In this explainer, we will learn how to perform calculations with complex numbers in polar form.
We recall that when we multiply a pair of complex numbers in the Cartesian form, we can simplify the resulting complex number in the Cartesian form by multiplying through the parentheses and collecting the real and imaginary parts separately. Also, when we divide a pair of complex numbers in the Cartesian form, we can make the denominator of the fraction real by multiplying the top and bottom of the fraction by the complex conjugate of the denominator. We can then multiply through the parentheses and collect the real and imaginary parts separately to express the resulting complex number in the Cartesian form.
This process is much simpler if we understand the polar form of complex numbers when we use the properties of the multiplication and division of complex numbers in relation to the modulus and the argument of complex numbers. In this explainer, we will prove the relationships between the multiplication/division of complex numbers and their arguments and moduli using polar forms. We begin by recalling the polar form of a complex number.
Definition: Polar Form of a Complex Number
A nonzero complex number with modulus and argument can be expressed in the polar form as
We recall that we can convert the Cartesian form of a complex number to the polar form by computing its modulus and argument. We can also convert the polar form of a complex number to the Cartesian form by multiplying through the parentheses and evaluating the trigonometric ratios.
Let us begin by demonstrating the relationship in contexts of the multiplication of complex numbers.
Theorem: Multiplication of Complex Numbers in Polar Form
Let and be nonzero complex numbers. Then, their product in the polar form is written as
Let us prove this theorem. The product of and is written as
Multiplying through the parentheses, we obtain
Using and gathering the real and imaginary terms, we have
We state the summation formulae for the sine and cosine functions:
We can apply the cosine summation formula to the real part and the sine summation formula to the imaginary part inside the parentheses of equation (1). Then, we can rewrite
This proves the theorem.
In our first example, we will apply this theorem to multiply two complex numbers in the polar form.
Example 1: Multiplying Complex Numbers in Polar Form
Given that and , find .
Answer
Recall that, given a pair of nonzero complex numbers and , the product is
We are given the polar form of the complex numbers and . From the given polar form, we can obtain and for , while and . Substituting these values into the formula, we have
Rationalizing the denominator and summing the fractions, we have
In the previous example, we computed the product of two complex numbers in the polar form. We note that this process is simpler than the multiplication of complex numbers in the Cartesian form, which would involve multiplying through the parentheses and collecting the real and the imaginary terms.
Let us examine the polar form of the product
From this polar form, we can see that the modulus of the product is , which is the product of the moduli of and . Also, the argument of the product is the sum of the arguments of and . This leads to the following fact.
Fact: Relationship between Product of Complex Numbers and Their Moduli and Arguments
For any pair of nonzero complex numbers and , we have
In the next example, we will apply this fact to find the modulus of the product of two complex numbers.
Example 2: Multiplying Complex Numbers in Polar Form
If , , and , then what is ?
Answer
Recall that, for any pair of nonzero complex numbers and ,
In this example, we are given the complex numbers and in the polar form. We recall that a nonzero complex number has the polar form
From the given polar form, we can see that and , which means
We can also see that and . So,
Since we are given that , we have . This leads to the polar form of the product
We know that and . Substituting these values into the equation above, we have
Hence, is the real number .
In previous examples, we found the products of complex numbers in the polar form. We now turn our attention to the division and quotient of complex numbers in the polar form.
Definition: The Quotient of Complex Numbers in Polar Form
Given a pair of nonzero complex numbers in the polar form and , their quotient can be written in the polar form as
Let us prove this theorem. We begin with the quotient of and written as
To make the denominator of this fraction real, we multiply the numerator and denominator by the conjugate of the denominator to obtain
Multiplying through the parentheses, we obtain
Using and gathering the real and imaginary terms, we have
Using the trigonometric identity , we can simplify this expression to
Finally, we state the difference identities for sine and cosine:
We apply the cosine difference identity in the real part and the sine difference identity in the imaginary part of the complex number within the parentheses on the right-hand side of equation (2). Then, we can rewrite:
This proves the theorem.
Let us consider an example where we will find the quotient of two complex numbers in the polar form using this method.
Example 3: Finding the Quotient of Two Complex Numbers in Polar Form
Given that and , find in polar form.
Answer
Recall that, given a pair of nonzero complex numbers in the polar form and , the quotient can be written in the polar form as
In this example, we are given the polar form for the complex numbers and . From the given polar form, we can identify and for , while and . Substituting these values into the formula for the polar form of the quotient, we have
Simplifying, we have
In the previous example, we computed the quotient of two complex numbers in the polar form. We note that this process is simpler than the division of complex numbers in the Cartesian form, which would involve multiplying the numerator and denominator by the conjugate of the denominator and then multiplying through the parentheses. Using this method, we can see that the division of complex numbers is much simpler in the polar form.
Let us examine the polar form of the quotient:
From this polar form, we can see that the modulus of the quotient is , which is the quotient of the moduli of and . Also, the argument of the quotient is the difference of the arguments of and . This leads to the following fact.
Fact: Relationship between Quotient of Complex Numbers and Their Moduli and Arguments
For any pair of nonzero complex numbers and , we have
In the next example, we will use these facts to find the polar form of a quotient of two complex numbers.
Example 4: Dividing Complex Numbers in Polar Form and Finding Their Quotient in Cartesian Form
Given that , , , and , find .
Answer
Recall that, given any pair of nonzero complex numbers and ,
In this example, the complex numbers and are given in the polar form. We recall that a nonzero complex number has the polar form
From the given polar form, we can find the moduli , . Hence,
Also from the given polar form, we can find the arguments , . Therefore,
Hence, the polar form of the quotient is
To finish the problem, we need to find the trigonometric ratios and from the provided information about the tangent function. We are given that . Since the tangent function is defined at , while the tangent function is not defined at , we know that . This means that . In other words, is an acute angle. For an acute angle, we can relate trigonometric ratios to right triangle trigonometry. Recall the trigonometric ratios for an acute angle :
We are given that , so we can draw the right triangle with an angle theta whose opposite side is of length 4 and whose adjacent side is of length 3. Using the Pythagorean theorem, the length of the hypotenuse must be
Using this triangle, we find
Substituting these values into the polar form for the quotient, we have
This leads to option B.
In the previous two examples, we found the quotients of complex numbers using the polar form. We can also use the rules of division to find a general form for the reciprocal of a complex number as the next example will show.
Example 5: The Reciprocal of a Complex Number in Polar Form
Given that , find .
Answer
Recall that, given a pair of nonzero complex numbers in the polar form and , the quotient can be written in the polar form as
In this example, we need to find the reciprocal . Note that the reciprocal is also a fraction where and . In this case, is the real number, which means that it has modulus 1 and argument 0. In other words, 1 can be expressed in the polar form as
This leads to and . On the other hand, the denominator is given in the polar form; hence, we can obtain and . Substituting these values into the equation for the polar form of the quotient, we have
While this is a correct answer, we also recall that, by convention, the argument of a complex number should lie in the range in radians. Such an argument is called the principal argument. The argument given in the polar form above, , does not lie in the range , so we need to add or subtract a multiple of the full revolution . Since the given argument is below the lower bound , we add to obtain an equivalent argument:
This argument lies in the range , which makes it the principal argument. Using the principal argument, the polar form of the reciprocal is
In the previous example, we found the reciprocal of a complex number in the polar form using the formula for the quotient using the polar form. Applying an analogous method, we can find a general formula for the polar form of the reciprocal of a complex number.
Definition: Polar Form of a Reciprocal of a Complex Number
Given a nonzero complex number in the polar form , the reciprocal can be written in the polar form as
The final example will demonstrate how we can use the formula for the product of complex numbers in polar form to find formulae for the powers of complex numbers.
Example 6: Using the Modulus and Argument to Calculate Powers of Complex Numbers in Cartesian Form
Consider the complex number .
- Find the modulus of .
- Find the argument of .
- Hence, use the properties of multiplication of complex numbers in polar form to find the modulus and argument of .
- Hence, find the value of .
Answer
Part 1
Recall that the modulus of a complex number in the Cartesian form is
In our example, and , so we obtain
Hence, the modulus of is 2.
Part 2
To calculate the argument, we first consider which quadrant of an Argand diagram the complex number lies in. Since both its real and imaginary parts are positive, the complex number lies in the first quadrant of an Argand diagram. We recall that the argument of a complex number in the first quadrant is given by . Then,
Hence, the argument of is .
Part 3
We recall the properties of the multiplication of complex numbers in relation to the moduli and the arguments of the complex numbers: for any pair of nonzero complex numbers and , we have
In this example, we need to compute , which can be obtained by multiplying three times:
Since we can obtain the modulus of a product by taking the product of the moduli of the complex numbers, we have that
In part 1, we obtained the modulus , so
Hence, the modulus of is 8.
Similarly, we know that the argument of the product of a complex number is the sum of the arguments of each complex number. So,
In part 2, we obtained that , so
Hence, the argument of is .
Part 4
We recall that a nonzero complex number with modulus and argument has the polar form
In the previous part, we have calculated that the modulus of is 8, so . We also obtained that the argument of is ; hence, . Substituting these values into the polar form,
Since and , we have
Hence, .
Let us finish by recapping a few important concepts from this explainer.
Key Points
- Multiplication and division of complex numbers are often simpler when we work with complex numbers in polar form.
- Given a pair of nonzero complex numbers in the polar form and ,
- the product of the complex numbers in the polar form is
- the quotient of the complex numbers in the polar form is
- For any pair of nonzero complex numbers and , we have
- Given a nonzero complex number in the polar form , the reciprocal can be written in the polar form as
