In this explainer, we will learn how to convert a complex number from the algebraic to the exponential form (Euler’s form) and vice versa.
Definition: Exponential Form of a Complex Number
A complex number can be written in the form where is the modulus and is the argument expressed in radians.
If we compare the exponential form to the polar form, we find canceling , we arrive at
This formula was derived by the famous mathematician Leonhard Euler (pronounced “Oiler”) and is called Euler’s formula or Euler’s relation. In many ways, it is a remarkable formula linking the exponential function , sine, cosine, , and the imaginary unit . We can visualize Euler’s formula on an Argand diagram as sweeping out a unit circle centered at the origin.
Another way to visualize it is to consider it in three dimensions, where the value of is along one axis and the plane perpendicular to the -axis is the complex plane.
This three-dimensional visualization is closely related to the electrodynamical concept of circular polarization for waves.
Returning to Euler’s formula, if we set , we get
By adding one to both sides, we derive the famous Euler’s 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 will begin by considering a method to derive Euler’s relation using power series. We start with the Maclaurin series expansion for :
Substituting into this equation gives
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
The Maclaurin series of sine and cosine are
Hence, we can see
To convert a complex number from algebraic to exponential form, we use a very similar technique as that used for converting between algebraic and trigonometric forms. A summary of how to do this is in the box below.
How To: Converting a Complex Number from Algebraic Form to Exponential Form
To convert a complex number in algebraic form, , to exponential form:
- Find the modulus, , of the complex number;
- Find the argument, , of the complex number;
- Write the number in exponential form: where and .
Example 1: Converting Complex Numbers from Algebraic to Exponential Form
Put the number in exponential form.
We start by finding the modulus of . Substituting the real and imaginary parts into the formula, we have
Simplifying, we get
We now find the argument of . Notice that since the real part is positive and the imaginary part is negative, lies in the fourth quadrant. Hence, we can find the argument by evaluating the inverse tangent as follows:
Simplifying this fraction gives
Hence, we can express in exponential form as
Given that the argument is -periodic, we could equally add to the argument and express in exponential form as
To convert from exponential form to algebraic form, we rewrite the number in polar form; then, we can convert from this form to algebraic form. In the next example, we will demonstrate this process.
Definition: Converting from Exponential to Algebraic Form
To convert a complex number to algebraic form, we first convert it to polar form:
Expanding the brackets and evaluating sine and cosine, we arrive at a complex number in algebraic form:
Using the properties of the modulus and argument, we can derive the rules for multiplication and division of complex numbers in polar form. Recall that for two complex numbers, and , their product can also be expressed in exponential form as , where is the modulus and is the argument. Using the properties of the modulus and argument for multiplication, we can see that and . Hence,
Likewise, using the properties of the modulus and argument for division, we can see that
Notice that if we express the argument as an exponential, we can write
Although the previous derivations might give the impression that all of the rules of exponents apply to complex numbers in general, this is, unfortunately, not true. For example, consider the rule , which applies for . If we let and be negative, this rule can no longer be applied in general; for example,
This just goes to demonstrate that we need to be careful when dealing with complex exponents and bases.
We will now look at some examples using the properties of multiplication and division.
Example 2: Multiplication of Complex Numbers in Exponential Form
Given and , express in the form .
Using the multiplicative properties of the exponential form of a complex number, we can write
To convert this to algebraic form, we first express it in polar form as follows:
Expanding the brackets and evaluating sine and cosine, we have
Example 3: Division of Complex Numbers
Given that , write in exponential form.
When faced with a question like this, we have one of two options: 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 have a division to evaluate, it is usually easier to do this with complex numbers in polar form. Hence, we will begin by converting each number to exponential form. Firstly, the numerator is . This is a purely imaginary number; hence, its argument will be . Additionally, its modulus is ; hence, we can express it in exponential form as . As for the denominator, its modulus is , and since it lies in the fourth quadrant, its argument can be calculated by evaluating . Hence, we can express this in exponential form as . Therefore,
Using the properties of division for complex numbers in exponential form, we can rewrite this as
Example 4: Converting Complex Numbers from Exponential to Algebraic Form
Given that , find the algebraic form of .
We begin by separating out the real and imaginary parts of the exponent as follows:
We can now convert this to polar form as follows:
Expanding the brackets and evaluating sine and cosine give us the algebraic form of as follows:
We can also use Euler’s formula to express sine and cosine in terms of the exponential function as the next example will demonstrate.
Example 5: Relationship between Sine, Cosine, and the Exponential Function
Use Euler’s formula to express and in terms of and .
We begin by using Euler’s formula to express in terms of sine and cosine:
Using the even/odd identities for sine and cosine, we can rewrite this as
Adding this to Euler’s formula, we have
Dividing by 2, we get
Similarly, we can derive a formula by sine by considering the difference of equation (1) with Euler’s formula as follows:
Dividing by , we arrive at
We will now consider a number of examples where we can use the properties of complex numbers in exponential form to solve problems.
Example 6: Solving Equations Involving Complex Numbers in Exponential Form
Given that , where and , find and .
To solve a problem like this, we want to get the left- and right-hand sides into the same forms. Currently, we have complex numbers in exponential form on the left and a complex number in algebraic form including sine and cosine on the right. We should, therefore, convert the complex number in exponential form to algebraic form. We start by expressing them in polar form:
Using the even/odd trigonometric identities, we can rewrite this as
Expanding the brackets and gathering like terms, we can express this as
We can now equate this with the right-hand side to get
Since we know that , we can equate the real and imaginary parts to get the simultaneous equations
Adding these equations together, we get . Hence, . Substituting this back into one of the equations yields .
Example 7: Properties of Complex Numbers in Exponential Form
Given , , and , find all possible values of , expressing them in exponential form.
We begin by considering what each equation tells us about the value of . Firstly, consider the equation
Using the properties of the modulus, we can rewrite the left-hand side as
Setting this equal to the right-hand side, we have
By dividing both sides by and multiplying by , we get
We now consider the equation
What does this tell us? It tells us that is a real number. Consequently, its argument, which we will denote , is either 0 (for a positive real number) or (for a negative real number). Hence, we can write
Using the properties of the argument, we can rewrite this as
Rearranging, we get an expression for the argument of :
We now find the argument of . Since has negative real and imaginary parts, it lies in the third quadrant. Hence, its argument
Therefore, we can write
Now we consider the two cases of and . When , we have
Hence, we can express in exponential form as
When , we have
Hence, we can express in exponential form as
Hence, the two possible values of are
Example 8: Working with Complex Numbers in Exponential Form
Find the numerical value of .
We could approach this problem by converting each number to algebraic form and working through everything. However, we can save ourselves some calculation by being able to recognize a complex conjugate pair when presented in exponential form. Recall that, for any complex number ,
Hence, . Therefore, we have the sum of a complex number and its conjugate. Now if we recall the properties of complex conjugates (i.e., for any complex number, ), we can simplify our expression:
From Euler’s formula, we know that the real part of . Hence,
- We can express a complex number in exponential form as where is the modulus and is the argument expressed in radians.
- Working with numbers in exponential form can simplify calculations involving multiplication, division, and powers.
- Using Euler’s formula, we can derive expressions for trigonometric functions such as
- For two complex numbers and ,