Arithmetic with Complex Numbers

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ARITHMETIC WITH COMPLEX NUMBERS

Jump right to the exercises!

Whenever you get a new mathematical object (like complex numbers), you need to develop tools to work with the new object.
In this section, you'll learn how to do basic arithmetic with complex numbers.
Remember the key fact: i:=- 1 , so that i2 =-1 .

Square Roots of Negative Numbers

Complex numbers allow us to make sense of the square root of negative numbers, like -4 :
Notice that (2i)² = (2i)(2i) = 4i² = -4 .
Thus, 2i is a (complex) number which, when squared, gives -4 .
We write: -4=2i .

SQUARE ROOTS OF NEGATIVE NUMBERS

Let p be a positive real number, so that -p is a negative real number.
Then: -p   =  ip

Some of my students like to think of it this way:
You can slide a minus sign out of a square root, and in the process, it turns into the imaginary number i !

sliding a minus sign out of a square root

sliding a minus sign out of a square root

Recall these two different questions, with two different answers:
What is 4 ?
Answer:   4= 2
The number 4 represents the nonnegative number which, when squared, gives 4 .

What are the solutions to the equation x2 =4 ?
Answer:   x=±2
There are two real numbers which, when squared, give 4 .

Similarly, there are two different questions about complex numbers, with two different answers:
What is -4 ?
Answer:   -4= 2i

What are the (complex) solutions to the equation x2 =-4 ?
Answer:   x=±-4 ,    so x=±2i
There are two complex numbers which, when squared, give -4 .

Powers of i

Let's look at powers of i :

i¹ = i	i² = -1	i³ = -i	i⁴ = 1
i⁵ = i	i⁶ = -1	i⁷ = -i	i⁸ = 1

Notice that the same four values occur over and over again: i , -1 , -i , and 1 .
Multiplying by i causes a 90° counter-clockwise rotation (about the origin) in the complex plane!

multiplying by i in the complex plane

This makes it easy to compute the value of i raised to any (whole number) power:
just write the power as a multiple of four plus remainder,
and notice that the remainder determines the final value, as illustrated below:

i2959 =i4 (739)+3 =i4( 739) i3= (i4 )739 i3 =1739 i 3=i 3=-i
Notice that the familiar laws of exponents are being used here.
Even though you'll be able to jump right to the final result based on the remainder,
you should be prepared to show all work leading to your answer on a problem like this.

Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers is easy:
just work with the real and imaginary parts separately.
Here are some examples:

(3+4i ) + (-7+ 5i) = (3 -7) + (4 i+5i) = -4+9i

i-3 +2i- 5i+6 = ( -3+6) + (i+2i- 5i) = 3- 2i

(2+3i ) - (1-4 i) = 2+3 i-1+4i =1+7i

You don't necessarily have to write out any intermediate steps.
Just make one pass through the expression combining all the real parts.
Make a second pass through the expression combining all the imaginary parts.

Precisely, we have:

ADDING AND SUBTRACTING COMPLEX NUMBERS

Let a , b , c , and d be real numbers, and let i:=- 1 .

Then, (a+bi) + (c+di) = (a+c) + (b+d) i
(a+bi) - (c+di) = (a-c) + (b-d) i

Multiplying Complex Numbers

Multiplying complex numbers is also easy: just use FOIL!
For example,

(3-4i )(2+5i )	=6+15i -8i-20 i2	(FOIL)
	= 6+7i-20 (-1)	(combine like terms; i² = -1 )
	=6 +7i+20	(simplify)
	=26+7i	(simplify)

Your goal should be to go from the original expression to the final answer in one step,
without having to write down any intermediate results.
Since you'll just think of using FOIL, you certainly won't memorize the following result.
It is included here only for completeness:

MULTIPLYING COMPLEX NUMBERS

Let a , b , c , and d be real numbers, and let i:=- 1 .

Then: (a+bi) (c+di) = (ac-bd) + (ad+bc) i

Finding all Complex Number Solutions for Simple Quadratic Equations

Question: Find all complex number solutions of the equation x2 +5=2 .
Be sure to write a nice, clean list of equivalent equations.

Solution:
x2 +5=2
x2 =-3
x=± -3
x=± i 3

Whereas it is conventional to write 2i (and not i2 ),
it is also a good idea to write i 3 (and not 3i ).
The reason is this: if you're not really careful about "closing the radical" when handwriting 3i ,
then it can lend itself to confusion about whether the i is inside the radical, or not.
Putting the i first completely eliminates any potential confusion.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.