DUMB
Eh, no. Basically, you define a number i such that √(-1), the square root of negative one, equals i. That's called an imaginary number because it doesn't "mesh" with normal (real) numbers, and is "outside" them; 1+i can't be reduced to a real number, for instance.
Then a complex number can be defined as any number a+bi, where a and b are real numbers, and i is as up there. So 1+i, 3+i/9, 4 (= 4 + 0i), and 10i (= 0 + 10i) are all complex numbers.
Hopefully that makes sense so far. Now the trickier part comes in when you multiply them. Since i = √(-1), i² = -1. Therefore, (a+bi)(c+di) = ac+adi+bci+bdi²=ac+adi+bci-bd=(ac-bd)+(ad+bc)i. Because of the subtraction, you lose the guarantee you get with real numbers, that multiplying two positive numbers gets you a bigger positive number.
Now the Mandelbrot set can be defined like this: For each point on the plane (x, y), call x+yi "c" (so c = x+yi). Then, starting with z_{0} = 0, say z_{n} = (z_{n-1})² + c. So for the point (1, 0), c = 1+0i = 1; z_{0} = 0; z_{1} = (z_{0})² + c = 0² + 1 = 1; z_{2} = 1²+1 = 2; z_{3} = 2²+1 = 5; etc. Now it should be pretty obvious that successively higher z values get farther and farther from zero. Because of this, we say that the point (1, 0) "escapes". In contrast, (0, 1) = 0+i = i results in the z sequence 0, i, -1+i, -i, -1+i, -i, etc. repeating, and never "escapes". We say that a point is in the Mandelbrot set if it does not escape. And from this relatively simple rule, you get the incredible complexity of the set.
Make any sense?
Not all there
I think so. I always wondered why it was called a set too. Thanks!
Forum Villain
I'm still in school, but ever since I aced pre-calc (over a year ago), I haven't taken a math class. Hopefully that will be rectified this year, seeing as there's a calculus class now available in my school.
I never thought I'd hear my self say this, but I really do miss math.
"Sarchasm: The gulf between the author of sarcastic wit and the person that doesn't get it."
I'm a grad student who'll (hopefully) have a Master's degree in it at the end of the school year. I'm focusing more on statistics though, since I want to ultimately become an economist.
I <3 Caps Lock!
Says dRoy:
edited 21st Aug '11 8:35:40 PM by BMeph It's always darkest after the lights go out.
^{ 81} Noaqiyeum, Wed, 24th Aug '11 2:20:35 PM from across the gulf of space
Relationship Status: Showing feelings of an almost human naturenobody knows
Sirs, madams, it-thingies, miscellanea, good day!
"I'm an odd person that likes to write odd people." - JHM
Bibliophile.
Me! I love Factorials! 7! 5040.
- Never be without a Hat!
paradox delivery service
looks like I'm a little late to the party, but I'm still here, just started a new school year and I'm a couple years ahead of what my grade should be at, seriously, I'm close to having to take the adult test to get an accurate measure of my skill, and I know most of my math book, this is going to be one boring year.
selling property in hell, lake-of-lava front timeshare with hitler or cheap 5th ring, only 250000 souls
^{ 84} Tuefel Hunden IV, Sun, 18th Sep '11 2:33:45 PM from Wandering.
Relationship Status: [TOP SECRET]Watchmen of the Apocalypse
It seems to happen in highschool with kids who are a bit head in some areas.
Ok i am not sure what sort of weird sorcery is at work but remember that problem I posted earlier? I built the structure in minecraft and it worked out with a flat top some how. I need to figure out what I am doing wrong there with my various formulas and set up. I have a feeling I am needlessly complicating it.
edited 18th Sep '11 3:08:05 PM by TuefelHundenIV "Who watches the watchmen?"
KVLFON
I LOVE CALCULUS
"Atheism is the religion whose followers are easiest to troll"
^{ 86} Noaqiyeum, Mon, 19th Sep '11 11:35:21 AM from across the gulf of space
Relationship Status: Showing feelings of an almost human naturenobody knows
Tuefel: According to your description you will never reach layer 50 (as Tzetze pointed out). Level n=1 is 68 blocks per side on the outer square and 66 for the inner square, level n=2 is 66 blocks wide outside and 64 inside, and so on, right? So the outside width on level n is 70-2n, which is -30 when n=50. Level 35 should have no blocks in it, so the highest, actual, physical level of the pyramid will be 34 (a four-block square).
If you built a hollow pyramid whose sides are one block thick, all the way to level 34... the number of blocks in any given level n is a_{n} = 4((70-2n)-1), which reduces to a_{n} = 276-8n. And you want the sum of a_{n} for all n from 1 to 34... ^{34}Σ_{n=1}(276-8n) is, happily, an arithmetic series, where each level has 8 fewer blocks than the one underneath it! Yay, predetermined formulas! S_{n} = (n/2)(a_{1} + a_{n}) = (34/2)(268 + 4) = 4624 for the outside. The inside is the same pyramid, short the bottom layer of 268, so the total number of blocks in the 34-layer pyramid is 2(4624)-268 = 8980. If I did everything right this time.
Did you possibly mean for the second level to be 67 blocks wide outside? (That would match your calculation that the 50th level would be 19 blocks wide.) The maths' different, but along the same lines...
"I'm an odd person that likes to write odd people." - JHM
DUMB
He's making a Minecraft pyramid, so reduction by one block doesn't result in a pyramid.
^{ 88} Noaqiyeum, Mon, 19th Sep '11 11:49:04 AM from across the gulf of space
Relationship Status: Showing feelings of an almost human naturenobody knows
Err... 'yes, I saw' to the first part, 'what?' to the second.
EDIT: Oh, you mean that the blocks don't stack on the corners of the layer underneath? In that case, that's probably the first point where the maths went wrong.
edited 19th Sep '11 11:51:53 AM by Noaqiyeum "I'm an odd person that likes to write odd people." - JHM
^{ 89} Tuefel Hunden IV, Fri, 23rd Sep '11 9:56:06 PM from Wandering.
Relationship Status: [TOP SECRET]Watchmen of the Apocalypse
I am going to try and get access to map again and double check my count. That has to be the only explination I can think of. The count must be higher for this to work.
"Who watches the watchmen?"
literary masochist
I recently realized Computer Science is math. Computery math, anyway.
I don't care, I'm still free
You can't take the sky from me
DUMB
Which is why the early pioneers had a lot of applied mathematics degrees between them and why I think the current computer engineering halfway education stuff is terrible.
o murr
I've always really liked math. I'm only a high school sophomore, though; I'm taking an honors Algebra II class.
I'll admit that I wasn't overly fond of geometry last year, but I think that was due to a teacher who didn't know what he was doing (making the work miserably difficult until I started doing my own research to learn the theorems better), rather than the work itself.
"Censorship offends me." -Author unknown
Currently taking AP Calculus. I'm not as enthusiastic about it as I was for my past math classes though...
f(x) = tan ^{-1}(x^{2}+2x)
f'(x) = (2x+2)/(1+[x^{2}+2x]^{2})
edited 30th Sep '11 7:40:36 AM by Pokénatic literary masochist
Ahh, differentiation and integration of trig functions. I always hated that part.
I don't care, I'm still free
You can't take the sky from me
Is that cake frosting?
I think I want to learn a little about dynamical systems. I have some vague memories of a very simple course I took a long time ago about Lyapunov Functions and so on, but for all intents and purposes I am starting from almost zero.
But I have started thinking about some fun ideas about the relationship between formal logic, Markov processes and dynamical systems, and I get the impression I lack some of the basics to really see them through.
First I'll have to wrap up the thesis, though — sigh. I really enjoy doing research, but putting everything together in a coherent, readable form is boring...
But they seem to
know where they are going, the ones who walk away from Omelas.
I want to ask the math-inclined people to review an essay I typed up here (OTC). It's basically to say that deductive and inductive reasoning aren't so "different", and it involves looking at mathematical induction to help see that.
edited 2nd Oct '11 5:19:01 PM by abstractematics Now using Trivialis handle.
Lurks too much
Tze: There's a lot of math involved in electrical engineering, too, especially if you're dealing with stuff like circuit board design or, well, pretty much anything dealing with RF. I actually kind of regret never having taken any formal higher math courses, since I'm sure it would have helped me understand things like 3D rendering better.
That said, IT is a skilled trade, not strictly a science, and so knowing the higher math isn't an absolute requirement. It does help, though. :P
Taking suggestions for a new sig. PM me or something!
^{ 98} Tuefel Hunden IV, Sat, 15th Oct '11 4:13:15 PM from Wandering.
Relationship Status: [TOP SECRET]Watchmen of the Apocalypse
Ok I have figured out why the diagram is not working with the structure. It is drawn wrong. Ok using the cubes draw a square with the sides touching with 6 blocks. The inside layer will be four blocks. Now draw the shape in a diamond with only the corners touching. The outside edge is still 6 by 6 but the inside is 5x5. When I drew the pyramid with diagonal scheme on a grid it worked. The final 55th layer leaves a 11 x 11 flat space at the top.
This works out with bottom layer being 66 x 66, 55 layers down.
edited 15th Oct '11 4:15:29 PM by TuefelHundenIV "Who watches the watchmen?"
Is that cake frosting?
OK, now I really want to learn some Complex System Theory/Chaos Theory. I was talking about it today with a colleague, and we agreed that we should try to set up a reading group on the topic — that is, pick a text, a (reasonably forgiving) schedule and agree to meet once in a while to discuss it.
The only problem is that neither of us is really in that area — he is a theoretical computer scientist, I am a logician — so we are not sure which text to use. We will ask around in the department, of course; but is there any good introductory book in the field that you all could recommend?
edited 15th Oct '11 4:30:16 PM by Carciofus But they seem to
know where they are going, the ones who walk away from Omelas.
^{ 100} Enthryn, Tue, 27th Dec '11 7:53:39 PM from Earth
Relationship Status: Having tea with CthulhuDidn't notice this thread before. I'm an undergraduate majoring and planning on getting a PhD in mathematics. Some of you have probably seen me geeking out about math elsewhere on the forum.
Taking a class on Lie groups next semester... I hope I won't get too crushed by this class — differential geometry definitely isn't my best subject.
Also, this is awesome.
edited 27th Dec '11 7:55:11 PM by Enthryn Prendre le bien, le mal et sans trier, accepter Sans couvrir tes yeux, tout regarder.
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