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Complex Numbers

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides practice problems related to development of concepts related to complex numbers.

A complex number is written in the form a + b i a+bi where a a and b b are real numbers. a a is the “real part” and b i bi is the “imaginary part.”

Examples are: 3 + 4 i 3+4i ( aa is 3, b b is 4) and 3 – 4 i 3–4i ( a a is 3, b b is –4).

Exercise 1

Is 4 a complex number? If so, what are a a and b b? If not, why not?

Exercise 2

Is i i a complex number? If so, what are a a and b b? If not, why not?

Exercise 3

Is 0 0 a complex number? If so, what are a a and b b? If not, why not?

All four operations—addition, subtraction, multiplication, and division—can be done to complex numbers, and the answer is always another complex number. So, for the following problems, let X=3+4⁢i X 3 4 i and Y=5−12⁢i Y 5 12 i . In each case, your answer should be a complex number, in the form a+b⁢i a b i .

Exercise 4

Add: X + Y X+Y.

Exercise 5

Subtract: X – Y X–Y.

Exercise 6

Multiply: X Y XY

Exercise 7

Divide: X / Y X/Y. (*To get the answer in a + b i a+bi form, you will need to use a trick we learned yesterday.)

Exercise 8

Square: X 2 X 2

The complex conjugate of a complex number a+b⁢i a b i is defined as a−b⁢i a b i . That is, the real part stays the same, and the imaginary part switches sign.

Exercise 9

What is the complex conjugate of ( 5 – 12 i ) (5–12i)?

Exercise 10

What do you get when you multiply ( 5 – 12 i ) (5–12i) by its complex conjugate?

Exercise 11

Where have we used complex conjugates before?

For two complex numbers to be equal, there are two requirements: the real parts must be the same, and the imaginary parts must be the same. In other words, 2 + 3 i 2+3i is only equal to 2 + 3 i 2+3i. It is not equal to 2 + 3 i 2+3i or to 3 + 2 i 3+2i or to anything else. So it is very easy to see if two complex numbers are the same, as long as they are both written in a + b i a+bi form: you just set the real parts equal, and the imaginary parts equal. (If they are not written in that form, it can be very tricky to tell: for instance, we saw earlier that 1i1i size 12{ { {1} over {i} } } {} is the same as –i–i!)

Exercise 12

If 2 + 3 i = m + n i 2+3i=m+ni, and mm and nn are both real numbers, what are mm and nn?

Exercise 13

Solve for the real numbers xx and yy: ( x – 6 y ) + ( x + 2 y ) i = 1 –3 i (x–6y)+(x+2y)i=1–3i

Finally, remember…rational expressions? We can have some of those with complex numbers as well!

Exercise 14

4+2i3+2i5−3i7−i4+2i3+2i5−3i7−i size 12{ { { left ( { {4+2i} over {3+2i} } right )} over { left ( { {5 - 3i} over {7 - i} } right )} } } {} Simplify. As always, your answer should be in the form a + b i a+bi.

Exercise 15

4+2i3+2i−5−3i7−i4+2i3+2i−5−3i7−i size 12{ { {4+2i} over {3+2i} } - { {5 - 3i} over {7 - i} } } {} Simplify.

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