School of Wisdom® Series: Volume 2
Math Appendix to
CHANCE AND CHOICEBy R.C.L.
The Mathematics of the Mandelbrot Formula and the Workings of Numbers and Vectors in the Complex Plane.
(Incomplete Draft)
Mathematics symbolizes the Complex Numbers with a letter z and defines a Complex Number as follows:
z = a + bi
a = real number, and bi = imaginary number
As the text in Chapter Two indicates both the real and imaginary parts of the complex number can be either positive or negative and either whole numbers or decimals. The complex numbers can be easily added and subtracted, and almost as easily multiplied and divided.
For instance to add the complex number 4 + 3i and the complex number 2 - 7i, you simply add the real and imaginary parts separately: 4 + 2 = 6; and 3i -7i = - 4i; the resulting complex number is 6 - 4i. Multiplication is a bit trickier for here you must remember the rule that each time i occurs you replace i squared with negative one (-1). For example, the product of 4 + 3i and 2 - 6i is 26 -18i. Here's how: you first multiply the real numbers times each other: 4 * 2 = 8; then multiply the real numbers times the imaginary numbers: 4 * -6i = -24i and 2 * 3i = 6i; then multiply the imaginary numbers times each other: 3i * -6i = -18i . Put these all together and you have: 8 - 24i + 6i - 18i . The last operation is the only tricky part: since imaginary numbers are a negative square root ( -1 * -1 = -1 not +1), the square of a negative imaginary number i is negative one: i = -1. So -18i = -18 * -1 which in turn equals positive 18. So now you have 8 + 18 - 24i + 6i which is easily reduced to 26 - 18i.
To illustrate how the calculation of z -> z + c works with a simple value for c, take the values of z = 0 and c = 1 - 1i. Start with 0 * 0 = 0 plus (1 -1i) which equals 1 -1i. 1 -1i then becomes the new z in the next iteration of the calculation. In the next step we start by squaring z so the calculation begins with (1 -1i) * (1 -1i) plus the same c again of 1 -1i. First find the value of the square of z. (1 -1i) * (1 -1i) is the same as (1 * 1 = 1) + (1 * -1i = -1i) + (1 * -1i = -1i) + (-1i * -1i = +1i ) (remember the i squared always equals negative one) then added up this equals 1 - 1i -1i -1 or 0 - 2i. If you don't see where the -1 came from, remember that 1i squared equals real number 1 times real number 1 (positive one), but the square of the imaginary number is always equal to negative one; so your left with negative one times positive one which of course equals negative one. Complete the calculation by adding the static value of c to the z squared value: (0 -2i) + (1 -1i) = 1 -3i.
In the next iteration of the calculation you start with z = 1 -3i. If you keep repeating this calculation you will find that it goes off into infinity. As you can see from the diagram of the Mandelbrot set in the Complex plane shown in Chapter Two, this starting value for c (1 -1i) is not within the Mandelbrot set (.8 is the outside right border of the Mandelbrot).
Other Complex numbers such as -1 +0i obviously stay stable from the beginning. Start with: z=0 + -1 +0i = -1; next step z= -1 +0i + -1 +0i = (-1 * -1 = 1) + (-1 *0i = 0) + (0i*0i=0) + (-1) = +1 -1 = 0; with z again equal to 0 the next step is again -1, the next is zero again, the next -1, and so forth ad infinitum. When c = -1 +0i then c is neither attracted to the infinitely large or infinitely small; instead it forever hops back and forth between 0 and -1.
Some iterations of complex numbers are like 1 -1i and run off into infinity from the start, just like all of the real numbers. Others are always stable like -1 +0i. Other complex numbers stay stable for many iterations, and then only further into the process do they unpredictably begin to start to increase or decrease exponentially (eg. .37 +4i stays stable for 12 iterations). These are the numbers on the edge of inclusion of the stable numbers shown in black. Chaos enters into the iteration because out of the potentially infinite number of complex numbers in the window -2.4 and .8 and -1.2 + 1.2, there are an infinite subset on the edge which are subject to the unpredictable strange attractor. All that we know about these edge numbers is that if the z produced by any iteration lies outside of a circle with a radius of 2 on the complex plane, then the subsequent z values will go to infinity, and there is no need to continue the process.
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- CHANCE AND CHOICE
- a Compendium of Ancient and Modern Wisdom Revealing the Meaning and Significance of the Myth of Science.
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