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Study Guides > Intermediate Algebra

Multiply and Divide Complex Numbers

Learning Outcomes

  • Multiply complex numbers
  • Find conjugates of complex numbers
  • Divide complex numbers
  • Simplify powers of i

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

 Multiplying a Complex Number by a Real Number

Showing how distribution works for complex numbers. For 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i) = 18 + 6i. Let us begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So,for 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i)(3)(6)+(3)(2i) = 18+6i18 + 6i.

How To: Given a complex number and a real number, multiply to find the product

  1. Use the distributive property.
  2. Simplify.

Example

Find the product 4(2+5i)4\left(2+5i\right).

Answer:

Distribute the 44.

4(2+5i)=(4⋅2)+(4⋅5i)=8+20i\begin{array}{cc}4\left(2+5i\right)=\left(4\cdot 2\right)+\left(4\cdot 5i\right)\hfill \\ =8+20i\hfill \end{array}

 Multiplying Complex Numbers Together

Now, let us 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

(a+bi)(c+di)=ac+adi+bci+bdi2\left(a+bi\right)\left(c+di\right)=ac+adi+bci+bd{i}^{2}

Because i2=−1{i}^{2}=-1, we have

(a+bi)(c+di)=ac+adi+bci−bd\left(a+bi\right)\left(c+di\right)=ac+adi+bci-bd

To simplify, we combine the real parts, and we combine the imaginary parts.

(a+bi)(c+di)=(ac−bd)+(ad+bc)i\left(a+bi\right)\left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i

How To: Given two complex numbers, multiply to find the product

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

Example

Multiply (4+3i)(2−5i)\left(4+3i\right)\left(2 - 5i\right).

Answer:

Use FOIL.

(4+3i)(2−5i)=(4⋅2)+(4⋅(−5i))+(3i⋅2)+((3i)⋅(−5i))=8−20i+6i−15i2=8−20i+6i−15(−1)Substituting -1 for i2.=8−20i+6i+15=(8+15)+(−20+6)iGrouping like terms together.=23−14iSimplifying.\begin{array}{lll}\left(4+3i\right)\left(2 - 5i\right) & =\left(4\cdot 2\right)+\left(4\cdot \left(-5i\right)\right)+\left(3i\cdot 2\right)+\left(\left(3i\right)\cdot \left(-5i\right)\right) & \\ & =8-20i+6i-15i^2 & \\ & = 8-20i+6i-15\left(-1\right) & \text{Substituting -1 for } {i}^{2} \text{.} \\ & = 8-20i+6i+15 & \\ & = \left(8+15\right)+\left(-20+6\right)i & \text{Grouping like terms together.} \\ & =23-14i & \text{Simplifying.}\end{array}

In the first video, we show more examples of multiplying complex numbers. https://youtu.be/Fmr3o2zkwLM

Simplifying Powers of i

There is a pattern to evaluating powers of i. Let us look at what happens when we raise i to increasing powers.

i1=ii2=−1i3=i2⋅i=−1⋅i=−ii4=i3⋅i=−i⋅i=−i2=−(−1)=1i5=i4⋅i=1⋅i=i\begin{array}{ll}{i}^{1}=i\\ {i}^{2}=-1\\ {i}^{3}={i}^{2}\cdot i=-1\cdot i=-i\\ {i}^{4}={i}^{3}\cdot i=-i\cdot i=-{i}^{2}=-\left(-1\right)=1\\ {i}^{5}={i}^{4}\cdot i=1\cdot i=i\end{array}

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 44. Let us examine the next 44 powers of i.

i6=i5⋅i=i⋅i=i2=−1i7=i6⋅i=i2⋅i=i3=−ii8=i7⋅i=i3⋅i=i4=1i9=i8⋅i=i4⋅i=i5=i\begin{array}{l}{i}^{6}={i}^{5}\cdot i=i\cdot i={i}^{2}=-1\\ {i}^{7}={i}^{6}\cdot i={i}^{2}\cdot i={i}^{3}=-i\\ {i}^{8}={i}^{7}\cdot i={i}^{3}\cdot i={i}^{4}=1\\ {i}^{9}={i}^{8}\cdot i={i}^{4}\cdot i={i}^{5}=i\end{array}

Example

Evaluate i35{i}^{35}.

Answer:

Since i4=1{i}^{4}=1, we can simplify the problem by factoring out as many factors of i4{i}^{4} as possible. To do so, first determine how many times 44 goes into 3535: 35=4⋅8+335=4\cdot 8+3.

i35=i4⋅8+3=i4⋅8⋅i3=(i4)8⋅i3=18⋅i3=i3=−i{i}^{35}={i}^{4\cdot 8+3}={i}^{4\cdot 8}\cdot {i}^{3}={\left({i}^{4}\right)}^{8}\cdot {i}^{3}={1}^{8}\cdot {i}^{3}={i}^{3}=-i

 

Q & A

Can we write i35{i}^{35} in other helpful ways?

As we saw in the previous example, we reduced i35{i}^{35} to i3{i}^{3} by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of i35{i}^{35} may be more useful. The table below shows some other possible factorizations.

Factorization of i35{i}^{35} i34⋅i{i}^{34}\cdot i i33⋅i2{i}^{33}\cdot {i}^{2} i31⋅i4{i}^{31}\cdot {i}^{4} i19⋅i16{i}^{19}\cdot {i}^{16}
Reduced form (i2)17⋅i{\left({i}^{2}\right)}^{17}\cdot i i33⋅(−1){i}^{33}\cdot \left(-1\right) i31⋅1{i}^{31}\cdot 1 i19⋅(i4)4{i}^{19}\cdot {\left({i}^{4}\right)}^{4}
Simplified form (−1)17⋅i{\left(-1\right)}^{17}\cdot i −i33-{i}^{33} i31{i}^{31} i19{i}^{19}

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.

In the following video, you will see more examples of how to simplify powers of i. https://youtu.be/sfP6SmEYHRw

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. This idea is similar to rationalizing the denominator of a fraction that contains a radical. To eliminate the complex or imaginary number in the denominator, you multiply by 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+bia+bi is a−bia-bi.

Note that complex conjugates have a reciprocal relationship: The complex conjugate of a+bia+bi is a−bia-bi, and the complex conjugate of a−bia-bi is a+bia+bi. Furthermore, 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+dic+di by a+bia+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.

c+dia+bi where a≠0 and b≠0\frac{c+di}{a+bi}\text{ where }a\ne 0\text{ and }b\ne 0

Multiply the numerator and denominator by the complex conjugate of the denominator.

(c+di)(a+bi)⋅(a−bi)(a−bi)=(c+di)(a−bi)(a+bi)(a−bi)\frac{\left(c+di\right)}{\left(a+bi\right)}\cdot \frac{\left(a-bi\right)}{\left(a-bi\right)}=\frac{\left(c+di\right)\left(a-bi\right)}{\left(a+bi\right)\left(a-bi\right)}

Apply the distributive property.

=ca−cbi+adi−bdi2a2−abi+abi−b2i2=\frac{ca-cbi+adi-bd{i}^{2}}{{a}^{2}-abi+abi-{b}^{2}{i}^{2}}

Simplify, remembering that i2=−1{i}^{2}=-1.

\begin{array}=\frac{ca-cbi+adi-bd\left(-1\right)}{{a}^{2}-abi+abi-{b}^{2}\left(-1\right)}\hfill \\ =\frac{\left(ca+bd\right)+\left(ad-cb\right)i}{{a}^{2}+{b}^{2}}\hfill \end{array}

A General Note: The Complex Conjugate

The complex conjugate of a complex number a+bia+bi is a−bia-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

Find the complex conjugate of each number.

  1. 2+i52+i\sqrt{5}
  2. −12i-\frac{1}{2}i

Answer:

  1. The number is already in the form a+bia+bi. The complex conjugate is a−bia-bi, or 2−i52-i\sqrt{5}.
  2. We can rewrite this number in the form a−bia-bi as 0−12i0-\frac{1}{2}i. The complex conjugate is a+bia+bi, or 0+12i0+\frac{1}{2}i. This can be written simply as 12i\frac{1}{2}i.

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. When dividing by a purely imaginary number, we can simply multiply the numerator and denominator by ii instead of the complex conjugate.

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

Divide (2+5i)\left(2+5i\right) by (4−i)\left(4-i\right).

Answer:

We begin by writing the problem as a fraction.

(2+5i)(4−i)\frac{\left(2+5i\right)}{\left(4-i\right)}

Then we multiply the numerator and denominator by the complex conjugate of the denominator.

(2+5i)(4−i)⋅(4+i)(4+i)\frac{\left(2+5i\right)}{\left(4-i\right)}\cdot \frac{\left(4+i\right)}{\left(4+i\right)}

To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).

(2+5i)(4−i)⋅(4+i)4+i)=8+2i+20i+5i216+4i−4i−i2=8+2i+20i+5(−1)16+4i−4i−(−1)Because i2=−1=3+22i17=317+2217iSeparate real and imaginary parts\begin{array}{rcll}\Large{\frac{(2+5i)}{(4-i)}}\cdot\frac{(4+i)}{4+i)}&=&\Large{\frac{8+2i+20i+5{i}^{2}}{16+4i - 4i-{i}^{2}}}& \\&=&\Large{\frac{8+2i+20i+5(-1)}{16+4i - 4i-(-1)}}&\quad\text{Because }i^2=-1\\&=&\Large{\frac{3+22i}{17}}& \\&=&\Large{\frac{3}{17}+\frac{22}{17}i}&\quad\text{Separate real and imaginary parts}\end{array}

 

Note that this expresses the quotient in standard form.

In the last video, you will see more examples of dividing complex numbers. https://youtu.be/XBJjbJAwM1c

Summary

Multiplying complex numbers is similar to multiplying polynomials. Remember that an imaginary number times another imaginary number gives a real result. When you divide complex numbers, you must first multiply by the complex conjugate to eliminate any imaginary parts, and then you can divide.

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