How to Multiply and Divide Complex Numbers

How to Multiply and Divide Complex Numbers

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Multiplying complex numbers

The process of multiplying complex numbers is quite similar to multiplying binomials. The most significant difference is that we work separately on the real and imaginary parts.

We have two methods for multiplying complex numbers: the distributive property and the FOIL technique. Remember that FOIL stands for multiplying together the First, Outer, Inner, and Last terms. We get the same answer whether we use the distributive property or the FOIL method:

(a + bi) × (c + di) = ac + adi + bci + bdi2

We know that i2 = 1, so we have: ac + adi + bci  bd

Now simplify to get the final answer: ac + adi + bci  bd = (ac  bd) + i(ad + bc)

Dividing complex numbers

Adding, subtracting, and multiplying is easier than dividing two complex numbers. This is because we can't divide by an imaginary number, so any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator to eliminate the imaginary part of the denominator and get a real number as the denominator. This term is called the complex conjugate of the denominator. It is found by changing the sign of the imaginary part of the complex number. In other words, the conjugated form of a + bi is a  bi, and the conjugated form of a  bi is a + bi

Also, complex solutions to a quadratic equation with real coefficients are always complex conjugates of each other.

Let's say we want to divide a + bi by c + di when c, d  0. First, we write the division as a fraction. Then, we find the complex conjugate of the denominator and multiply:

a + bic + di

Multiply both the numerator and the denominator by the denominator's complex conjugate.

a + bic + di×c  dic  di = (a + bi) (c  di)(c + di) (c  di)

Use the distributive property: ac  adi + bci  bdi2c2  d2i2

Now Simplify, (i2 = 1)

ac  adi + bci + bdc2 + d2 = (ac + bd)+ i(ad + bc)c2 + d2

Free printable Worksheets

Exercises for Multiplying and Dividing Complex Numbers

1) Find the answer: (2i) × (5i)

2) Find the answer: (2 + 3i) × (4 + 2i)

3) Find the answer: (5  i) × (2 + 3i)

4) Find the answer: (7 + 4i) × (8  i)

5) Find the answer: 3i × (7 + 12i)

6) Find the answer: 3  2i4 + 3i

7) Find the answer: 3  i2 + 4i

8) Find the answer: 5  9i3 + 2i

9) Find the answer: 9i5 + 6i

10) Find the answer: 5 + i3i

 

1) Find the answer: (2i) × (5i)

(2i) × (5i) = 10

2) Find the answer: (2 + 3i) × (4 + 2i)

2(4) + 2(2i) + (3i)(4) + (3i)(2i) = 8 + 4i + 12i + 6(1) = 2 + 16i

3) Find the answer: (5  i) × (2 + 3i)

5(2) + 5(3i)  (i)(2)  (i)(3i) = 10 + 15i  2i  3(1) = 13 + 13i

4) Find the answer: (7 + 4i) × (8  i)

(7)(8) + (7)(i) + (4i)(8) + (4i)(i) = 56 + 7i + 32i  4(1) = 52 + 39i

5) Find the answer: 3i × (7 + 12i)

(3i)(7) + (3i)(12i) = 21i + 36(i2) = 21i + 36(1) = 21i  36

6) Find the answer: 3  2i4 + 3i

3  2i4 + 3i = 3  2i4 + 3i×4  3i4  3i = 12 + 9i  8i + 6(1)16  9(1) = 18 + i25 = 1825 + i25

7) Find the answer: 3  i2 + 4i

3  i2 + 4i = 3  i2 + 4i×2  4i2  4i = 6  12i + 2i  44  (16) = 10  10i20 = 1020  10i20 = 12  i2

8) Find the answer: 5  9i3 + 2i

5  9i3 + 2i = 5  9i3 + 2i×3  2i3  2i = 15  10i + 27i + 18(1)9  4(1) = 33 + 17i13 = 3313 + 17i13

9) Find the answer: 9i5 + 6i

9i5 + 6i = 9i5 + 6i×5  6i5  6i = 45i + 54(1)25  36(1) = 45i  5461 = 45i61  5461

10) Find the answer: 5 + i3i

5 + i3i = 5 + i3i×ii = 5i + (1)3(1) = 5i3  13

Related Topics

How to Multiply and Divide Complex Numbers
How to Graph Complex Numbers
How to Solve Rationalizing Imaginary Denominators

Multiplying and Dividing Complex Numbers Practice Quiz