Complex Numbers/Introduction
Notice: Incomplete
Prerequisites
[edit | edit source]All four operations (+, -, *, /) for the real numbers
The purpose of parentheses in math
Box method for multiplication and/or FOIL
Meaning of Complex Numbers
[edit | edit source]What does i mean?
[edit | edit source]i is a 'fake', or imaginary number such that i*i=-1. We need to make up a number because 1*1=1 and -1*-1=1.
Complex numbers
[edit | edit source]Complex numbers are numbers that 'look like' all of the following:
-1-2i
-3+10i
3-2i
4+3i
Math of Complex Numbers
[edit | edit source]Add
[edit | edit source]To add complex numbers, you add similar things, so (1+2i)+(3+4i)=(1+3)+(2+4)i=4+6i
Take away
[edit | edit source]To 'take away' complex numbers, you 'take away' similar things, so (1+2i)-(3+4i)=(1-3)+(2-4)i=-2-2i
Times
[edit | edit source]To 'multiply' complex numbers, you 'multiply' similar things, so (1+2i)(3+4i)=(1)(3)+(1)(4i)+(2i)(3)+(2i)(4i)
With Box Method
[edit | edit source]Using the example of (1+2i)*(3+4i), we get:
1 | 2i | |
3 | 1*3=3 | 2i*3=6i |
4i | 1*4i=4i | 2i*4i=8i*i=-8 |
Going across both diagonals, we get:
(3+-8)+(6+4)i=-5+10i
With FOIL
[edit | edit source]To multiply complex numbers, you multiply using FOIL, like this: (1+2i)*(3+4i)=(1*3)+(1*4i)+(2i*3)+(2i*4i)=3+4i+6i-8=-5+10i
Divide
[edit | edit source]To divide complex numbers, you turn it into a multiplication problem, like this: , which can be solved normally. This works because
and dividing is just multiplication by the reciprocal.
3 | 4i | |
Conjugate
[edit | edit source]The conjugate of a complex number is the number you get if you replace the + linking the real and imaginary parts with - or vice versa. Some examples include:
Complex number | Conjugate |
---|---|
1+2i | 1-2i |
1-2i | 1+2i |
-1+2i | -1-2i |
-1-2i | -1+2i |