Imaginary and Complex Numbers
Imginary Numbers
Definition of a imaginary number:
“i” is the symbol for an imaginary number. There is not really an “exact definition for an imaginary number”. I mean I could give you the exact definition, but what does that really matter in math?
“i” is the symbol for an imaginary number. There is not really an “exact definition for an imaginary number”. I mean I could give you the exact definition, but what does that really matter in math?
The best way to describe an imaginary number is through an example. Lets say you are asked to determine the square root for -16. What would you do? Well, before you answer with 4, remember that you can not find the square root for a negative number… there for this is where imaginary numbers come into place-
Example 1:
Question: What is the square root of -25?
Question: What is the square root of -25?
1st step: Take the root of the absolute value of -25. Which would be 5.
2nd step: Then multiply it by “i”.
3rd step: So square root of -25 = 5i
Side note: I forgot to mention earlier that i2 = -1.So lets test it out! 5i*5i = 25i2 = 25(-1)= -25
Complex Numbers
3rd step: So square root of -25 = 5i
Side note: I forgot to mention earlier that i2 = -1.So lets test it out! 5i*5i = 25i2 = 25(-1)= -25
Complex Numbers
Definition of a complex number: When a number has the form a + bi (a real number plus an imaginary number)
Now I bet you are wondering when is this needed in mathematics?
Well, for example to find the roots of the quadratic equation x2-6x+25= 0
The original equation was:
(X - 3 -4i) * (X - 3 + 4i) = X2 -3x +4Xi -3X +9 -12i -4Xi +12i -(4i)2
side note: x2 means x squared. every time 2 is to the right of the number it is squaring the number before it. I cant figure out how to square a number on here, so that is what we will use for now.
This reduces to:
X2 -3x -3X +9 -(16)*i2
Since i2= -1 then -(16)*i2 becomes -(-16) = 16 and so:
X2 -6X +25 =0
Mandelbrot Set
In mathematics the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal.
Mathematically the Mandelbrot set can be defined as: the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial:
Now, how do we actual apply this?
1st: We need you need to know how to graph complex and imginary numbers.
Graping Real Numbers: Real numbers can be represented on a one dimensional line called the real number line. Negative numbers like -2 are plotted to the left of zero and positive numbers like 2 are plotted to the right of zero. Any real number can be graphed on the real number line.
Graphing complex numbers:
Since complex numbers have two parts, a real one and an imaginary one, we need a second dimension to graph them. We simply add a vertical dimension to the real number line for the imaginary part. Since our graph is now two-dimensional, it is a plane, the complex number plane. We can graph any complex number on this plane. The colored dots on this graph represent the complex numbers [2 + 1i], [-1.5 + 0.5i], [2 - 2i], [-0.5 - 0.5i], [0 + 1i], and [2 + 0i].
How to graph the Mandelbrot Set:
The Mandelbrot set is a set of complex numbers, so we graph it on the complex number plane. However, first we have to find many numbers that are part of the set. To do this we need a test that will determine if a given number is inside the set or outside the set. The test is based on the equation:
Z = Z^2 + C
C represents a constant number, meaning that it does not change during the testing process. C is the number we are testing, the point on the complex plane that will be plotted when testing is complete. Z starts out as zero, but it changes as we repeatedly iterate this equation. With each iteration we create a new Z that is equal to the old Z squared plus the constant C. So the number Z keeps changing throughout the test.
Z = Z^2 + C
C represents a constant number, meaning that it does not change during the testing process. C is the number we are testing, the point on the complex plane that will be plotted when testing is complete. Z starts out as zero, but it changes as we repeatedly iterate this equation. With each iteration we create a new Z that is equal to the old Z squared plus the constant C. So the number Z keeps changing throughout the test.
We're not really interested in the actual value of Z as it changes, we just look at its magnitude. The magnitude of a number is its distance from zero. For example, the number -9 is a distance of 9 from zero, so it has a magnitude of 9. The magnitude of a complex number is harder to measure. To calculate it, we add the square of the number's distance from the x-axis (the horizontal real axis) to the square of the number's distance from the y-axis (the imaginary vertical axis) and take the square root of the result. In this illustration, a is the distance from the y-axis, b is the distance from the x-axis, and d is the magnitude, the distance from zero.
As we iterate our equation, Z changes and the magnitude of Z also changes. The magnitude of Z will do one of two things. It will either stay equal to or below 2 forever, or it will eventually surpass two. Once the magnitude of Z surpasses 2, it will increase forever. In the first case, where the magnitude of Z stays small, the number we are testing is part of the Mandelbrot set. If the magnitude of Z eventually surpasses 2, the number is not part of the Mandelbrot set. This sounds confusing, but just try the concept out. Before we do that, lets look at something else thats kind of interseting.
Now you have learned:
1- what a complex number is
2- how to graph a complex number
3- how to use your knowledge to use the Mandelbrot and apply it.
...now before we go to our last lesson, do you understand the 3 concepts I just listed? If not, go back over the part you don't get and THEN come back to the last lesson on using the Mandelbrot set and mapping it out and testing it out, as well.
How to determine if a point is a part of the Mandelbrot set:
Example one:
The (random) point ,0+i, on a Complex plane.
c= i (constant)
z= is the point you are testing
z1 means the first in the series.
z1= i^2 + i = -1 + i
z2= (-1 + i)^2 + i = -i
z3= (-i)^2 + i = -1 + i
Z3 then becomes z1 again and the cycle continues. This cyclical nature means that c is bounded and in the Mandelbrot set.
One that doesn't work:
The point, say, 2+2i on a complex plane. c=-3i
z1= (2 + 2i)^2 - 3i= i
z2= i^2 - 3i= -1 - 3i
z3= (-1 - 3i)^2 - 3i= -5 + 3i
z4= (-5 + 3i)^2 - 3i= 16 - 33i
It just got even farther away without any inclination towards or around the origin. c is unbounded and is therefore not part of the Mandelbrot set.
Now here is a picture of the Mandelbrot Set:
