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.

Recall that the modulus of a complex number is the distance, in an Argand diagram, between the origin and the complex number, and also that the argument (also called the amplitude) of a complex number is the counterclockwise angle between the positive real axis of an Argand diagram and the line segment between the origin and the complex number. We also recall that the principal argument of a complex number is the argument that lies in the range in radians or .

If we know the real part and the imaginary part of a complex number, we can represent the complex number in the Cartesian form, , which can then be plotted in an Argand diagram with coordinates . In this explainer, we will introduce the polar form of a complex number, which is used to represent a complex number using its modulus and argument.

For instance, let us consider the complex number , which is represented by a point with coordinates on an Argand diagram as shown below.

We recall that the modulus of a complex number is given by , which leads to

In the Argand diagram above, the modulus of is the length of the blue line segment between the origin and the complex number . The argument of a complex number that lies in the second quadrant is given by ; hence, we have

In the Argand diagram above, the argument of is the counterclockwise angle between the positive real axis and the blue line segment. Let us add these values to the diagram.

In the Argand diagram above, we have indicated the location of the complex number by only using its modulus and argument. We can see that knowing the modulus and argument of a complex number is sufficient to identify the location of a complex number in an Argand diagram. The polar form of a complex number provides a way to write down the expression for the complex number only using its modulus and argument.

In our first example, we will now consider how to write the expression of a complex number by only using its modulus and argument.

### Example 1: Using Trigonometry to Write Complex Numbers in Polar Form

Consider the diagram.

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

### Answer

We are given an Argand diagram of the complex number , where the modulus of is labeled as and the argument of is labeled as . In the diagram, is the hypotenuse of a right triangle whose two other sides have lengths and respectively. Because the complex number shown is in the first quadrant, and are positive, so they give the lengths of the two sides of this triangle. Also, is an angle in this right triangle, which is adjacent to the side with length and opposite to the side with length . Keeping this in mind, we can apply right triangle trigonometry to answer each part.

**Part 1**

The side with length is adjacent to the angle . Recall that the cosine function gives the trigonometric ratio between the adjacent side and the hypotenuse of a right triangle:

Rearranging this equation so that is the subject, we obtain

Hence, the correct relationship between , , and is option D.

**Part 2**

Similar to the previous part, the side of length is opposite to the angle . Recall that the sine function gives the trigonometric ratio between the opposite side and the hypotenuse of a right triangle. Hence, we can write

Rearranging this equation so that is the subject, we obtain

Hence, the correct relationship between , , and is option C.

**Part 3**

Since we have obtained the expressions for and in terms of and , we can substitute these expressions into to obtain

In the previous example, we discovered that a complex number in the first quadrant with modulus and argument can be written as

If we factor out the in the right-hand side of the equation above, we obtain the polar form of the complex number .

### Definition: Polar Form of a Complex Number

Consider a nonzero complex number that has the modulus and the argument (or amplitude) . The polar form of the complex number is

This form is also referred to as the trigonometric form or the modulusβargument form.

Previously, we noted that the complex number has the modulus and argument . Hence, the polar form of this complex number is written as

Let us consider another example of writing the polar form of a complex number given its modulus and argument.

### Example 2: Writing a Complex Number in Polar Form given Its Modulus and Principal Argument

Given that and the argument of is , find , giving your answer in polar form.

### Answer

Recall that the polar form of a complex number with modulus and argument is

In this example, we are given that the modulus is equal to 9; hence, . We are also given that the argument of this complex number is . Substituting these values into the formula above, we obtain

Let us consider an example where we write the complex number given in an Argand diagram in the polar form.

### Example 3: Finding the Polar Form of Complex Numbers Represented on the Argand Diagram

Find the polar form of the complex number represented by the given Argand diagram.

### Answer

Recall that the polar form of a complex number with modulus and argument is

The modulus of a complex number is the distance between the origin and the complex number in an Argand diagram. In the diagram above, we are provided with the fact that the modulus is 4; hence, .

The argument of a complex number is the angle between the positive real axis of an Argand diagram and the line segment between the origin and the complex number, measured counterclockwise. By convention, the argument is given in radians in the range . In the diagram, we are given the complementary angle in degrees to the argument of the complex number . Subtracting the provided angle from , we obtain

Pictured below is the angle.

We can convert this angle to radians by multiplying by the angle:

This is the clockwise angle from the positive . Since the argument is measured counterclockwise, we can write it as , which is in the range as desired. Hence, . Substituting these values into the polar form, we have

In the next example, we will modify a given form of a complex number into the polar form and identify the modulus and the principal amplitude (argument) of a complex number from the polar form.

### Example 4: Finding the Modulus and Principal Argument of Complex Numbers in Polar Form

Find the modulus and the principal amplitude of the number .

### Answer

Recall that the polar form of a complex number with modulus and amplitude (or argument) is

Recall also that the principal amplitude of a complex number is an amplitude in the range in radians.

We note that the provided form of the complex number is not the polar form. In particular, it differs from the polar form in the following ways:

- The number in front of the parenthesis is negative.
- The real part is given by the sine function and the imaginary part by the cosine function.
- There is a minus sign between the real and imaginary parts.

We will first convert the given form into the polar form to identify the modulus and the principal amplitude. To correct the negative in , we will distribute to the terms inside the parenthesis:

Now, this operation also moved the negative sign to the sine function. We know that the sine function is odd and the cosine function is even. This leads to the identities:

Applying these identities, we have

Lastly, to switch the sine and cosine functions, we can apply the cofunction identities:

Applying these identities, we obtain

This is the polar form of . From the polar form, we note that the modulus of the complex number is 37, and the amplitude of the complex number is . However, since this amplitude does not lie in the range , we need to add or subtract multiples of the full revolution to find the principal amplitude. Since this number is larger than the upper bound of the range , we will subtract from this amplitude to obtain an equivalent amplitude

We can see that this amplitude lies in the range and it is the principal amplitude.

Hence, we have the modulus and the principal amplitude .

When we are given the Cartesian form of a complex number, we can find its modulus and argument. By computing these two characteristics, we can then write the polar form of the complex number. This is how the Cartesian form is converted to the polar form. Let us look at a few examples where we convert the Cartesian form of a complex number to the polar form.

### Example 5: Converting a Complex Number from the Cartesian Form to the Polar Form

Express the complex number in polar form.

### Answer

Recall that the polar form of a complex number with modulus and amplitude (or argument) is

Notice that is a purely imaginary number with no real part. Hence, this number lies in the positive imaginary axis of an Argand diagram. Plotting this number in an Argand diagram immediately reveals both the modulus and argument of this complex number.

From the diagram, we can see that the modulus of is 4; hence, . Also, we can see that the argument of is radians.

Substituting these values into polar form, we have

In the previous example, we converted a purely imaginary number from the Cartesian form to the polar form. If a complex number is either purely real or purely imaginary, the task of conversion to the polar form is simple because we can immediately see the modulus and the argument of these complex numbers from an Argand diagram.

When we convert a complex number that is neither purely real or purely imaginary into the polar form, we must first compute the modulus and the argument of the complex number. We recall the method for finding the modulus and the argument of a complex number in the Cartesian form.

### How To: Finding the Modulus and the Argument of a Complex Number

The modulus of a complex number is given by

The argument of a complex number can be obtained using the inverse tangent function in each quadrant of an Argand diagram as follows:

- If lies in the first or the fourth quadrant, then
- If lies in the second quadrant, then
- If lies in the third quadrant, then

We will highlight this process in the next example.

### Example 6: Converting a Complex Number from the Cartesian to the Polar Form

- Find the modulus of the complex number .
- Find the argument of the complex number .
- Write the complex number in polar form.

### Answer

**Part 1**

Recall that the modulus of a complex number is given by

The given complex number can also be written as , which means and . 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.

We can see that lies in the first quadrant. Recall that the argument of a complex number in the first quadrant is given by . Hence,

The argument of is .

**Part 3**

Recall that the polar form of a complex number with modulus and amplitude (or argument) is

We obtained in part 1 that the modulus of is ; hence, . In part 2, we obtained that the argument of is ; hence, . Then, the polar form of is

So far, we have considered how to write the polar form of a complex number. Converting the polar form of a complex number to the Cartesian form is much simpler. To do this, we need to multiply through the parenthesis and evaluate the trigonometric ratios. The next example will deal with converting a complex number from the polar form to the Cartesian form.

### Example 7: Converting Complex Numbers from the Polar Form to the Cartesian Form

- Find .
- Find .
- Express the complex number in Cartesian form.

### Answer

**Part 1**

Recall that is a βspecialβ angle. Hence, we recall that

**Part 2**

Similarly,

**Part 3**

Recall that the Cartesian form of a complex number is , where and are real numbers. Substituting the trigonometric ratios we obtained in the previous parts, we get

Multiplying through the parenthesis, we have

Hence, the Cartesian form of the given complex number is .

In our final example, we will find the polar form of the conjugate of a given complex number to find its modulus.

### Example 8: Finding the Modulus of a Complex Number

Given that , find .

### Answer

Recall that the polar form of a complex number with modulus and amplitude (or argument) is

We are given the complex number in the polar form. We will first find the polar form of the conjugate and obtain the modulus of the conjugate from the polar form.

We recall that the conjugate of a complex number is obtained by changing the sign of the imaginary part of the number. We can first multiply through the parenthesis of the polar form of to write

Then, we can change the sign of the imaginary part to obtain the conjugate:

Let us put this back into a polar form. Factoring 6 from both terms, we get

We can see that there is a negative sign between the real and imaginary parts inside the parenthesis. This needs to change to a sum for this to be the polar form. We know that the sine function is odd and the cosine function is even. This leads to the identities below:

Applying these identities, we have

This is the polar form of . In particular, we can see that the modulus of this number is 6.

Hence, .

In the previous and final example, we found the polar form of the conjugate of a complex number to find the modulus of the conjugate. In fact, the polar form of the conjugate of a complex number can be obtained easily from the polar form of the original complex number. For this purpose, let us recall how the complex conjugates are represented in an Argand diagram.

The conjugate of is the reflection of in the real axis in an Argand diagram. Since distances are conserved under reflections, the moduli of and are the same. Geometric angles are also conserved under reflection, but the orientation of the angle changes as seen in the diagram above. We can see that has the equal size and the opposite orientation of . This means that the sign of the argument changes under conjugation. This leads to the following relationship:

Using these relationships, we can write the polar form of the conjugate of a complex number given the polar form of the original complex number.

### Definition: Polar Form of the Conjugate of a Complex Number

Consider a nonzero complex number that has the modulus and the argument (or the amplitude) . The polar form of the conjugate is

In other words, the polar form of the conjugate is obtained by replacing by in the polar form of the original complex number.

Let us finish by recapping a few important points.

### Key Points

- Given a nonzero complex number that has the modulus and the argument (or amplitude) , the polar form of the complex number is This form is also referred to as the trigonometric form or the modulusβargument form.
- To convert the Cartesian form of a complex number to the polar form, we need to compute its modulus and argument.
- To convert the polar form of a complex number to the Cartesian form, we multiply through the parenthesis and evaluate the trigonometric ratios.
- Given a nonzero complex number that has the modulus and the argument (or amplitude) , the polar form of the conjugate is