# What is the most beautiful mathematical concept ever created by man?



## Philip

What is the most beautiful and useful mathematical concept ever created or discovered by man?


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## mmsbls

I wonder what your answer will be? I will have to think a bit. When you include both beauty and usefulness that makes it harder.

In the meantime I will say the most astonishing mathematical fact I've ever encountered is that:

e^(i*pi) = -1

Even though I understand the identity, I find it almost incomprehensible as though reality is playing tricks with us.


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## ComposerOfAvantGarde

1+1=2

.

.


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## Moira

The Zero. 

Could you imagine working with Roman numerals for ever?


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## Philip

mmsbls said:


> e^(i*pi) = -1


It is indeed the most beautiful relation in mathematics. It literally brings tears to my eyes when i try to make sense of it.


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## Kopachris

Limits. Those were the most beautiful things to me when I figured them out and it all clicked in my mind, and I instantly understood derivation and integration. Can't have your blessed _e_ without limits. 

Regarding Euler's identity: it shall be my quest to go from _a_[sup]2[/sup]+_b_[sup]2[/sup]=_c_[sup]2[/sup] to _e[sup]iπ[/sup]_-1=0.


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## aleazk

mmm, hard. I would say the concept of a _differentiable manifold_, which is used to describe spacetime, for example.


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## aleazk

what about the _derivative_?, certainly is one of the most useful concepts.


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## ComposerOfAvantGarde

Here's a new one that I just worked out:

1+1=3


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## aleazk

ComposerOfAvantGarde said:


> Here's a new one that I just worked out:
> 
> 1+1=3


Oh, what a surprise, a thread derailed by CoAG.


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## Ukko

Energy = mass times the square of the velocity. Changed baseball.


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## ComposerOfAvantGarde

^ That's physics. Stop before my brain explodes.


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## Philip

Kopachris said:


> Limits. Those were the most beautiful things to me when I figured them out and it all clicked in my mind, and I instantly understood derivation and integration. Can't have your blessed _e_ without limits.
> 
> Regarding Euler's identity: it shall be my quest to go from _a_[SUP]2[/SUP]+_b_[SUP]2[/SUP]=_c_[SUP]2[/SUP] to _e[SUP]iπ[/SUP]_-1=0.


Well, the derivative is certainly one of the most useful things. You don't really need limits to understand it, though. I don't know if the expression "rate of change" is common in english, but once you've understood that the derivative is the _taux de variation_ (in french), you've basically grasped derivation and integration.

In the same way, _e_ can be thought of simply as the exponential base which yields a rate of change (slope) equal to its power. That's certainly the most intuitive way of looking at it.

Sums are quite revealing when you realize that an infinite sum can add up to a finite number. But concepts like sums and limits remind me too much of intro calculus exams 

In algebra it's also insightful to know that a[SUP]2[/SUP] + b[SUP]2[/SUP] = c[SUP]2[/SUP] can be generalized to n dimensions.


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## aleazk

Philip said:


> Well, the derivative is certainly one of the most useful things. You don't really need limits to understand it, though. I don't know if the expression "rate of change" is common in english, but once you've understood that the derivative is the _taux de variation_ (in french), you've basically grasped derivation and integration.
> 
> In the same way, _e_ can be thought of simply as the exponential base which yields a rate of change (slope) equal to its power. That's certainly the most intuitive way of looking at it.
> 
> Sums are quite revealing when you realize that an infinite sum can add up to a finite number. But concepts like sums and limits remind me too much of intro calculus exams
> 
> In algebra it's also insightful to know that a[SUP]2[/SUP] + b[SUP]2[/SUP] = c[SUP]2[/SUP] can be generalized to n dimensions.


In differential geometry, for example, derivative operators are defined axiomatically by some properties like linearity and the Leibniz rule. So, basically, those properties define the capability of a operator for being a 'rate of change' operator.


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## Kopachris

Philip said:


> Well, the derivative is certainly one of the most useful things. You don't really need limits to understand it, though. I don't know if the expression "rate of change" is common in english, but once you've understood that the derivative is the _taux de variation_ (in french), you've basically grasped derivation and integration.
> 
> In the same way, _e_ can be thought of simply as the exponential base which yields a rate of change (slope) equal to its power. That's certainly the most intuitive way of looking at it.
> 
> Sums are quite revealing when you realize that an infinite sum can add up to a finite number. But concepts like sums and limits remind me too much of intro calculus exams
> 
> In algebra it's also insightful to know that a[SUP]2[/SUP] + b[SUP]2[/SUP] = c[SUP]2[/SUP] can be generalized to n dimensions.


When I was in high school, I had a hard time figuring out derivatives because I just couldn't wrap my head around the fact that I was basically finding the slope of each and every point. Then I read about limits and it all clicked together. Ah, oh yes, figuring out that the Pythagorean theorem could be generalized to however many dimensions I wanted was another big moment for me in high school, because that meant the distance formula could be generalized to _n_ dimensions as well.


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## Art Rock

The golden ratio. Big help in composition of photographs.


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## Couchie

Well are you asking beautiful, or useful?

_Useful _is easily numerical analysis... but I'm sure pure math snobs will cry at its lack of beauty.


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## Philip

Couchie said:


> Well are you asking beautiful, or useful?
> 
> _Useful _is easily numerical analysis... but I'm sure pure math snobs will cry at its lack of beauty.


But numerical analysis is so broad...

I said both beautiful and useful because i had in mind the Fourier transform, which excels at both. I guess everything is useful... but it's astonishing how it can be useful in so many fields while being such a high level concept. All its babies (DFT, FFT, DCT, MDCT, etc.) are everywhere... and its mother, the Laplace transform, is very noble and powerful.

Perhaps i meant elegant.


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## aleazk

Couchie said:


> Well are you asking beautiful, or useful?
> 
> _Useful _is easily numerical analysis... but I'm sure pure math snobs will cry at its lack of beauty.


, is awful.


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## cwarchc

Pythagoras theorem. Surely both simple and elegant?


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## HarpsichordConcerto

Euler's identity, the relationship between the derivative and the integral, the radian (more a definition), Lagrange multiplier; these are the more practical ones that I come across.


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## Lenfer

The fact that numbers are infinite provides me with some solace, I don't know why. :tiphat:


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## Rangstrom

For those that want a little mystery in life, I would say Goedel's Incompleteness Theorem fits the bill.


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## TxllxT

Philip said:


> What is the most beautiful and useful mathematical concept ever created or discovered by man?


I would like to have A) an example of a mathematical concept *created* by man, B) an example of a mathematical concept *discovered* by man. I can understand that Leonardo da Vinci created the Mona Lisa. Does there exist a parallel in mathematics?


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## Philip

TxllxT said:


> I would like to have A) an example of a mathematical concept *created* by man, B) an example of a mathematical concept *discovered* by man. I can understand that Leonardo da Vinci created the Mona Lisa. Does there exist a parallel in mathematics?


It is debated whether maths are discovered or created, that's why i included both terms.

You could argue that the decimal numeral system was _created_, but the fact that 1+1=2 was _discovered_.


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## Krisena

Mathematical concepts are not created by man, they're inherent to the universe.


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## Krisena

I'm sorry, I didn't see that someone had already pointed this out.

...Why can't I edit my posts?


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## aleazk

Krisena said:


> Mathematical concepts are not created by man, they're inherent to the universe.


Mathematics is a field created by man in which you use logic to investigate the properties of some objects and concepts defined and invented by man.
On the other hand, in physics, you assume that the physical reality obeys logic. So, when you find that some mathematical concept is useful to describe some physical entity, you can use the mathematical properties of this mathematical concept to make new physical predictions (because those properties follow logically from the mathematical concept and you assume that the physical reality obeys logic). That's the reason why mathematics is useful in physics.


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## Philip

aleazk said:


> Mathematics is a field created by man in which you use logic to investigate the properties of some objects and concepts defined and invented by man.


...and whether or not these concepts are inherently _applicable_ to the universe is an entirely different story.


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## TxllxT

aleazk said:


> Mathematics is a field created by man in which you use logic to investigate the properties of some objects and concepts defined and invented by man.
> On the other hand, in physics, you assume that the physical reality obeys logic. So, when you find that some mathematical concept is useful to describe some physical entity, you can use the mathematical properties of this mathematical concept to make new physical predictions (because those properties follow logically from the mathematical concept and you assume that the physical reality obeys logic). That's the reason why mathematics is useful in physics.


Interesting! But the 'field' of mathematics does look more like a field of chess than a field of onions, doesn't it? The reason why I tend to dislike mathematics is this similarity with playing a game.


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## Chris

Euler's relation? Fourier transforms? Pah! What could be more elegant than the eleven times table? What other table has such beautiful symmetry? (Up to 99 anyway. It gets untidy above 100)

1 * 11 = 11
2 * 11 = 22
3 * 11 = 33
4 * 11 = 44
5 * 11 = 55
6 * 11 = 66
7 * 11 = 77
8 * 11 = 88
9 * 11 = 99
10 * 11 = 110
11 * 11 = 121
12 * 11 = 132


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## ComposerOfAvantGarde

Who does up to 12x tables these days? I thought that the metric system was better.


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## Philip

I've got a message from a friend:



Dodecaplex said:


> The most beautiful concept in mathematics is Abraham Robinson's use of the Transfer Principle, which finally made it possible to present a rigorous construction of the hyperreals, which finally gave us the beautiful infinitesimals that Leibniz dreamed of, which finally did away with the boring concept of limits.


Cheers


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## Chris

Philip said:


> I've got a message from a friend:
> 
> 
> 
> 
> The most beautiful concept in mathematics is Abraham Robinson's use of the Transfer Principle, which finally made it possible to present a rigorous construction of the hyperreals, which finally gave us the beautiful infinitesimals that Leibniz dreamed of, which finally did away with the boring concept of limits.
> 
> 
> 
> Cheers
Click to expand...

Is that harder than the eleven times table?


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## Philip

Chris said:


> Is that harder than the eleven times table?


The 11x table is merely an artifact of these numbers being expressed in decimal form, try it in binary or hexadecimal and the symmetry completely crumbles. Sorry.


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## Philip

...that's one mistake beginners often make when trying to analyze patterns in prime numbers. You must make abstraction of the number system and treat the numbers for what they are, for they are not simple drawings!

I'm surprised prime numbers haven't been mentioned yet.


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## Chris

Philip said:


> I'm surprised prime numbers haven't been mentioned yet.


It's going back a long time but I remember when PCs first became available somebody joined several of them together to search for beastly palindromic primes; that is, palindromic prime numbers containing 666.

A quick google search reveals the knowledge has not been lost. Here is a beastly palindromic prime:

*1000000000000066600000000000001*


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## science

Philip said:


> The 11x table is merely an artifact of these numbers being expressed in decimal form, try it in binary or hexadecimal and the symmetry completely crumbles. Sorry.


But other apparent symmetries appear.

When I was in high school, I wasted many hours trying to do math in base-7. I thought it would make something easier. But no, it made nothing easier. It did impress upon me the arbitrariness of the meaning of symbols like "1001" though.


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## science

My favorite is the Euler identity, but earlier someone mentioned the Pythagorean Theorem, and that is a good one that is easily overlooked. 

I teach history and literature, but fortunately in my classes we have time to talk about ancient Greek geometry, zero, Descartes and Euler. One of the things I really enjoy showing my students (most of them have had at least a year of algebra) is how all those curves - circles, parabolas, etc. - are actually just applications of the Pythagorean Theorem. One of the things that I most enjoy about math is how it hangs together like that. 

(When I teach Euler I show them the Euler identity, and then I tell them that I won't be able to tell them anything about math for the rest of history because I don't understand it. Someday I'd like to get myself up to be able to show them something about Maxwell's equations, which for now I can't understand.)


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## Couchie

The most profound:

0.9999999999999... = 1


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## Toddlertoddy

http://forums.xkcd.com/viewtopic.php?t=4507


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## emiellucifuge

Maybe a little too broad but I LOVE complex numbers. The fact that you can take a problem "out of reality" so to speak, and then back in order to solve it is just so awesome.


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## Couchie

Just joking of course.


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## Chris

Couchie said:


>


I was just doing that in my head when you posted


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## graaf

Lobachevskian geometry.


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## clavichorder

ComposerOfAvantGarde said:


> 1+1=2
> 
> .
> 
> .


Plenty of creatures that came before man created that concept though!


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## Renaissance

Hyperbolic partial differential equations.









Well...all music is first of all, sound.


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## dionisio

Couchie said:


> Just joking of course.


Hehehe i wonder how many members here belong to science/engineering...

Navier-Stokes general equation (in this case for cylindrical axes) is the most important equation for Momentum Transfer Phenomena. But it comes from Newton's 2nd law. There's no general solution and one most always look for barrier conditions.

The most important concept for modern world is surely Differential Calculus. Everything comes afterwards, specially mechanics, quantum mechanics, EDO's, optimization, integral calculus, numerical analysis and all fields in engineering.


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## etkearne

As a holder of a ****** in Mathematics who almost got a Ph.D. In "pure" (as they call it- it just means theoretical) mathematics, I would say the most beautiful mathematical concept is abstraction in general. A let down, yes, but it is at the root of the most beautiful mathematics.

To take a real world scenario, understand it's every essence, then to create hypothetical "out of this world" versions, which spur into ENTIRE fields of mathematics (think Vector Spaces and Functional Analysis), which then are stripped down to THE ABSOLUTE bare bones to the point where such a theorem could exist in any hypothetical platform, even infinite dimensional, is the holy grail of pure Mathematics, and is best exemplified by something like the definition of a Compact Set and Point-Set Toplogy in general. It makes me miss it a bit. 

But then I think of the favoritism and elitism in mathematical academia and it makes me want to projectile vomit!


Edit: the asterisked word is the abbreviation for Bachelor's of Science, which must have been seen as meaning "male cow feces" by the forum.


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## Philip

All math is beautiful until the exam.


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## Novelette

Poisson processes, used a certain way, can be used to turn a continuous gamma distribution into a discrete Poisson distribution with variable Alpha/Beta.

I used that little trick in Calculus 3 in college to get through my final exam in half the time.


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## PetrB

Zero, as in '0.'


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## Bone

etkearne said:


> As a holder of a ****** in Mathematics who almost got a Ph.D. In "pure" (as they call it- it just means theoretical) mathematics, I would say the most beautiful mathematical concept is abstraction in general. A let down, yes, but it is at the root of the most beautiful mathematics.
> 
> To take a real world scenario, understand it's every essence, then to create hypothetical "out of this world" versions, which spur into ENTIRE fields of mathematics (think Vector Spaces and Functional Analysis), which then are stripped down to THE ABSOLUTE bare bones to the point where such a theorem could exist in any hypothetical platform, even infinite dimensional, is the holy grail of pure Mathematics, and is best exemplified by something like the definition of a Compact Set and Point-Set Toplogy in general. It makes me miss it a bit.
> 
> But then I think of the favoritism and elitism in mathematical academia and it makes me want to projectile vomit!
> 
> Edit: the asterisked word is the abbreviation for Bachelor's of Science, which must have been seen as meaning "male cow feces" by the forum.


Agreed. Abstraction is the transformative concept behind human thought.


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## Kieran

I'd say the best of them, or right up there, was the guy who was taking a bath when an apple fell on his head...


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## Ryan

The minimal surface equation


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## Taggart

Kieran said:


> I'd say the best of them, or right up there, was the guy who was taking a bath when an apple fell on his head...


Reminds one of the Dr Who episode "The Talons of WengChiang" where The Doctor (Tom Baker) tells to his companion, Leela, that "Eureka" is Greek for "this bath is too hot."


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## Ingélou

My violin teacher feels a fascination for *prime numbers* and has passed it on to me - unless it's because I was born on the 13th of the 5th?


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## KenOC

Fourier transform? And here I thought he was a cellist. 

BTW if you like primes, you may want to read about Srinivasa Ramanujan. He was one of the greatest "natural" self-taught mathematicians of all time, who died in 1920 at the age of 32. He did a lot of totally original and unexpected work with prime numbers.

Not related to primes, but the British mathematician G.H. Hardy recalled, 'I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." '

In fact, 1729 is still known as the Ramanujan Number and has its own Wiki entry!


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## Crudblud

KenOC said:


> Fourier transform? And here I thought he was a cellist.


He only transforms in his spare time.


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## Huilunsoittaja

This is quite beautiful to me:


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## KenOC

Huilunsoittaja said:


> This is quite beautiful to me:
> 
> View attachment 16998


Not sure what this is. It appears to be a magic square, with each row and column containing all the number's from 0 through 11 (and thus adding to 66). Other magical properties?


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## Kopachris

KenOC said:


> Not sure what this is. It appears to be a magic square, with each row and column containing all the number's from 0 through 11 (and thus adding to 66). Other magical properties?


It's a tone matrix, showing a tone row with all its transpositions, inversions, and reversions. Used in 12-tone composition.


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## KenOC

Kopachris said:


> It's a tone matrix, showing a tone row with all its transpositions, inversions, and reversions. Used in 12-tone composition.


Well, Bach might have got his jollies off that. But he would have insisted that the two major diagonals have the same characteristics as the rows and columns (they don't). He was a bit fussy.


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## ProudSquire

Math...... my arch-nemesis.


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## BlazeGlory

I realize that in the final part of the video one could substitute one's own words or phrases and come up with different results but overall I think it's quite interesting. The music alone is worth the view.


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## Kieran

Huilunsoittaja said:


> This is quite beautiful to me:
> 
> View attachment 16998


Does it come in blue?


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## Novelette

Recursion is an incredibly useful tool! Recursive probability methods have saved my life a number of times.

Euler's Method of converting continuous forward Markov Chains into discrete, incremental transition probabilities is also very useful. It helps skip some very tedious integration.  I like calculus, but it really can be time-consuming and tedious.

What has never ceased to amaze me is the mysticism with which most branches of advanced mathematics are held. Calculus is no more difficult than high school algebra. In fact, Euclidean Geometry is a good analogy. The ends are impressive and the referents are such that make the whole field appear difficult. But the procedures of calculating are almost exclusively algebraic and arithmetic. It's nothing but algebra in a certain sequence of steps. Calculus is often no different. Integrating and differentiating is an algebraic process.

Learn algebra and all you need to do is learn certain steps and concepts, then you are easily equipped to handle calculus like a pro.  

And maybe that's the ultimate beauty in mathematics: to take the most disgustingly complicated figure, and find that it simplifies into the most elegantly simple thing! Math is all about finding an easier way. Euler's method is, especially, a useful way to approximate integral values by means of a little geometry. It's good stuff.


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## millionrainbows

Calculus is where most people lose the trail. It is emphasized after algebra because of its usefulness in our culture of war for calculating trajectories of projectiles, artillery shells, and missiles.

There are many other interesting areas of math. My favorites are number and set theory, statistics, and ratios/fractions, especially as applied to music.


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## millionrainbows

Kopachris said:


> It's a tone matrix, showing a tone row with all its transpositions, inversions, and reversions. Used in 12-tone composition.


Yeah, but usually in early 12-tone, pitch names are used, but here they are replaced by numbers, so it's probably a newer serial chart.

If P0 (top row) were pitches and C=0, then the top row would be C-Db-Eb-A-D-B-E-Bb-G-Ab-F-F#.

Some observations: the row begins with C, and ends on F#, a tritone relation.

The first hexad has stepwise relations, and spans a fifth, A-E, and yields A-B-C-D-Eb-E if rearranged.

The second hexad is very chromatic, and also spans a fifth, E-B, and yields E-F-F#-G-Ab-B if rearranged.

These "boundary" intervals of fifths will aid in creating a stable harmonic effect. The two hexads will provide contrast, one giving chromatic relations, the other giving stepwise.

It looks like a row with possibilities.


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## Novelette

millionrainbows said:


> Calculus is where most people lose the trail. It is emphasized after algebra because of its usefulness in our culture of war for calculating trajectories of projectiles, artillery shells, and missiles.
> 
> There are many other interesting areas of math. My favorites are number and set theory, statistics, and ratios/fractions, especially as applied to music.


It's fun to integrate over whole distributions to calculate perfect probabilities for statistics. It's fun to ask people to try to integrate the Gaussian function for normal distribution values.

My work calls for calculus almost exclusively for probability calculation, both deterministic and stochastic. I used to be completely indifferent to math when I was in high school and college. I never imagined that I would take a liking to it.


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## millionrainbows

Novelette said:


> It's fun to integrate over whole distributions to calculate perfect probabilities for statistics. It's fun to ask people to try to integrate the Gaussian function for normal distribution values.
> 
> My work calls for calculus almost exclusively for probability calculation, both deterministic and stochastic. I used to be completely indifferent to math when I was in high school and college. I never imagined that I would take a liking to it.


Yes, that's interesting. BTW, were you in high school band? Do you know how to march? Could you teach other people to march?


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## Guest

Well, I looked up Fourier Transform, but it just goes to show that an appreciation of beauty can't come before understanding...


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## KenOC

I suppose it's the Pythagorean theorem. The area of a square formed on the hypotenuse of a right triangle is equal to the sum of the squares formed on the other two sides. Pythagoras was able to figure this out, and to prove it, without even a decent numbering system!


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## Guest

KenOC said:


> I suppose it's the Pythagorean theorem. The area of a square formed on the hypotenuse of a right triangle is equal to the sum of the squares formed on the other two sides. Pythagoras was able to figure this out, and to prove it, without even a decent numbering system!


I love watching good teachers demonstrate the proof to their classes. It's like magic!


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## millionrainbows

KenOC said:


> I suppose it's the Pythagorean theorem. The area of a square formed on the hypotenuse of a right triangle is equal to the sum of the squares formed on the other two sides. Pythagoras was able to figure this out, and to prove it, without even a decent numbering system!


Well, since Geometry is visual...


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## millionrainbows

http://amzn.com/0802778135

"The quadrivium-the classical curriculum-comprises the four liberal arts of number, geometry, music, and cosmology (astronomy). It was studied from antiquity to the Renaissance as a way of glimpsing the nature of reality. Geometry is number in space; music is number in time; and cosmology expresses number in space and time. Number, music, and geometry are metaphysical truths: life across the universe investigates them; they foreshadow the physical sciences."


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## Tristan

I love the properties of the number 142857. Not exactly useful, but beautiful nonetheless.

1/7 = 0.142857142857...

Notice 14, 28, 5(6)...(multiples of 7).

142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142
142857 x 7 = 999999

Even multiplying it by larger numbers will often get some part of the original number inside it:

142857 x 39 = 5*57142*3
142857 x 652 = 93*142*764


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## EddieRUKiddingVarese

433, it just has some sort of mystical properties.


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## Couchie

dionisio said:


> The most important concept for modern world is surely Differential Calculus. Everything comes afterwards, specially mechanics, quantum mechanics, EDO's, optimization, integral calculus, numerical analysis and all fields in engineering.


Surely arithmetic and algebra are vastly more important than differential calculus. Most people have no idea what calculus even is. You cannot function in modern society without arithmetic, and most practical engineering problems are solved with a basic application of algebra. As for differential equations, the vast majority of engineers will never revisit them after graduation.


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## Stavrogin

I remember from my University days a wonderful graphical demonstration of the innumerability of the numbers between 0 and 1.

Too bad I don't remember the name of this demonstration or of the scientist 

Any of you can help me?

The demonstration had to to with a circle, tangent to the line at 0.5 and with a diameter of 1, and the line segments that connect the centre of the circle to all the points in the line.


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## Stavrogin

Also the TREE function is a very beautiful and funny "little" thing.


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