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kevCast
December 5th, 2009, 12:36 AM
I'm partial to Euler's identity.

A video that relates a little:
http://www.ted.com/talks/murray_gell_mann_on_beauty_and_truth_in_physics.ht ml

What are some that you're partial to?

Viranh
December 5th, 2009, 12:44 AM
I like the heat equation.
http://en.wikipedia.org/wiki/Heat_equation

I actually know how to use that and have derived things with it, which scares me. It's pretty though. Look at all those partial derivatives. :)

soni1770
December 5th, 2009, 12:46 AM
i like bernoulli's theorem,

applied to a jet engine

(1 - M^2)du/u = - dA/A

with M - mach number:popcorn:

it's the best, one simple equation to explain the shape of a jet exhaust.

sgosnell
December 5th, 2009, 12:52 AM
e=mc^2

dbbolton
December 5th, 2009, 01:02 AM
The Leibniz formula for pi probably intrigues me the most. f(x) = e^x is up there too.

phrostbyte
December 5th, 2009, 01:22 AM
Fourier transform.

Keyper7
December 5th, 2009, 01:23 AM
Euler's identity is a strong contender for me to... just so damn beautiful.

phrostbyte
December 5th, 2009, 01:24 AM
euler's identity is a strong contender for me to... Just so damn beautiful.

+1

phrostbyte
December 5th, 2009, 01:26 AM
F = d/dt(m/v)

ZankerH
December 5th, 2009, 01:32 AM
Mine would be the Taylor series. Just something about watching rational functions converge towards a non-elementary rule.

vambo
December 5th, 2009, 01:45 AM
For me, Maxwell's equations - curl, div and del and all that!

BuffaloX
December 5th, 2009, 01:55 AM
I'm not very good at math, but still I find it fascinating what can be done with it.

20 years ago I was fascinated with the Mandelbrot set., and made a very heavily optimized program, examined black areas, how calculations progressed for single points, figured out how to determine if a point truly was black, or just needed more iterations... :roll:

Fourier transformations are also amazing especially for audio.
I don't understand how to do it myself, but I've used quite a lot of programs that depend on it heavily.

In the mid 80's it was considered impossible to compress audio, without a near equal loss of quality, audio was considered too complex and chaotic for compression. Seems the necessary math to do it, was discovered already in the early 18th century. :oops:

papangul
December 5th, 2009, 05:55 AM
0 = ∞

(I am not good with maths :D)

xuCGC002
December 5th, 2009, 06:00 AM
COS♥=?

...My normal approach is useless here.

JDShu
December 5th, 2009, 06:07 AM
I don't think you can get much better than Euler's Identity, it boggles the mind.

On the other hand, the Pythagorean Theorem has fascinated all the ancient civilizations.

phrostbyte
December 5th, 2009, 06:08 AM
I'm not very good at math, but still I find it fascinating what can be done with it.

20 years ago I was fascinated with the Mandelbrot set., and made a very heavily optimized program, examined black areas, how calculations progressed for single points, figured out how to determine if a point truly was black, or just needed more iterations... :roll:

Fourier transformations are also amazing especially for audio.
I don't understand how to do it myself, but I've used quite a lot of programs that depend on it heavily.

In the mid 80's it was considered impossible to compress audio, without a near equal loss of quality, audio was considered too complex and chaotic for compression. Seems the necessary math to do it, was discovered already in the early 18th century. :oops:

Fourier transforms are useful in SO MANY PLACES! Not just audio by a long shot. In computer vision, it's a very common operation (also wavelet transforms) :)

JPEG (and many/most video codecs) use a discrete cosine transform to remove high frequency ranges from the image. This increases the image entropy and improves compression ratios considerably.

chronus_aurelius
December 5th, 2009, 06:10 AM
The Mandelbrot set, because it's concept somewhat links discrete math with the calculus math. It also is very eye-opening in much the same way as Einsteins formula.

P_c: z\to z^2 + c,

http://en.wikipedia.org/wiki/Mandelbrot_set

phrostbyte
December 5th, 2009, 06:13 AM
The Mandelbrot set, because it's concept somewhat links discrete math with the calculus math. It also is very eye-opening in much the same way as Einsteins formula.

P_c: z\to z^2 + c,

http://en.wikipedia.org/wiki/Mandelbrot_set

It's amazing how so much (infinite) complexity can be uncovered with such a simple formula. :)

markthecarp
December 5th, 2009, 06:22 AM
I am a carpenter. A squared plus B squared equal C squared.

The smart engineer types forget the basics _often.

-mark
"Think, there must be a harder way." Joe Gessler 1954-2003
my old boss

Flying caveman
December 5th, 2009, 06:34 AM
all the time -value-of-money ones http://en.wikipedia.org/wiki/Time_value_of_money#Present_value_of_a_perpetuity

but especially the perpetuity one.

lisati
December 5th, 2009, 06:35 AM
It's amazing how so much (infinite) complexity can be uncovered with such a simple formula. :)

I was about to say something similar but probably more verbosely.

The special case of the Pythagorean theorem (3*3+4*4=5*5) is a contender too.

Marvin666
December 5th, 2009, 06:41 AM
I like the infinite series equation.
http://en.wikipedia.org/wiki/Infinite_series

ve4cib
December 5th, 2009, 08:50 AM
Euler's Identity is just plain weird, and whenever you show it to anyone they always react with confusion. It's great!

The equations for the Mandelbrot and Julia sets are also both pretty nice. Or the results they generate are nice when properly-rendered.

And finally there's the Fibonacci sequence. Maybe not the most beautiful of recursive equations, but it's still not bad.

wilee-nilee
December 5th, 2009, 10:43 AM
138625

soni1770
December 5th, 2009, 02:16 PM
138625

nice

BuffaloX
December 5th, 2009, 05:11 PM
Fourier transforms are useful in SO MANY PLACES! Not just audio by a long shot. In computer vision, it's a very common operation (also wavelet transforms) :)

JPEG (and many/most video codecs) use a discrete cosine transform to remove high frequency ranges from the image. This increases the image entropy and improves compression ratios considerably.

OK that's cool too, but graphics was always considered possible to compress and enhance.
For audio it made all kinds of cool stuff possible, that was previously considered impossible.

I am a carpenter. A squared plus B squared equal C squared.

The smart engineer types forget the basics _often.

-mark
"Think, there must be a harder way." Joe Gessler 1954-2003
my old boss

That brings back memories, my mother taught me that. :D

Euler's Identity is just plain weird, and whenever you show it to anyone they always react with confusion. It's great!

The equations for the Mandelbrot and Julia sets are also both pretty nice. Or the results they generate are nice when properly-rendered.

And finally there's the Fibonacci sequence. Maybe not the most beautiful of recursive equations, but it's still not bad.

What is this Euler thing good for?

I had some fun, when I found out I could make Julia sets from coordinates in the Mandelbrot set, and I succesfully made a realtime Julia set tracer for my mandelbrot program, on an old 486 system.

I have never seen the Fibonacci sequence used for anything practical, except as a programming exercise and in IQ tests.
I find it a little annoying, and sometimes I suspect people for using it to determine if people have any math knowledge at all.

koleoptero
December 5th, 2009, 05:21 PM
Excercise #52 in my measure theory notes attached.

phrostbyte
December 5th, 2009, 05:25 PM
What is this Euler thing good for?

Euler's identify uncovers very simple relationship between the numbers pi, e, 1, 0 and i. IMO it's quite profound.

3rag0n
December 5th, 2009, 05:28 PM
Cauchy's Integral Theorem. There is something immensely beautiful about being able to predict the value of an integral just by looking at the path.

LinuxFanBoi
December 5th, 2009, 05:45 PM
I like "PC - MS + X = True"

Hyporeal
December 5th, 2009, 05:45 PM
What is this Euler thing good for?

Euler's identity is the basis for linear circuit analysis in the presence of alternating current. A single complex number represents both the DC and AC components of a measurement.

I have never seen the Fibonacci sequence used for anything practical, except as a programming exercise and in IQ tests.
I find it a little annoying, and sometimes I suspect people for using it to determine if people have any math knowledge at all.

The Fibonacci sequence describes a number of natural phenomena, such as population growth and the curl of a nautilus shell. In a simple recursive function, it explains why the golden ratio appears so much in nature. That's as practical as I can get from my knowledge, but Wikipedia has a list of applications of the fibonacci sequence (http://en.wikipedia.org/wiki/Fibonacci_sequence#Applications).

SuperSonic4
December 5th, 2009, 05:48 PM
Euler's Equation

Nepherte
December 5th, 2009, 05:56 PM
The Navier-Stokes equations.

bobbob1016
December 5th, 2009, 06:10 PM
sqrt(-1)

But I like the one someone came up with in my HS Calc class:
Since you can algebraically do anything to one side you do to the other
0*anything = 0
Therefore 0*integral of (something I can't integrate) = ?*0
0 = 0
You only prove 0 = 0, but it is algebraically accurate

I've also written this on tests when I had no clue or no time to finish a problem:
LUE=42, this question falls under "everything" therefore the answer is 42
(LUE is Life, the Universe, and Everything from Hitch Hikers Guide)

JDShu
December 5th, 2009, 06:12 PM
What is this Euler thing good for?

According to my math prof back in uni, not a whole lot practically. Its a special case. However, the idea that it includes every kind of number makes it amazing even to the common man.

NoaHall
December 5th, 2009, 06:12 PM
I like all equations, except 1/2absinC.

jomiolto
December 5th, 2009, 06:27 PM
What is this Euler thing good for?

It's magic! ;)

Euler's identity itself is not terribly useful (I believe?), but it is a special case of Euler's formula (http://en.wikipedia.org/wiki/Euler%27s_formula), which is quite useful -- for example, it is used in Fourier transform (IIRC).

3rag0n
December 5th, 2009, 07:04 PM

Every first-year undergrad student who learns either electronics or math or signal processing or communication systems has to learn about this. Practically every single identity, formula, theorem flows from knowledge of it. FM radio, wave modulation, speech recognition in your phones/computers, audio editing, image processing ( the cool effects you use in GIMP ) - everything uses THAT identity [ it's general form, actually - although the general form is even more beautiful, i think ].

chucky chuckaluck
December 5th, 2009, 07:32 PM
"two plus two equals five"

mathyou
December 5th, 2009, 07:50 PM
+1 for Euler's identity.

Surprised no-one has mentioned Schrodinger's equation, pretty amazing.

rajeev1204
December 5th, 2009, 07:53 PM
THis thread should become the star thread on the forums some day.Its fascinating.

Too bad i failed in maths. :D

phrostbyte
December 5th, 2009, 08:26 PM
I'll mention Euler's formula, which is related to Euler's identity:

e^(ix) = cos(x) + i*sin(x)

This allows you to use e to the power of a complex number as replacement for sine or cosine, ie. in Fourier analysis.

BuffaloX
December 5th, 2009, 08:36 PM
Euler's identify uncovers very simple relationship between the numbers pi, e, 1, 0 and i. IMO it's quite profound.

Wow that sounds weird, like impossible. :shock:

Euler's identity is the basis for linear circuit analysis in the presence of alternating current. A single complex number represents both the DC and AC components of a measurement.

The Fibonacci sequence describes a number of natural phenomena, such as population growth and the curl of a nautilus shell. In a simple recursive function, it explains why the golden ratio appears so much in nature. That's as practical as I can get from my knowledge, but Wikipedia has a list of applications of the fibonacci sequence (http://en.wikipedia.org/wiki/Fibonacci_sequence#Applications).

Thanx on both counts, not that I'm now going to use either a whole lot now, but I like to know how this stuff is actually useful.

According to my math prof back in uni, not a whole lot practically. Its a special case. However, the idea that it includes every kind of number makes it amazing even to the common man.

Every kind of number? OK doesn't matter, I can see how it's cool.

It's magic! ;)

Euler's identity itself is not terribly useful (I believe?), but it is a special case of Euler's formula (http://en.wikipedia.org/wiki/Euler%27s_formula), which is quite useful -- for example, it is used in Fourier transform (IIRC).

Someone once said that sufficiently sophisticated technology will be indistinguishable from magic, I guess that goes for math too. :p

Thanx guys a lot of nice answers. :popcorn:

phrostbyte
December 5th, 2009, 08:42 PM
Wow that sounds weird, like impossible. :shock:

Definitely shocking. That's part of what it makes it so cool. :) I don't know of any explanation of "why it is", other then "it is".

phrostbyte
December 5th, 2009, 08:56 PM
Another cool equation is that of the Riemann zeta function. There is a powerful relation between the zeta function and prime numbers which I don't quite understand.

zeta(s) = sigma from 1 to infinity of n in 1/n^s

It's similar in idea to the infinite p-series in Calculus. The parameter (s) to the zeta function is actually the p in the p-series.

Keyper7
December 5th, 2009, 11:53 PM
Wow that sounds weird, like impossible. :shock:

Euler's identity is simply amazing. It uses:

- pi, the fundamental constant of geometry
- e, the fundamental constant of analysis
- 1, the real unit
- i, the imaginary unit
- the basic operation of addition
- the basic operation of multiplication
- the basic operation of exponentiation

By combining one and only one instance of each, it arrives at... ZERO.

Can't get any more beautiful than that.

Zoot7
December 5th, 2009, 11:56 PM
My favourite is the FFT (Fast Fourier Transform).

Not an equation, rather an algorithm, but really nifty and efficient.

Zoot7
December 6th, 2009, 12:03 AM
Euler's identity is simply amazing. It uses:

- pi, the fundamental constant of geometry
- e, the fundamental constant of analysis
- 1, the real unit
- i, the imaginary unit
- the basic operation of addition
- the basic operation of multiplication
- the basic operation of exponentiation

By combining one and only one instance of each, it arrives at... ZERO.

Can't get any more beautiful than that.
Yup, got to love

e^(j*pi) + 1 = 0

:)

papangul
January 10th, 2010, 05:20 AM
'Most beautiful' math structure appears in lab for first time:

http://www.newscientist.com/article/dn18356-most-beautiful-math-structure-appears-in-lab-for-first-time.html

judge jankum
January 10th, 2010, 05:22 AM
2+2 for me....and I finally learned it lol!!!