle between $0^\circ$ and $360^\circ$.

Q: What do I do if the real part 'a' is zero?

If $a=0$, the number is on the imaginary axis, like $z=5i$. You can't use $\tan\theta = b/a$ because you can't divide by zero. Instead, you can see from the graph that the angle must be $90^\circ$ (if $b$ is positive) or $270^\circ$ (if $b$ is negative).

Q: What is the polar form of a real number like z=7?

A real number like $7$ can be written as $7 + 0i$. It lies on the positive real axis, so its distance from the origin is $r=7$ and its angle is $\theta=0^\circ$. The polar form is $z = 7(\cos(0^\circ) + i\sin(0^\circ))$.

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Ever wondered if there's another way to write a complex number besides $a + b i$ ? There is! The polar form uses a distance and an angle to pinpoint a number on the complex plane, unlocking powerful new ways to understand and manipulate these fascinating numbers.

The Polar Form Of A Complex Number — an original Algebra911 reference diagram defining the polar form of a complex number with its key formula and a worked example. — The Polar Form Of A Complex Number

What Is the Polar Form of a Complex Number?

The polar form of a complex number is a method of representing it using its distance from the origin and the angle it makes with the positive real axis. While you're likely familiar with the standard rectangular form $z = a + b i$ , the polar form provides a different, and often very useful, perspective.

Imagine a complex number plotted on the complex plane, which is just like a standard Cartesian coordinate plane. The horizontal axis is the 'real' axis, and the vertical axis is the 'imaginary' axis. A number like $z = 3 + 4 i$ corresponds to the point $(3, 4)$ .

To get to that point, you can go $3$ units right and $4$ units up. That's the rectangular way of thinking. But what if you described it differently? You could also get to that point by stating:

How far you have to travel in a straight line from the origin $(0, 0)$ .
What angle your path makes with the positive x-axis (the real axis).

This distance and angle are the two key components of polar form.

The distance is called the modulus, usually written as $r$ or $| z |$ .
The angle is called the argument, written as $θ$ (the Greek letter theta).

So, instead of using $a$ and $b$ to define our complex number, we use $r$ and $θ$ . This gives us a new way to write the same number, which is incredibly useful for multiplication, division, and finding roots of complex numbers.

How Do You Calculate the Modulus (r)?

The modulus, $r$ , is simply the distance from the origin to the point representing the complex number on the complex plane. Finding it is a direct application of a familiar tool: the Pythagorean theorem.

Think about our complex number $z = a + b i$ as the point $(a, b)$ . If you draw a line from the origin to this point, and then drop a vertical line to the real axis, you create a right-angled triangle. The two legs of this triangle have lengths equal to the absolute values of $a$ and $b$ , and the hypotenuse is the distance we want, $r$ .

According to the Pythagorean theorem ( $l e g_{1}^{2} + l e g_{2}^{2} = h y p o t e n u s e^{2}$ ), we have:

$a^{2} + b^{2} = r^{2}$

To find $r$ , we just take the square root of both sides. Since $r$ represents a distance, it must always be non-negative.

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

Let's find the modulus for the complex number $z = 3 + 4 i$ . Here, $a = 3$ and $b = 4$ .

$r = \sqrt{3^{2} + 4^{2}}$ $r = \sqrt{9 + 16}$ $r = \sqrt{25}$ $r = 5$

So, the modulus of $3 + 4 i$ is $5$ . This means the point $(3, 4)$ is exactly $5$ units away from the origin.

How Do You Find the Argument (θ)?

The argument, $θ$ , is the angle between the positive real axis and the line segment connecting the origin to our complex number. We measure this angle counter-clockwise. To find it, we turn to basic trigonometry (SOH CAH TOA).

In our right-angled triangle from the previous section:

The side adjacent to the angle $θ$ is $a$ .
The side opposite to the angle $θ$ is $b$ .
The hypotenuse is $r$ .

The tangent function relates the opposite and adjacent sides:

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a}

To find the angle $θ$ itself, we use the inverse tangent function (often written as $\arctan$ or $\tan^{- 1}$ on calculators).

CRITICAL NOTE: The calculator's $\arctan (b / a)$ result isn't always the final answer! The calculator only gives angles in Quadrant I (for positive inputs) or Quadrant IV (for negative inputs). You must look at the signs of $a$ and $b$ to determine the correct quadrant and adjust the angle if necessary. The angle you get from the calculator is called the reference angle.

Here is a guide to adjusting the angle based on the quadrant where $(a, b)$ lies:

Quadrant	Sign of a (Real Part)	Sign of b (Imaginary Part)	Argument $θ$ Calculation
I	+	+	$θ = \arctan (\| b / a \|)$
II	-	+	$θ = 180^{\circ} - \arctan (\| b / a \|)$ or $π - \arctan (\| b / a \|)$
III	-	-	$θ = 180^{\circ} + \arctan (\| b / a \|)$ or $π + \arctan (\| b / a \|)$
IV	+	-	$θ = 360^{\circ} - \arctan (\| b / a \|)$ or $2 π - \arctan (\| b / a \|)$

For example, for $z = - 1 + i$ , we have $a = - 1$ and $b = 1$ . This is in Quadrant II. The reference angle is $\arctan (| 1 / - 1 |) = \arctan (1) = 45^{\circ}$ . Since it's in Quadrant II, the actual argument is $θ = 180^{\circ} - 45^{\circ} = 135^{\circ}$ .

How Do You Convert a Complex Number to Polar Form?

Now that we can find both the modulus $r$ and the argument $θ$ , we can write the number in its full polar form. The standard polar form connects $a, b, r,$ and $θ$ using cosine and sine.

From our right-triangle trigonometry, we know:

$\cos (θ) = \frac{a}{r} ⟹ a = r \cos (θ)$

$\sin (θ) = \frac{b}{r} ⟹ b = r \sin (θ)$

Now, we can substitute these expressions for $a$ and $b$ back into the standard form $z = a + b i$ :

$z = (r \cos θ) + (r \sin θ) i$

By factoring out the $r$ , we arrive at the final polar form:

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

Sometimes, you will see this abbreviated as $z = r cis θ$ , where 'cis' stands for 'cosine + i sine'.

Example 1

Convert the complex number $z = 1 + \sqrt{3} i$ to polar form.

Step 1: Find the modulus (r).
Here, $a = 1$ and $b = \sqrt{3}$ .
$r = \sqrt{a^{2} + b^{2}} = \sqrt{1^{2} + (\sqrt{3})^{2}} = \sqrt{1 + 3} = \sqrt{4} = 2$

Step 2: Find the argument (θ).
First, determine the quadrant. Since $a = 1$ (positive) and $b = \sqrt{3}$ (positive), the number is in Quadrant I.
$\tan (θ) = \frac{b}{a} = \frac{\sqrt{3}}{1} = \sqrt{3}$
The angle whose tangent is $\sqrt{3}$ is $60^{\circ}$ or $\frac{π}{3}$ radians. Since we are in Quadrant I, this is our final angle.
$θ = 60^{\circ}$

Step 3: Write the number in polar form.
Using the formula $z = r (\cos θ + i \sin θ)$ :
$z = 2 (\cos (60^{\circ}) + i \sin (60^{\circ}))$

How Do You Convert from Polar Form Back to Rectangular?

Converting from polar form back to the standard $a + b i$ form is generally more straightforward. You are given $r$ and $θ$ , and your goal is to find $a$ and $b$ .

You simply use the same relationships we discovered earlier:

$a = r \cos θ$
$b = r \sin θ$

All you need to do is evaluate the cosine and sine of the given angle, multiply by the modulus $r$ , and you'll have your $a$ and $b$ values directly.

Example 2

Convert the complex number $z = 8 (\cos (210^{\circ}) + i \sin (210^{\circ}))$ to rectangular form $a + b i$ .

Step 1: Identify r and θ.
From the given form, we can see that $r = 8$ and $θ = 210^{\circ}$ .

Step 2: Calculate a and b.
We use the conversion formulas. You may need a calculator or knowledge of the unit circle for these trigonometric values.
$a = r \cos θ = 8 \cos (210^{\circ}) = 8 (- \frac{\sqrt{3}}{2}) = - 4 \sqrt{3}$ $b = r \sin θ = 8 \sin (210^{\circ}) = 8 (- \frac{1}{2}) = - 4$

Step 3: Write the number in rectangular form.
Now we substitute our calculated $a$ and $b$ into $z = a + b i$ .
$z = - 4 \sqrt{3} - 4 i$

Key formulas for the polar form of a complex number by Algebra911.

A Complete Example in a Different Quadrant

Let's work through one more example from rectangular to polar form, this time with a number in Quadrant III. This will highlight the importance of checking the quadrant when finding the argument $θ$ .

Example 3

Convert the complex number $z = - \sqrt{6} - \sqrt{2} i$ to polar form.

Step 1: Find the modulus (r).
Here, $a = - \sqrt{6}$ and $b = - \sqrt{2}$ .
$r = \sqrt{(- \sqrt{6})^{2} + (- \sqrt{2})^{2}} = \sqrt{6 + 2} = \sqrt{8} = 2 \sqrt{2}$

Step 2: Find the argument (θ).
First, determine the quadrant. Since $a$ is negative and $b$ is negative, the number is in Quadrant III.
Now, let's find the reference angle using the absolute values of $a$ and $b$ .
$\tan (θ_{r e f}) = \frac{| b |}{| a |} = \frac{\sqrt{2}}{\sqrt{6}} = \frac{1}{\sqrt{3}}$
The reference angle whose tangent is $\frac{1}{\sqrt{3}}$ is $30^{\circ}$ .
Since we are in Quadrant III, we must adjust this angle:
$θ = 180^{\circ} + θ_{r e f} = 180^{\circ} + 30^{\circ} = 210^{\circ}$

Step 3: Write the number in polar form.
Using the formula $z = r (\cos θ + i \sin θ)$ with our calculated $r$ and $θ$ :
$z = 2 \sqrt{2} (\cos (210^{\circ}) + i \sin (210^{\circ}))$

What Are Some Common Mistakes to Avoid?

When working with polar forms, a few common errors can trip students up. Being aware of them is the first step to avoiding them!

Ignoring the Quadrant: This is the most common mistake. Simply taking $θ = \arctan (b / a)$ without checking the signs of $a$ and $b$ will give you the wrong angle for any number in Quadrants II or III. Always plot the point mentally to confirm you're in the right place.
Degree/Radian Mix-ups: Some problems use degrees ( $0^{\circ}$ to $360^{\circ}$ ) and others use radians ( $0$ to $2 π$ ). Make sure your calculator is in the correct mode (DEG or RAD) to match the problem. Mixing them up will lead to incorrect answers.
Forgetting to Square Negative Signs: When calculating the modulus $r = \sqrt{a^{2} + b^{2}}$ , remember that squaring a negative number results in a positive one. For example, if $a = - 3$ , then $a^{2} = (- 3)^{2} = 9$ , not $- 9$ .
Incorrectly Distributing $i$ : The polar form is $r (\cos θ + i \sin θ)$ . The imaginary unit $i$ is only attached to the $\sin θ$ term. A common error is writing $r (i \cos θ + i \sin θ)$ or some other incorrect variation.
Simplification Errors: Be careful when simplifying radicals for the modulus or evaluating trigonometric functions for the argument. Double-check your arithmetic.

Quick Summary and Key Formulas

This lesson covered a new way to represent complex numbers. Here are the most important takeaways and formulas in one place for easy reference.

The Two Forms:

Rectangular Form: $z = a + b i$
Polar Form: $z = r (\cos θ + i \sin θ)$

Converting from Rectangular to Polar:

Find the modulus (distance):
r = \sqrt{a^2 + b^2}
Find the argument (angle):
\theta = \arctan\left(\frac{b}{a}\right) \text{, adjusted for the correct quadrant!}
Write the final form: $z = r (\cos θ + i \sin θ)$

Converting from Polar to Rectangular:

Find the real part:
a = r\cos\theta
Find the imaginary part:
b = r\sin\theta
Write the final form: $z = a + b i$

Frequently Asked Questions

Why is polar form useful?

Polar form makes multiplying and dividing complex numbers incredibly simple. Instead of using the FOIL method, you just multiply the moduli and add the arguments. It's also essential in higher math, physics, and engineering for describing rotations, waves, and oscillations.

Can the modulus (r) be negative?

No, the modulus represents a distance from the origin, so it is always non-negative (zero or positive). The formula $r = \sqrt{a^{2} + b^{2}}$ uses the principal square root, which by definition is never negative.

Can there be more than one angle (θ) for the same number?

Yes. Since angles on a circle repeat every $360^{\circ}$ (or $2 π$ radians), you can add or subtract any multiple of $360^{\circ}$ to an angle and it will point to the same location. We usually use the 'principal value', which is the angle between $0^{\circ}$ and $360^{\circ}$ .

What do I do if the real part 'a' is zero?

If $a = 0$ , the number is on the imaginary axis, like $z = 5 i$ . You can't use $\tan θ = b / a$ because you can't divide by zero. Instead, you can see from the graph that the angle must be $90^{\circ}$ (if $b$ is positive) or $270^{\circ}$ (if $b$ is negative).

What's the difference between radians and degrees?

They are two different units for measuring angles, just like inches and centimeters are for length. A full circle is $360$ degrees or $2 π$ radians. Always check which unit a problem requires and make sure your calculator is in the correct mode (DEG or RAD).

What is the polar form of a real number like z=7?

A real number like $7$ can be written as $7 + 0 i$ . It lies on the positive real axis, so its distance from the origin is $r = 7$ and its angle is $θ = 0^{\circ}$ . The polar form is $z = 7 (\cos (0^{\circ}) + i \sin (0^{\circ}))$ .

What does 'cis' mean in some textbooks?

The term 'cis' is a convenient shorthand. It stands for 'cosine + i sine'. So, writing $z = r cis θ$ is just a more compact way of writing the full polar form, $z = r (\cos θ + i \sin θ)$ .