Algebra 2 Imaginary Numbers
Math-Prof HOME Algebra 2 Table of Contents Ask A Question PREV NEXT

It's Just Your Imagination
Imaginary Numbers

 

 Remember when you were solving equations using the quadratic formula?

 

 

 
 There was a special piece of the equation called the discriminant ...
 

 

 
 When it turned out to be positive, we get two values for X.
 

 

 
 When the discriminant turned out to be zero,
 we get one answer.

 

 
 But when (b 2 - 4ac) turned out to be a negative number,
 we were stuck.
 
 A square root means 
 "what number times itself equals the number under the radical?"
 and no number times itself equals a negative number.
 This was a problem we couldn't solve.
 
 Math people HATE IT when that happens.
 They hate it so much that they usually invent a new kind of math to deal with it.
 
 That's what they did here.
 They invented a number that times itself equaled negative one.
 They called it "i"
 They said:

 1 1 = -1

 That also means that:

 

 
 Now "i" is not a real number, it was made up.
 Maybe they should have called it a "made up" number,
 but that didn't sound fancy enough.
 So they called it an "imaginary" number.
 
 Now that we have "i", if you have something like:
 

 

 
 you can deal with it.
 
 The rules of roots say that you can do this ...
 

 

 
 We know that ...
 

 

 So:

 

 
 Question:
 Is "3i" a real number or an imaginary number?
 
 Hey, it's just a name. It's not that important.
 But this is the deal ...
 If you have a real number times "i," you get an imaginary number.
 
 Enough already, let's do one!

 

 X2 + X + 1 = 0

 
 Solve for X.

 a = 1   b = 1   c = 1

 

 
 This is all correct, 
 but math people like to see it written a little differently ...
 

 
 Take a minute and make sure you understand how we did that.
 
 Whenever you have an answer that has an "i" in it,
 math people want it written in this special way.
 If we let "a" and 'b" stand for any numbers we might have,
 the special way to write this is:

 a + bi

 
 This is called a complex number.
 
 So:

 

 
 are the places where the equation is equal to zero.
 
 That means the factors of X 2 + X + 1 are:
 

 

 
 Don't believe me?
 Multiply it out ...
 WARNING! It's pretty messy ...
 

 

 
 We got the last term, 3/4 by multiplying ...
 

 

 Make sure you can multiply that out.
 
 Now combine terms. Watch all the nasty stuff drop out ...
 

 

 
 Let's try another one.
 
 Example:
 

 2X2 + 4X + 3 = 0

 

A = 2   B = 4   C = 3

 

 

 
 Now we write this as our complex numbers ...
 

 

 
 are the factors of:

 2X2 + 4X + 3

 
 Are they the only factors?
 Let's multiply them out and see ...

 

 
 So we have ...

 and we want ...

 2X2 + 4X + 3

 
 That means the missing factor is a 2.
 
 So:

 

 

   copyright 2005 Bruce Kirkpatrick

Math-Prof HOME Algebra 2 Table of Contents Ask A Question PREV NEXT