In this explainer, we will learn how to represent a complex number in polar form, calculate the modulus and argument, and use this to change the form of a complex number.

We can represent a complex number such as (where is the square root of negative one) on an Argand diagram as shown below.

We can refer to points in the plane in both Cartesian and polar forms. In a similar way, complex numbers can be written in both algebraic and polar forms.

Recall, when changing the coordinates of a point to polar form , that we calculate using the Pythagorean theorem as follows: and we calculate using the inverse tangent function

Applying the same method to the point , we calculate and we calculate

For complex numbers, we have special names for and : we refer to as the modulus of the complex number (which we write as ) and as the argument (which we write as ). Using and , we can express as

A complex number expressed in this form is said to be in polar form.

### Definition: Polar Form of a Complex Number

A complex number written in the form where the modulus and argument are said to be in polar form. This form is also referred to as the trigonometric form or the modulusβargument form.

### Example 1: Recognizing the Polar Form of a Complex Number

Which of the following complex numbers are correctly expressed in polar form?

### Answer

A complex number is said to be in polar form if where is the modulus of and is the argument. We will consider each option in turn and see whether it is correctly expressed in this form.

- Initially, this number looks like it is in polar form. However, closer inspection reveals that it is not actually in polar form. In fact, it demonstrates one common error students make when writing complex numbers in polar form: getting sine and cosine the wrong way around. To correctly write this number in polar form, we can use the cofunction identities: Hence, setting , we can correctly express this number in polar form as
- This number is in the form , where and . Since , this number is correctly written in polar form.
- One again, this number is suspiciously close to being in the correct form. However, the minus sign before means that the number is not in the correct form. To correct this, we can use the even/odd identities, to rewrite the number as which is now correctly expressed in polar form.
- It would be easy to mistakenly assume that this number it not in polar form since it
looks like and are the wrong way around.
However, can be any positive real number; hence,
a perfectly legitimate value for .
Similarly, can take any real value, which means that
is a legitimate value of the argument.

Hence, this number is correctly written in polar form. - It can be easy to miss the fact that this number is not in polar form. However, a careful look reveals that the sine and cosine functions have different arguments. To rewrite this in the correct form, we would need to calculate its actual modulus and argument.

Therefore, the only two numbers that are correctly expressed in polar form are and .

Gaining fluency in converting between the algebraic form and the polar form of complex numbers will prove to be extremely useful. We will now look at an example of converting a complex number from algebraic to polar form.

### Example 2: Converting a Complex Number from Algebraic Form to Polar Form

- Find the modulus of the complex number .
- Find the argument of the complex number .
- Hence, write the complex number in polar form.

### Answer

**Part 1**

Recall that the modulus of a complex number is given by

Hence, the modulus of is .

**Part 2**

When calculating the argument of a complex number, we need to be careful to check in which quadrant of the Argand diagram the complex number lies.

Since we are in the first quadrant, we can just use the inverse tangent function to find the argument. Hence,

**Part 3**

Finally, using the definition of polar form, we can write

### How to Convert a Complex Number from Algebraic Form to Polar Form

To convert a complex number in algebraic form to polar form, follow the following steps.

- Find the modulus, , of the complex number using the formula .
- Find the argument, , of the complex number. There is more than one technique for finding the argument of a complex number; one such technique is presented here. If is in the first or fourth quadrant of the Argand diagram , we can simply use the inverse tangent function and calculate However, if the complex number is in the second quadrant ( and ), we need to add to the value we get using the inverse tangent function. Hence, But if the complex number is in the third quadrant ( and ), we need to subtract from the value we get using the inverse tangent function. Hence, Finally, if the complex number is purely imaginary , then if , and if . Note that when , the argument is undefined.
- Write the number in polar form:

where and .

Now we will look at an example where we cannot use the inverse tangent function to find the argument.

### Example 3: Converting a Complex Number from Algebraic Form to Trigonometric Form

Express the complex number in trigonometric form.

### Answer

Remember that trigonometric form and polar form are two different names for the same thing. We begin by finding the modulus of the complex number . Applying the formula to find the modulus, noting that there is no real part, gives .

Now we would like to find the argument. Notice that is a purely imaginary number with no real part.

In this case, we are not able to apply the inverse tangent function to calculate the modulus sincewe cannot divide by zero. However, given that all complex numbers that lie on the positive imaginary axis have an argument of , we can conclude that .

Finally, we can write the number in its trigonometric (or polar) form as follows:

We will nowconsider an example that highlights the relationship between the polar and algebraic forms of a complex number.

### Example 4: Relation between Polar and Algebraic Forms of a Complex Number

Consider the diagram.

- Which of the following correctly describes the relationship between
, and ?
- Which of the following correctly describes the relationship between
, and ?
- Hence, express in terms of and .

### Answer

**Part 1**

The triangle with sides , and is a right triangle with hypotenuse . Hence, to write a relationship between , and , we can apply basic trigonometry. The side of length is adjacent to angle . Hence, we can use the cosine as follows:

Rearranging, we can write

Hence, the correct relationship between , and is ().

**Part 2**

Similarly, the side of length is opposite angle . Hence, using the sine function, we can write

Rearranging, we can write

Hence, the correct relationship between , , and is ().

**Part 3**

Since , we can substitute the values of and into this equation to get

If we factor out the in the final part of the previous example, we find that we have an expression for in polar form, whereas in its current form, we have

which is expressed in algebraic form: .We will now have a look at a further example of converting from polar form to algebraic from.

### Example 5: Converting Complex Numbers from Trigonometric to Rectangular Form

- Find .
- Find .
- Hence, express the complex number in rectangular form.

### Answer

**Part 1**

Recall that is a βspecialβ angle. Hence, we should have committed its sine, cosine, and tangent values to memory and simply be able to recall

**Part 2**

Similarly,

**Part 3**

Recall that rectangular form is another name for algebraic form. Hence, using the two values we have just given, we can rewrite the complex number as follows:

Expanding the brackets, we have

As we saw in the previous example, converting from polar form to algebraic form is as simple as evaluating sine and cosine and then expanding the brackets.

### Example 6: Argument of a Complex Number

Given that , where , find the principal argument of .

### Answer

Since is given in a form that is similar to polar form, the simplest way to solve this problem will be using trigonometric identities to rewrite in polar form. Once we have in polar form, we can simply read off its argument. First, sine and cosine are the wrong way around for polar form. To resolve this, we can use the cofunction identities, which relate sine and cosine as follows:

Using these, we can rewrite as follows:

This is not quite in polar form yet: we have a negative sign before sine, whereas, for polar form, it should be positive. To correct this, we can use the even/odd identities, to rewrite as

Simplifying, we have

Now that is expressed in polar form, we can simply read off its argument as . Finally, we need to check whether this is the principal argument. Recall that, for any given argument to be considered the principal argument, it must be given in the range . We are given that . Hence,

Subtracting , we get

Hence, is in the range , which is contained in and is, therefore, the principal argument of .

### Key Points

- Similar to howpoints in the plane can be written in Cartesian and polar coordinates, we can write complex numbers in algebraic and polar forms.
- The polar form of a complex number is where is the modulus and is the argument.
- To convert a complex number to polar form, we simply calculate its modulus and its argument .
- To convert a complex number from polar form to algebraic form, we can expand the brackets and evaluate sine and cosine.