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Working with Complex Numbers

This is an easy step forward, trust that and you will not make this harder than

it needs to be. Often, when people see a new symbol in math, they believe it to

be something fundamentally new. This is not.

1st : How would you simplify (3 – x) + (6 + 2x)? We can use the associativeand commutative properties to change the expression to (3 + 6) + (–x + 2x)?

With like terms together we can combine to leave only 9 + x. Simple right?

2nd: You may be wondering why we just spent time talking about such a

simple example. Here is why. How would you simplify (3 – i ) + (6 + 2i )? We

can use the associative and commutative properties to change the

expression to (3 + 6) + (–i + 2 i )? With like terms together we can combine

to leave only 9 + i . Just like above.

Try another example: (–2 + i ) – (3 – 7i )

After you try this on your own, look at the back side of this paper (or

end of the document) to check your answer.

But what is this new symbol i ?

3rd: What we have seen is that  i behaves the same as x for addition and

subtraction. This is because in equations x represents any number. And this

new symbol just represents another number. This symbol came into

existence when people were trying to define numbers such as √ . So far

we have pretended that numbers such as these did not exist. Now we are

changing the game (as happens repeatedly in K-12 math) by simply

defining a new number for the square root of negative one, √  = i . Using

rules of radicals we see that i 2 = i i   √ √  = √  =   = 1.

So above √ , by the product rule for radicals also equals √ √ . And

this can be simplified (by substitution) with our definition of √  = i to give

i √ .

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4th: In this last note we will look at multiplication. On your own, try to

simplify . Next replace the x with i . Look at the bottom

of this page to check your answer.

Now that we know that i 2 = i i   √ √  = √  =   = 1. We

can see that if we will have one more step of simplification. The term

 will turn into . By substitution and knowing that  , we will

have . Combine this with the other constant piece and we

are done.

Let’s look at one done out. Use the FOIL method to expand the product:

 

But since we know that 

we can substitute and get  .

Now , so rewrite as .

And now we see the like terms 40 and 3. Combine these to get 43 + 19i .

DONE!  

Congratulations, you now know how to work with Imaginary and

Complex numbers. An imaginary number is one that just has i s.

Examples: 3i , 4i , -77i , 137i   .

A complex number is a combination of both real and imaginary numbers.

Examples: .   

Combining with “addition”: (–2 + i ) – (3 – 7i ) should turn into –2 – 3 + i + 7i 

after you distribute the minus sign into the second set of parentheses.

Combining the like terms will leave you with–

5 + 8i 

.Combining with “multiplication”:

 

In the next step, when we substitute using x = i we get  which

reduces to .

NOW FIND YOURSELF SOME PRACTICE!!