In this explainer, we will learn how to convert a complex number from the algebraic to the exponential form (Eulerβs form) and vice versa.
Let us begin by recalling the polar form of a complex number. If a complex number has modulus and argument , then the polar form of the complex number is
This gives us a way to represent the complex number using only the modulus and the argument. The exponential form of a complex number is an alternative and simpler way to represent a complex number using its modulus and argument. To obtain the exponential form of a complex number, we need to understand how to simplify the expression , which appears in the polar form. For this purpose, we introduce a very important formula credited to a Swiss mathematician, Leonhard Euler (1707β1783).
Formula: Eulerβs Formula
For any real number ,
This equation is known as Eulerβs identity or Eulerβs formula. This remarkable identity links the exponential function , trigonometric functions, and the imaginary unit . We can visualize Eulerβs formula on an Argand diagram as sweeping out a unit circle centered at the origin.
A particular case of Eulerβs formula is when , we have
By adding 1 to both sides, we derive the famous identity
This equation is considered by many to be an example of mathematical beauty because by using three of the most fundamental operations in mathematics (addition, multiplication, and exponentiation) just once, it links the five fundamental constants of mathematics: 0, 1, , , and .
We can prove Eulerβs formula using Maclaurin series. The Maclaurin or Taylor series of a function allows us to extend a real variable function to take complex-valued inputs, as long as the series converges. We begin with the Maclaurin series expansion for :
Since the left-hand side of Eulerβs formula has , we substitute into this equation to write
Recall that the integer powers of form a cycle for an integer as follows:
Hence, we have
Gathering the real and imaginary parts separately, we have
Recall that the Maclaurin series of sine and cosine are
Replacing the power series in the real and imaginary parts of with suitable Maclaurin series, we obtain
This proves Eulerβs formula.
In our first example, we will use Eulerβs formula to write and in terms of complex exponential functions.
Example 1: Relationship between Sine, Cosine, and the Exponential Function
Use Eulerβs formula to express and in terms of and .
Answer
Recall Eulerβs formula:
We begin by applying Eulerβs formula to express by substituting in place of in the formula above:
Recall the even/odd identities for sine and cosine:
Substituting these identities above, we can rewrite
Now, we have two identities:
We can think of these as simultaneous equations where the unknowns are and . If we add these two equations, we can remove the sine function and find the expression for in terms of and . Also, if we subtract these equations, we can remove the cosine function and find the expression for in terms of and .
Let us first find the expression for in terms of and by adding the two simultaneous equations:
Dividing by 2, we get
This gives the expression for in terms of and .
Next, let us find a similar formula for by taking the difference of the simultaneous equations:
Dividing by , we arrive at
This gives the expression for in terms of and .
Hence, we have obtained
In the previous example, we applied Eulerβs formula to express the sine and cosine functions in terms of complex exponential functions. The primary application of Eulerβs formula in this explainer is to convert the polar form of a complex number to the exponential form.
Recall that the polar form of a complex number with modulus and argument is
Eulerβs formula tells us that the expression inside the parentheses is equal to . When we make this substitution, we obtain the exponential form of a complex number.
Definition: Exponential Form of a Complex Number
The exponential form of a nonzero complex number with modulus and argument is
The standard range of the argument in the exponential form is radians.
To write the exponential form of a complex number, we need to know its modulus and argument, which are the same characteristics required for the polar form of the complex number. Hence, converting complex numbers between the polar and exponential forms is a simple task.
Let us consider an example where we will convert a complex number from the polar form to the exponential form.
Example 2: Converting Complex Numbers from Polar to Exponential Form
Put in exponential form.
Answer
Recall that the exponential form of a nonzero complex number with modulus and argument is
We are given the complex number in a form that is close to the polar form. We know that the polar form of a nonzero complex number with modulus and argument is
Since we know the polar form of a complex number, we can identify the modulus and argument of the complex number, which then would lead to the exponential form of the complex number.
However, the given form of our complex number has a negative sign in front of the imaginary part, which means that it is not exactly the polar form. We need to first convert this number to the polar form so that we can identify its modulus and argument. For this purpose, we need to recall that the sine function is odd and the cosine function is even. This leads to the identities
Applying these identities to our number, we can write
This is the polar form of . This means that the modulus of is , and the argument of is . We recall that the standard range of arguments is , so our argument is within the correct range. This leads to the exponential form
In the previous example, we converted a complex number from the polar form to the exponential form. This process is simple since the modulus and argument of a complex number are already provided in the polar form. If we start with the Cartesian form of a complex number, we must first compute the modulus and argument of the complex number in order to obtain the exponential form. Let us consider an example where we convert a complex number from the Cartesian form to the exponential form.
Example 3: Converting Complex Numbers from Cartesian to Exponential Form
Put the number in exponential form.
Answer
Recall that the exponential form of a nonzero complex number with modulus and argument is
We start by finding the modulus of . Recall that the modulus of a complex number is given by . Substituting the real part and imaginary part into this formula and simplifying, we have
This tells us that the modulus of is .
Let us now find the argument of . Notice that since the real part is positive and the imaginary part is negative, lies in the fourth quadrant of an Argand diagram. Recall that the argument of a complex number in the fourth quadrant is given by . Hence,
This means that the argument of is . We recall that the standard range of arguments is , so our argument is within the correct range.
Using the modulus and the argument , we can express in the exponential form as
In the previous example, we converted a complex number in the Cartesian form to the exponential form. We now consider the converse process. To convert a complex number in the exponential form to the Cartesian form, we need to first convert it to the polar form. We can then convert the polar form of the complex number to the Cartesian form. In the next example, we will demonstrate this process.
Example 4: Converting Complex Numbers from Exponential to Cartesian Form
Put in Cartesian form.
Answer
We are given the exponential form of a complex number. To convert a complex number in the exponential form to the Cartesian form, we need to first convert it to the polar form. We can then convert the polar form of the complex number to the Cartesian form.
Recall that the exponential form of a nonzero complex number with modulus and argument is
We also recall that the polar form of a nonzero complex number with modulus and argument is
We can see that the exponential and polar forms share the same parameters and , which makes the conversion simpler.
From the given exponential form, we obtain modulus and argument . Hence, the polar form of is
To convert the polar form of a complex number to the Cartesian form, we need to multiply through the parentheses and evaluate the trigonometric ratios. Since is a special angle, we recall the trigonometric ratios
Substituting these values into the polar form of and simplifying, we obtain
Hence, the Cartesian form of the given complex number is
Using the properties of the modulus and argument under multiplication, the multiplication of a pair of nonzero complex numbers is simpler in the polar or exponential form. Let us recall these properties:
This leads to the multiplication rule for complex numbers in the exponential form.
Rule: Multiplication of Complex Numbers in Exponential Form
Let and be complex numbers in the exponential form. The product in the exponential form is
If we recall the law of exponents that states then the multiplication rule for the exponential form is very intuitive. In fact, this rule states that the law of exponents still holds when the exponents are complex numbers. This is true as long as the base of the exponent is a positive real number.
Let us consider an example where we will multiply a pair of complex numbers in the exponential form.
Example 5: Multiplication of Complex Numbers in Exponential Form
Given that and , express in the form .
Answer
Recall the multiplication rule for complex numbers in the exponential form,
Since we are given both complex numbers and in the exponential form, we can apply this rule with the values
Substituting these values into the multiplication rule, we obtain
This leads to the polar form of the complex number .
To convert a complex number in the exponential form to the Cartesian form, we need to first convert it to the polar form. We can then convert the polar form of the complex number to the Cartesian form.
Recall that the exponential form of a nonzero complex number with modulus and argument is
We also recall that the polar form of a nonzero complex number with modulus and argument is
We can see that the exponential and polar forms share the same parameters and , which makes the conversion simpler.
From the exponential form , we obtain modulus 30 and argument . Hence, the polar form of is
To convert the polar form of a complex number to the Cartesian form, we need to multiply through the parentheses and evaluate the trigonometric ratios. Using the unit circle for trigonometric ratios, we can find the trigonometric ratios
Substituting these values into the polar form of and simplifying, we obtain
Hence, the Cartesian form of the given complex number is
In the previous example, we multiplied complex numbers in the exponential form. Let us now turn our attention to the division, or the quotient, of a pair of nonzero complex numbers in the exponential form. We recall the properties of the modulus and argument under division,
This leads to the division rule for complex numbers in the exponential form.
Rule: Division of Complex Numbers in Exponential Form
Let and be nonzero complex numbers in the exponential form. The quotient in the exponential form is
Similar to the multiplication rule for complex numbers in the exponential form, this rule can be seen as a generalization of the law of exponents that states
This rule also is valid when the exponent is complex valued, as long as the base of the exponent is a positive real number.
In the next example, we will consider a quotient of complex numbers in the Cartesian form. We will first convert the complex number into the exponential form and carry out the division using this rule.
Example 6: Division of Complex Numbers
Given that , write in exponential form.
Answer
We have two methods for this example: we can either convert each number to exponential form and then use their properties to carry out the division, or we can carry out the division with the complex numbers in their current form and then convert the result. Since we know that the division of complex numbers is simpler in the exponential form, we will choose the first method. We recall the division rule for a pair of nonzero complex numbers in the exponential form:
We begin by converting the complex numbers in the numerator and the denominator of the given quotient. First, consider the complex number in the numerator, . This is a purely imaginary number, where the imaginary part is positive. In an Argand diagram, this number is plotted in the positive imaginary axis, which means that its argument is . Also, since it is purely imaginary, the absolute value of the imaginary part of this number is equal to the modulus. This means that the modulus is . Hence, we can express the numerator in the exponential form as
Next, let us convert the denominator of the quotient to the exponential form. Recall that the modulus of a complex number is . For the denominator , we have and ; thus, its modulus is
For the argument of this number, we first note that lies in the fourth quadrant in an Argand diagram since it has a positive real part and a negative imaginary part. Recall that the argument of a complex number in the fourth quadrant is given by . Thus,
Hence, we can express the denominator in the exponential form as
Now that we have obtained the exponential form of both complex numbers, we can write
Therefore, we can now apply the division rule with the values
Substituting these into the division rule above, we obtain
We recall that the standard range of arguments is , so our argument is within the correct range.
Hence, the exponential form of the given quotient is .
We have considered the properties of the exponential form under the multiplication and the division of complex numbers. Let us now consider the property of the exponential form relating to the complex conjugate.
We recall the properties of the modulus and argument of a complex number under the conjugate operation. Given any nonzero complex number , we have
These properties lead to the following rule.
Rule: Conjugate of Complex Numbers in Exponential Form
Given a nonzero complex number in the polar form , its conjugate in the polar form is
This statement tells us that the complex conjugate operation can go through the exponential function. We can see that the right-hand side of the statement can be obtained by taking the complex conjugate of the exponent only. This also is true as long as the base of the exponent is a positive real number.
Let us consider an example involving the conjugate property of the exponential form.
Example 7: Adding Complex Numbers in Exponential Form
Find the numerical value of
Answer
In this example, we are adding two complex numbers in the exponential form. We can notice that the modulus of both complex numbers is the same, and their arguments have opposite signs. We recall the conjugate rule of complex numbers in the exponential form:
Using this rule, we can notice that the second number in the sum is the conjugate of the first number in the sum. We recall the property of complex conjugates: for any complex number ,
Hence, we can simplify our expression:
We only need to compute the real part of this number. We recall Eulerβs formula:
From Eulerβs formula, we know that
Hence,
In our final example, we will use various properties of the exponential form to solve a problem.
Example 8: Properties of Complex Numbers in Exponential Form
Given that , , and , find all possible values of , expressing them in exponential form.
Answer
In this example, we must find all possible expressions of a complex number from given information about a quotient. The quotient also includes a square, which is a multiplication of complex numbers. We know that the multiplication and division are simplified when using the exponential form, so we will solve the problem using various properties of the exponential form.
Recall that the exponential form of a nonzero complex number with modulus and argument is
Let us begin by converting into the exponential form. For this, we need to find its modulus and argumment. Recall that the modulus of a complex number is . For , we have and . Hence,
We now find the argument of . Since has negative real and imaginary parts, it lies in the third quadrant. Hence, its argument is
Hence, we can write in the exponential form:
Now, let us say that the complex number has modulus and argument , so that its exponential form is written as
We will compute the quotient to relate to the given information. We begin with the denominator by recalling the multiplication rule for complex numbers in the exponential form,
Since and , we can write
Next, to compute the quotient , we recall the division rule for a pair of nonzero complex numbers in the exponential form:
Since we have already obtained and , we can write
This gives us the quotient in the exponential form. In particular, this tells us that the modulus of this quotient is . We are given that this modulus is equal to . Since , we know . Hence,
Simplifying so that the subject of the equation is ,
Since is the modulus of the complex number , it cannot be negative. Thus, we have .
Now, the only information we have not used is . This tells us that the imaginary part of the quotient is equal to zero, which means that the complex number on the Argand diagram would lie on the real axis. In order for the argument to lie on the real axis, the argument of this complex number must be an integer multiple of . Hence,
Let us rearrange the equation so that is the subject:
Since is the argument of the complex number , it should be in the range . We need to find all possible values of in this range. That is, we need to identify integer values satisfying
Adding to each part of the inequality above,
We can multiply to each part of the inequality, reversing the direction of the inequalities:
Hence, the set of integer values satisfying above is . Substituting these values of into (1), we obtain
We recall that the standard range of arguments is , so all arguments of complex numbers above are within the correct range. Together with the modulus , the exponential forms of all possible values of are
Let us finish by recapping a few important concepts from this explainer.
Key Points
- For any real number , Eulerβs formula tells us that The standard range of the argument in the exponential form is radians.
- A complex number with modulus and argument is written in the exponential form as
- The exponential form of complex numbers simplifies
multiplication, division, and conjugation by the following rules:
- Multiplication rule:
- Division rule:
- Conjugation rule: