Imaginary Numbers

Mathematics created imaginary numbers to “pick up” where real

numbers stopped

For example: Remember how we can’t take the square root

of a negative number….

UNTIL NOW…..

An imaginary number is easy to identify….

We use an “i”

For example: 3 is a real numberBut 3i is an imaginary number.

Imaginary numbers are defined as the

Yeah, there are still plus or minus roots.

x

x xi

4 i2

16 i4

So quick review so far….

Students think the “-”sign just turns into an “ i”

SORTA…..

You already know the rules that apply to imaginary numbers except for one…

That’s right . It equals a plain old number -1. So 2i2 is the same as -

2…

2i

Are you starting to get the new rule?

= -1And -3i2 is the same as

3…since we can/must replace the i2 with a – 1 and that would make it

(-3)(-1)

And that would make: 2i2 -3i2 the same as -2+

3 or 1…

Whenever else you work with these numbers you apply the same rules you

would for variables.For example: 2i + 3i would be the same

as 5i similar to 2x + 3x which is 5x.

And 2i times 3i would be 6i2.

And 2i times 3i times 4 would be 24i2.

But…..

Why is there always a “but”?2i times 3i plus 4 would be 4 + 6i2.

Yeah, that’s right just like 2x times 3x plus 4

would be 6x2 + 4BUT what about the new rule?

YEAH!!! The new rule that defines

“i2 “ as – 1.

But….

So 6i2 is and MUST be written -624i2 = -24

And 4 + 6i2 is the same as 4 + -6

I told you there was always a “but”….

4 + -6 = -2

4i(5 – 6i) would be20i -24i2 which simplifies to

This is called a complex number

Yeah, the real part comes first followed

by the imaginary part…

20i +24 which must be written as24 – 20i

4i – 5 – 2 + 3i would be-7 + 7i

(2 – 3i)(1 + 5i) would beYeah, FOILING and yeah, factoring is

coming back tooooo.

2 +10i – 3i – 15i2 but collecting like termsMakes it: 2 +7i– 15i2 but since i2 = -1

This becomes: 2 +7i– 15(-1) or2 + 7i + 15 or

17 + 7i

OKAY, that will get you started for now… turn

to page 277.Read example #3. That’s how “real”

imaginary numbers are!

You’d be in THE DARK without them!Yeah, I know sad, sad, sad….

Asignment: p. 279 12 -27 all