Mandelbrot Set

Fatou and Julia

Benoit Mandelbrot

M-Set How?

Complex Numbers

Imaginary Numbers

Iteration

M-Set Algorithm

Julia Sets

J-Set Algorithm

Part III

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Part III

Part IV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Imaginary Numbers

Imaginary Numbers

Imaginary numbers were defined by Rafael Bombelli in 1572. At the time such numbers were regarded as fictitious or even useless much as zero was once regarded as nonsense in the past.

Imaginary numbers arise when you try to take the square root of a negative number. This of course leads to all sorts of problems since you cannot take the square root of a negative real number and end up with a real number. In other words, there is no such real number that will satisfy this problem.

What Bombelli did, was he defined a new KIND of number by taking the square root of –1 and calling it “i” for imaginary. This “i” became the (multiplicative) IDENTITY for the imaginary numbers, much in the same way that ONE is IDENTITY for the integer and real numbers.

In other words, any real number multiplied by the square root of negative 1 (or i) is called an imaginary number. So, six(6) times “i” is an imaginary number and PI times “i” is also an imaginary number. Any real number, times the square root of –1 is imaginary.

Interestingly, imaginary numbers seem to pop up all over the place and can be found littered throughout the equations used in electrical engineering, quantum mechanics and Einstein’s theory of relativity.

With that in mind, it becomes clear that "imaginary numbers" are an integral part of our physical universe and therefore, must play a big role in the creation and persistence of our universe.