Could anyone explain about 'complex number'?

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Permalink Reply by Daniel Piker on November 25, 2009 at 4:38am

ok, I'll have a go...


With the real numbers any number squared gives a positive answer. eg:

5 * 5 = 25 , -5 * -5 = 25

because a negative multiplied by a negative gives a positive.

The imaginary numbers behave differently from the reals, in that when they are squared they give a negative result. They are written as multiples of the imaginary unit i, which is defined so that:

i*i=-1

Complex numbers are numbers which have two parts (hence the name complex) - a real part and an imaginary part.
For example:

3+4i,

or more generally:

a+bi, where a and b are some real numbers.

Well that's a definition, but I guess you might be wondering what is the point of them - I've not said anything yet about why they are interesting and useful...

Solving cubic equations was one of their first uses, but I doubt that is what most of you are interested in.

Where they really get fun is when you start looking at them geometrically.
The Argand plane is a setting that allows us to treat complex numbers a bit like vectors.
Each complex number a+bi defines a point relative to an origin (0,0), much the same as a vector with an x and y component.
Like vectors we can add and subtract them to get a new point.

But when we multiply them, unlike vectors, we add the angles (measured anti-clockwise from the positive real axis, also called the argument) and multiply the lengths (or the modulus of each number).
This all follows naturally as a consequence of the definition of i as the square root of minus one.

........

That is just dipping a toe into the great depths.
Complex number math, and in particular complex Analysis (calculus with complex numbers) is a vast subject that I obviously can't cover much of here.

If you are interested in learning more :

The Math department at Cal State Fullerton has some very nice Complex Analysis pages.
Chapters 5 and 6 of the film Dimensions covers complex numbers very visually. You can watch it online here, or read the description here.
Complex numbers on Wikipedia
on MathWorld
Hans Lundmark's complex analysis pages
The book Indra's Pearls is about making certain types of fractals with complex numbers, and includes a good introduction, along with lots of pseudocode.

To really engage with some of the true depth and power of complex numbers I particularly recommend the beautiful Visual Complex Analysis. This was the book that made me love this subject.

I'm really looking forward to seeing more designers make use of complex numbers. I think it is a wonderful tool. It is an advanced branch of mathematics, requiring some serious study to understand, but because of its strong geometric connections, I think relatively accessible to those who tend to think more visually. Now that David has included them in Grasshopper, starting to explore them should be easier than ever.

Permalink Reply by Damien Alomar on November 25, 2009 at 6:53am

Thanks for the info David. I actually just got back from London and while I was there stopped by the AA bookstore. I wound up thumbing through that book, so now that you're recommending it, it will definitely be one that I pick up.

Permalink Reply by Vicente Soler on November 25, 2009 at 8:05am

Daniel, have you seen this thread i posted some time ago?:
http://www.grasshopper3d.com/forum/topics/3d-mandelbrot

Maybe you should try to have a go at it :)

Permalink Reply by Daniel Piker on November 25, 2009 at 8:30am

Thanks Vicente.
Yes, I've been following some of the recent developments with this 'Mandelbulb' and it is very interesting indeed. I might try and play around with it a bit over christmas if i get time.

I did start to look at plotting the Hubbard Douady potential using grasshopper, based on Inigo Quilez's approach.

(He's one of those demoscene people that do amazing things with insanely small bits of code. You might have seen his 4k demo elevated. The 256b Puls by Rrrola is just incredible too. That's animated geometry and a 3d renderer from scratch in less data than it takes to write this comment. A lot of very clever math tricks go into making those things, and Inigo's site has some interesting and useful info.)

Permalink Reply by Daniel da Rocha on August 16, 2010 at 10:49am

nice intro!
it would be great if anyone could post a simple definition in GH which uses complex numbers... just as a kick start to start using them...
I searched everywhere on the web but could not find anything!

I am having a hard time to translate the idea of complex numbers into something tangible and doable in GH.....

anyone know of any simple complex numbers definition?

Permalink Reply by YJ on November 25, 2009 at 3:55pm

My jaw dropped! Thank you so much opening new world!!