Dividing Complex Numbers

Dividing complex numbers is a little more complicated than addition, subtraction, and multiplication of complex numbers because it is difficult to divide a number by an imaginary number. For dividing complex numbers, we need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary part of the denominator so that we end up with a real number in the denominator.

In this article, we will learn about the division of complex numbers, dividing complex numbers in polar form, the division of imaginary numbers, and dividing complex fractions.

What is Dividing Complex Numbers?

Dividing complex numbers is mathematically similar to the division of

Steps for Dividing Complex Numbers

Now, we know what dividing complex numbers is, let us discuss the steps for dividing complex numbers. To divide the two complex numbers, follow the given steps:

1. First, calculate the conjugate of the complex number that is at the denominator of the fraction.
2. Multiply the conjugate with the numerator and the denominator of the complex fraction.
3. Apply the algebraic identity (a+b)(a-b)=a² – b² in the denominator and substitute i² = -1.
4. Apply the distributive property in the numerator and simplify.
5. Separate the real part and the imaginary part of the resultant complex number.

Division of Complex Numbers in Polar Form

Important Notes on Dividing Complex Numbers

• To divide a complex number a+ib by c+id, multiply the numerator and denominator of the fraction a+ib/c+id by c−id and simplify.
• The conjugate of the complex z = a+ib is a−ib.
• The modulus of the complex number z = a+ib is |z| = √(a² + b²)

Dividing Complex Numbers Examples

How to Simplify Dividing Complex Numbers?

How to do Division of Complex Numbers?

How Do you Write the Division of Complex Numbers by a Real Number?

What is the Quotient when Dividing Complex Numbers 4+8i by 1+3i?

Multiply and divide complex numbers

Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

EXAMPLE 4: MULTIPLYING A COMPLEX NUMBER BY A REAL NUMBER

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,

HOW TO: GIVEN A COMPLEX NUMBER AND A REAL NUMBER, MULTIPLY TO FIND THE PRODUCT.

1. Use the distributive property.
2. Simplify.

EXAMPLE 5: MULTIPLYING A COMPLEX NUMBER BY A REAL NUMBER

Find the product 4(2+5i).

Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get

HOW TO: GIVEN TWO COMPLEX NUMBERS, MULTIPLY TO FIND THE PRODUCT.

1. Use the distributive property or the FOIL method.
2. Simplify.

EXAMPLE 6: MULTIPLYING A COMPLEX NUMBER BY A COMPLEX NUMBER

Dividing Complex Numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of a+bi is a−bi.

Note that complex conjugates have a reciprocal relationship: The complex conjugate of a+bi is a−bi, and the complex conjugate of a−bi is a+bi. Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.

Suppose we want to divide c+di by a+bi, where neither a nor b equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.

A GENERAL NOTE: THE COMPLEX CONJUGATE

The complex conjugate of a complex number a+bi is a−bi. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

• When a complex number is multiplied by its complex conjugate, the result is a real number.
• When a complex number is added to its complex conjugate, the result is a real number.

EXAMPLE 7: FINDING COMPLEX CONJUGATES

Find the complex conjugate of each number.

Analysis of the Solution

Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by i.

HOW TO: GIVEN TWO COMPLEX NUMBERS, DIVIDE ONE BY THE OTHER.

1. Write the division problem as a fraction.
2. Determine the complex conjugate of the denominator.
3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
4. Simplify.

EXAMPLE 10: SUBSTITUTING AN IMAGINARY NUMBER IN A RATIONAL FUNCTION

Simplifying Powers of i

The powers of i are cyclic. Let’s look at what happens when we raise i to increasing powers.

We can see that when we get to the fifth power of i, it is equal to the first power. As we continue to multiply i by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of i.

EXAMPLE 11: SIMPLIFYING POWERS OF I

Lesson Explainer: Dividing Complex Numbers

In this explainer, we will learn how to perform division on complex numbers.

When a student first encounters complex numbers, expressions like

can seem a little mysterious or, at least, it can seem difficult to understand how one might compute the result. This explainer will connect this idea to more familiar areas of mathematics and help you understand how to evaluate expressions like this. Before we deal with division of complex numbers in general, we will consider the two simpler cases of division by a real number and division by a purely imaginary number.

In many ways, dividing a complex number by a real number is a rather trivial exercise. However, dividing a complex number by an imaginary number is not so trivial as the next example will demonstrate.

Example 2: Dividing a Complex Number by an Imaginary Number

The technique we used above can be generalized to help us understand how to divide any two complex numbers. The first thing we need to do is identify a complex number which when multiplied by the denominator gives a real number. Then, we can multiply both the numerator and the denominator by this number and simplify. The question is, given a complex number 𝑧 ,what number when multiplied by 𝑧 results in a real number? This is the point where we should recall the properties of the complex conjugate, in particular, that for a complex number 𝑧=𝑎+𝑏𝑖,

which is a real number. Hence, by multiplying the numerator and the denominator by the complex conjugate of the denominator, we can eliminate the imaginary part from the denominator and then simplify the result. This technique should not be new to most people. When learning about radicals, we face a similar problem trying to simplify expressions of the form

In this case, we multiply the numerator and the denominator by the conjugate of the denominator. This technique is often called rationalizing the denominator. With complex numbers, in many ways we are using the same technique in the special case where 𝑓 is a negative number.

Now, let’s consider an example where we have to simplify the division of two complex numbers, in a similar way to how we rationalize the denominator with radicals.

Example 3: Dividing Complex Numbers

Answer

We begin by identifying a complex number that when multiplied by the denominator results in a real number. We usually use the complex conjugate of the denominator: 1+5𝑖.Now we multiply both the numerator and the denominator by this number as follows:

How To: Dividing Complex Numbers

To divide complex numbers, we use the following technique (sometimes referred to as “realizing” the denominator):

1. Multiply the numerator and denominator by the complex conjugate of the denominator.
2. Expand the parenthesis in the numerator and denominator.
3. Gather like terms (real and imaginary) remembering that 𝑖²=−1.
4. Express the answer in the form 𝑎+𝑏𝑖 reducing any fractions.

Using this technique, we can actually derive a general form for the division of complex numbers as the next example will demonstrate.

Example 4: General Form of Complex Division

Even though we have derived a general formula for complex division, it is preferable to be familiar with the technique rather than simply memorize the formula.

The fact that 𝑎²+𝑏²=1 in the previous question is no accident. In fact, this is an example of a general rule that if

for some complex number 𝑧,then 𝑎²+𝑏²=1.This can be proved by working through the algebra. However, this is not very enlightening. Instead, results like this are best understood once we learn about the modulus and argument.

Example 6: Solving Complex Division Equations

Solve the equation 𝑧(2+𝑖)=3−𝑖 for 𝑧.

Answer

We begin by dividing both sides of the equation by 2+𝑖 which results in the following equation:

Due to the fact that multiplying and dividing complex numbers in this way can be fairly time consuming, it is useful to consider which approach will be the most efficient. This often involves using properties of complex numbers or noticing factors that we can quickly cancel. The next two examples will demonstrate how we can simplify our calculations.

Example 7: Complex Division

Answer

When presented with an expression like this, it is good to first consider what approach we should take to solving it. We could expand the parenthesis in the numerator and the denominator and then multiply both the numerator and the denominator by the complex conjugate of the denominator. Alternatively, we could split the fraction in two and try to simplify each part then multiply the resulting complex numbers. The approach taken will generally depend on the particular expression given; however, it is good to look for features of the expression that might simplify the calculation. In this case, it is good to notice that we have a common factor of (1+𝑖) in both the numerator and the denominator. By canceling this factor first, we can simplify our calculation. Hence,

We can now multiply both the numerator and the denominator by the complex conjugate of the denominator as follows:

For the next question, we will again look at an example where applying the properties of complex numbers can simplify out calculations.

Example 8: Complex Expressions Involving Division

Answer

It is possible to solve this problem by performing the complex division on both of the fractions and then adding their results. However, we can simplify our calculation by first noticing that we can factor out 3−4𝑖 from both terms. Hence, we can rewrite the expression as

Now we consider the expression in the parentheses; notice that the denominators of the two fractions are a complex conjugate pair; that is, the expression is in the form

Finally, let’s consider an example where we have to find missing values in an equation by dividing complex numbers.

Example 9: Solving a Two-Variable Linear Equation with Complex Coefficients

Answer

In this example, we want to determine the missing values 𝑥 and 𝑦 in a two-variable linear equation with complex coefficients.

The given equation contains 3 separate complex divisions, two on the left hand side and one on the right hand side of the equation. We begin by simplifying each term by performing the complex division. This is achieved by multiplying the numerator and denominator by the complex conjugate of the denominator, which results in a real number in the denominator after distributing over the parentheses.

Let’s summarize some of the key points that we covered in this explainer.

Key Points

• Dividing complex numbers uses the same technique as for rationalizing the denominator.
• To divide complex numbers, we multiply the numerator and the denominator by the complex conjugate of the denominator, and then we expand the brackets and simplify using 𝑖=−1.
• In expressions involving multiplication and division of multiple complex numbers, it is useful to look for common factors or whether we can apply some of the properties of complex numbers to simplify our calculations.

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