Imaginary Numbers Essential Question: How do we take the square root of negative numbers?
Imaginary Numbers. Mathematics created imaginary numbers to “pick up” where real numbers stopped...
-
Upload
garry-gardner -
Category
Documents
-
view
213 -
download
0
Transcript of Imaginary Numbers. Mathematics created imaginary numbers to “pick up” where real numbers stopped...
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