Complex Numbers
Complex Numbers are, in fact, a simple idea:
Examples:
| 1 + i |
39 + 3i |
0.8 - 2.2i |
-2 + πi |
√2 + i/2 |
A Number is a Combination of Two Numbers?
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Can you make up a number from two other numbers? Sure you can!
You do it with fractions all the time. The fraction 3/8 is a number made up of a 3 and an 8. We know it means "3 of 8 equal parts".
Well, a Complex Number is just two numbers added together (a Real and an Imaginary Number). |
Either Part Can Be Zero
So, a Complex Number has a real and an imaginary part.
But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.
| Complex Number |
Real Part |
Imaginary Part |
| 3 + 2i |
3 |
2 |
| 5 |
5 |
0 |
| -6i |
0 |
-6 |
Adding
To add two complex numbers we add each element separately:
(a+bi) + (c+di) = (a+c) + (b+d)i
Example: (3 + 2i) + (1 + 7i) = (4 + 9i)
Multiplying
But to multiply we follow a more interesting rule:
(a+bi)(c+di) = (ac-bd) + (ad+bc)i
Example: (3 + 2i)(1 + 7i) = (3×1 - 2×7) + (3×7 + 2×1)i = -11 + 23i
You can try it yourself: enter (3 + 2i)(1 + 7i) into the Complex Number Calculator.
Here is another example:
Example: (1 + i)2 = (1 + i)(1 + i) = 1 + i + i + i2 = 1 + 2i - 1 = 0 + 2i
Why does it work?
Just try binomial multiplication (the "FOIL" method) to see:
| (a+bi)(c+di) |
= |
ac + adi + cbi + bdi2 |
|
|
| |
= |
ac + (ad + bc)i - bd |
|
(because i2=-1) |
| |
= |
(ac - bd) + (ad + bc)i |
|
(gathering like terms) |
And there you have the (ac - bd) + (ad + bc)i pattern.
How About i2
Just for fun, let's see what happens when we calculate i2, with i as a complex number 0+i:
Example: i2 = (0 + i)(0 + i)
= (0×0 - 1×1) + (0×1 + 1×0)i = -1 + 0i
= -1
And that agrees nicely with the definition that i2 = -1
So it all makes sense!
Conjugates
The conjugate is where you change the sign in the middle like this:
A conjugate is often written with a bar over it:
Dividing
The conjugate is used to help division.
The trick is to multiply both top and bottom by the conjugate of the bottom.
Example: Do this Division:
Multiply top and bottom by the conjugate of 4 - 5i :
| |
2 + 3i |
· |
4 + 5i |
= |
8 + 10i + 12i + 15i2 |
|
|
|
| 4 - 5i |
4 + 5i |
16 + 20i - 20i - 25i2 |
Now remember that i2 = -1, so:
| |
= |
8 + 10i + 12i - 15 |
|
| 16 + 20i - 20i + 25 |
Add Like Terms (and notice how on the bottom 20i - 20i cancels out!):
We should then put the answer back into a + bi form:
DONE!
Multiplying By the Conjugate
What happened on the bottom was interesting:
(4 - 5i)(4 + 5i) = 16 + 20i - 20i - 25i2
The middle terms cancel out!
And since i2=-1 we ended up with this:
(4 - 5i)(4 + 5i) = 42 + 52
Which is really quite a simple result
In fact we can write a general rule like this:
(a + bi)(a - bi) = a2 + b2
Remember that when you do division ... it will save you time.
Mandelbrot Set
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The beautiful Mandelbrot Set (pictured here) is based on Complex Numbers.
It is a plot of what happens when you take the simple equation z2+c (both complex numbers) and feed the result back into z time and time again.
The color shows how fast z2+c grows, and black means it stays within a certain range. |
|
.Here is an image made by zooming into the Mandelbrot set |
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| And here is the center of the previous one zoomed in even further: |
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