View Full Version : The Euler Formula... what am I looking at?

meshpig

09-20-2009, 06:16 AM

I went to so many different secondary schools my mathematical understanding doesn't practically go much beyond Trigonometry.

Given that I'm no mathematician, what is Euler doing with his imaginary unit and why is it "beautiful"?

http://mathworld.wolfram.com/EulerFormula.html

I can see how it might be perfect but the penny hasn't dropped.

PS.http://en.wikipedia.org/wiki/Leonhard_Euler

jameswillmott

09-20-2009, 06:22 AM

I went to so many different secondary schools my mathematical understanding doesn't practically go much beyond Trigonometry.

Given that I'm no mathematician, what is Euler doing with his imaginary unit and why is it "beautiful"?

http://mathworld.wolfram.com/EulerFormula.html

I can see how it might be perfect but the penny hasn't dropped.

Beauty is mostly subjective, don't worry if you don't find it 'beautiful'.

meshpig

09-20-2009, 06:32 AM

"Beauty" as in something complete without need for addition... is enough for me without getting all neo-kantian. Bit of mathematical newb!

Mark The Great

09-20-2009, 07:46 AM

Well, I think that the beauty comes from two sources.

The first, is stated on the page:

'[This is] an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero), the fundamental operations +, ×, and exponentiation, the most important relation =, and nothing else. '

It's not often that you find something like that.

The second instance that I saw was in his integral proof (it's early, and I don't feel like messing with series one). He makes very clever use of the properties of i to transform a set of trigonometric functions into an exponential. Note all the clever cancellations that result from multiplying/dividing negative numbers by i. These occur because i = sqrt(-1), so when you divide 1 or -1 by i, it's like taking the square root, which results in i (or -i).

Note also the clever substitution of his original equation towards the end, which gets his result.

For math nerds everywhere, who struggle through tough formulas and processes, so see something this big done so easily is beautiful.

[edit] Also, if I'm not mistaken, this identity allows you to solve any other trig identity, which is nice too.

OnlineRender

09-20-2009, 01:22 PM

'[This is] an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero), the fundamental operations +, ×, and exponentiation, the most important relation =, and nothing else. '

It's not often that you find something like that.

:bowdown:

meshpig

09-21-2009, 02:01 AM

Well, I think that the beauty comes from two sources.

The first, is stated on the page:

'[This is] an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero), the fundamental operations +, ×, and exponentiation, the most important relation =, and nothing else. '

It's not often that you find something like that.

The second instance that I saw was in his integral proof (it's early, and I don't feel like messing with series one). He makes very clever use of the properties of i to transform a set of trigonometric functions into an exponential. Note all the clever cancellations that result from multiplying/dividing negative numbers by i. These occur because i = sqrt(-1), so when you divide 1 or -1 by i, it's like taking the square root, which results in i (or -i).

Note also the clever substitution of his original equation towards the end, which gets his result.

For math nerds everywhere, who struggle through tough formulas and processes, so see something this big done so easily is beautiful.

[edit] Also, if I'm not mistaken, this identity allows you to solve any other trig identity, which is nice too.

Thanks for that, didn't mean to do your head in before breakfast. Yes, I see your point now that I've done a bit more reading and stirred up the grey matter.

So, with i and -i you create an imaginary object by defining its properties in the real world and see what the logical consequences are and in the case of Euler's formula they turned out to be very neat and tidy?

Mmm, sounds like fun!:)

PS. I'm guessing but Im and Re are "real" and "imaginary"?

biliousfrog

09-21-2009, 08:08 AM

You lost me at trigonometry :D

aurora

09-21-2009, 08:19 AM

Oh trust me it truly is beautiful. Without it, solving many integrals for ODE's, PDE's (especially in Fourier series) and other things are dang near impossible, which hey, is often the case anyways.

Its not the only formula of this type either but it is the one most often seen. The others, sans other series types, are seen only in higher maths such as the Gamma, Beta, Bessel, ect functions.

meshpig

09-22-2009, 03:43 AM

Oh trust me it truly is beautiful. Without it, solving many integrals for ODE's, PDE's (especially in Fourier series) and other things are dang near impossible, which hey, is often the case anyways.

Its not the only formula of this type either but it is the one most often seen. The others, sans other series types, are seen only in higher maths such as the Gamma, Beta, Bessel, ect functions.

Must say I'm envious in a way. I feel a bit like I did when I first started playing classical violin when confronted with Maths now.

There one needed to make a whole lot of bad sounds and work on correcting them, which happens pretty quickly but the violin isn't like the guitar where you can get a good sound straight away:)

aurora

09-22-2009, 08:26 AM

PS. I'm guessing but Im and Re are "real" and "imaginary"?

Yep, Imaginary numbers are mapped to Z space but often can be transformed into real space with different types of conformal transforms. Imaginaries are usually treated as taboo, ignored and thus falsely labled, you know like in grade school when you learn you can't divide by zero (x/0) as its undefined. Sadly that's a lie it is defined, its infinity. Whats not a lie about it is what kind of infinity it is, that's still undefined as is what kinda of infinity you get when you add -infinity with +infinity.

The importance of Eulers formula is that you often get results from ODE (ordinary differential equations) and PDE's (partial differential equations) that has both real and imaginary parts. Since almost everything in the physical world is modeled with ODE's/PDE's imaginary results make little to no sense. But with Eulers we can convert the imaginary results into real results that do make sense and do actually match the data.

There are other great results achieved with Eulers formula but this is one of the more important ones.

Also don't confuse Euler formula with Eulers equation. Eulers equation is a numerical algorithm for solving ODE's its the simplest method to learn thus what most people know, sadly its also the worst method. Runge-Kutta methods are more accurate, thus also slower, and is the one most commonly use. But depending on you equations there are others that may work much better for your numerical solution.

Why is that important? One can find derivatives for any equation but integration, which is basically anti-derivation, not so much so. In fact there are more unsolvable equations then there are solvable. Meaning Many ODEs, and most non-linear ODE's as well as PDE's have no analytical solutions, but using alg's like Eulers equation we can get extremely accurate numerical solutions.

Why is that important? Its one of the bases of CG programs. Its also why it can take so long for some parts to go so slow, as numerically solving can require looping many times through complex equations and then you end up using lots of those and it all adds up and fast too.

Mark The Great

09-22-2009, 01:59 PM

I'm not sure about x/0 being undefined. The limit x/0 as x approaches zero doesn't exist, right? I mean, from the right, you get positive infinity, but from the left you get negative infinity.

I think I remember hearing something about positive and negative infinity being the same thing, but I'm a little foggy on that.

Andyjaggy

09-22-2009, 03:14 PM

I'm glad there are smart people out there.

jameswillmott

09-22-2009, 04:03 PM

I'm not sure about x/0 being undefined. The limit x/0 as x approaches zero doesn't exist, right? I mean, from the right, you get positive infinity, but from the left you get negative infinity.

I think I remember hearing something about positive and negative infinity being the same thing, but I'm a little foggy on that.

I think it is defined in 'higher' mathematics as unsigned infinity.

But if you try to work out how to distribute 10 apples among 0 people ( 10 divided by 0 ) ; how many apples does each person get becomes a nonsensical question, because there are no people.

aurora

09-22-2009, 05:33 PM

I think it is defined in 'higher' mathematics as unsigned infinity.

Yes, unsigned and uncountable. Its the uncountable part that always gets me, intuitively I always think that it should be countable even if you do assume you are in Reals, which s uncountable, you are still restricted to a smaller mapping meaning it should be countably infinite. I guess thats why I'm a lowly math grad 'student'.

jameswillmott

09-22-2009, 05:38 PM

Yes, unsigned and uncountable. Its the uncountable part that always gets me, intuitively I always think that it should be countable even if you do assume you are in Reals, which s uncountable, you are still restricted to a smaller mapping meaning it should be countably infinite. I guess thats why I'm a lowly math grad 'student'.

Uncountable? Wouldn't any infinite value be uncountable?

aurora

09-22-2009, 06:11 PM

Nope, there's both countable infinity and uncountable. In the simplest case of countable consider the mapping of Integers; 1,2,3,4,5, you can count them and if you did it infinitely long you would have a countable infinity. For the simplest case of uncountable consider the Reals. While you can count the integer portions think about the domain of just [0,1] you can subdivide it into smaller subdivision say 1/2, 2/2, 3/2, 4/2, 5/2, ect which now becomes an infinite series. Now think of doing that for the next domain [1,2], then [2,3] ect. Now consider going back and subdividing the domain between [0, 1/2] into an infinite series. Then expand that out over [0,1] then [1, ...], then going back and subdividing you subdomain again, and on and on, that's an uncountable infinity.

The fun part is there's other infinities, such as infinitesimals, which can have strange properties of their own.

meshpig

09-23-2009, 04:27 AM

aurora. That's very interesting and clear, thanks! I think I have a way to go yet, can't actually remember doing a differentiation.

... Meaning Many ODEs, and most non-linear ODE's as well as PDE's have no analytical solutions, but using alg's like Eulers equation we can get extremely accurate numerical solutions.

Is that because the solutions inherently don't exist or are in some way not yet conceivable? Presumably in Euler's day there were fewer unsolved equations than there are now.

- off topic. I woke up this morning and thought we'd been moved to Mars. Massive dust storm... terrible pic but represents the colouir of the sky adequately.

aurora

09-23-2009, 08:03 AM

Is that because the solutions inherently don't exist or are in some way not yet conceivable? Presumably in Euler's day there were fewer unsolved equations than there are now.

That's a Big/Little Endian kinda question. Some will say there is no analytical answer, end of discussion. Theorists will say we just don't know how to do it. In the end they both still eat the egg using numerical solutions so does it really matter? Well sometimes yes. It would awfully damn nice to be able to step from one system to the next without having to compound errors due to numerical solutions, welcome to the world of physics.

That image is cool. I woke up to snow yesterday meaning a gray sky, the red is so much cooler. But I have to admit I'm a winter lover so so is always welcome to me!

meshpig

09-24-2009, 03:50 AM

That's a Big/Little Endian kinda question. Some will say there is no analytical answer, end of discussion. Theorists will say we just don't know how to do it. In the end they both still eat the egg using numerical solutions so does it really matter? Well sometimes yes. It would awfully damn nice to be able to step from one system to the next without having to compound errors due to numerical solutions, welcome to the world of physics.

Sounds a bit like the same argument in Linguistics where the preeminence of the "signifier" over the "signified" is as contentious. In Gulliver's Travels the linguistic problem is that the big endians have a different word for the same thing as the little endians so the order of signifier over signified means the same thing... confusion.

But since spoken languages are better at expressing the negative; as in it's more functional and efficient to say no than yes, to say "you can't" rather than "you can"... the idea in modern linguistics is to unhinge the order altogether and say "all signs refer to other signs" in the signifying chain. A rose is a rose by any other name but the word is not a rose. "This is not a pipe" a'la Rene Magritte, kind of thing.

Yeah, the dust storm. Breathing red dirt from 2000 k's away, give me 5m of snow any day except it was all gone by midday:)

meshpig

09-25-2009, 05:39 AM

Sorry, that was totally unclear. Curious to know what mathematicians make of Philosophers and their metaphorical use of maths?

Given the Humanities aren't taken that seriously, sometimes with good reason, this guy Hjelmslev has something

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

and here http://en.wikipedia.org/wiki/Louis_Hjelmslev#Theoretical_work

which seems both as rigorous and as pertinent.

Forms of expression and forms of content must surely have correlates in maths?:)

Mark The Great

09-25-2009, 03:48 PM

Actually, many famous historic mathematicians were also philosophers.

Such as, Godel (http://en.wikipedia.org/wiki/Godel), Leibniz (http://en.wikipedia.org/wiki/Gottfried_Leibniz), and Descartes (http://en.wikipedia.org/wiki/Descartes), to name a few.

meshpig

09-26-2009, 03:29 AM

Actually, many famous historic mathematicians were also philosophers.

Such as, Godel (http://en.wikipedia.org/wiki/Godel), Leibniz (http://en.wikipedia.org/wiki/Gottfried_Leibniz), and Descartes (http://en.wikipedia.org/wiki/Descartes), to name a few.

Not sure about Descartes...

- At a fork in the road between two cities, you see 2 people. One always tells the truth, and comes from the city of safety. The other person always lies and comes from the city of cannibals, where they will eat you. They both look exactly the same. You must choose one of the persons, and ask him one and only one question (no compound questions either, such as "is this shirt red and which way to safety?"). What question could you ask to find out which path leads to the city of safety?

I can't fully grasp the underlying mathematical problem because I'm not conversant in maths. How would you solve this?

Mark The Great

09-26-2009, 09:02 AM

Well, Descartes is the "I think, therefore I am" guy, which I think qualifies him as a philosopher. :D

This problem isn't a math problem. It's actually a logic problem. There were attempts during the 19th century to reduce mathematics to logic, which Godel later proved to be impossible, I believe.

Well, as for the problem, I'll call them T and F (and we don't know which is which).

If we ask T which way F would lead us (We would have to phrase it, "Where would the other guy lead me?"), he would tell us the way to the bad city.

If we ask F where T would lead us (also phrased "Where would the other guy lead me?"), he would lie, and say that T would lead us down the way of the bad city.

So, we have one question which, when asked to either man, will give us the way to the bad city.

So, we just ask whichever man that question, and go on the path that he doesn't say to go on.

I hope that's clear.

You lost me at trigonometry :D

LOL, I went at Euler!

meshpig

09-27-2009, 03:13 AM

Well, Descartes is the "I think, therefore I am" guy, which I think qualifies him as a philosopher. :D

This problem isn't a math problem. It's actually a logic problem...

I hope that's clear.

Yes, very clear:), very neat. Thanks!

- At least you wouldn't call Descartes a "modern" philosopher but I don't know much about his contribution to math.

I mean "I think therefore I am" cogito ergo sum is a bit like saying, for all intents and purposes these days - only educated white men should be able to vote whereas modern philosophy tends toward anti Platonism... (Plato being the thinker of the State and establishing, as it were). Consider Sigmund Freud's discovery of the unconscious much as he hated philosophers as gratis for us in our understanding of consciousness.

With say a word application, Godel is proved right because my guess is an awful lot of maths is needed to accommodate the logic of words?

Mark The Great

09-27-2009, 07:52 AM

Whether Descartes ideas are discredited today or not (I don't know that they are), he still made significant contributions to philosophy, and should be counted as a philosopher.

As for Descartes' contributions in math, the only one I know off the top of my head is that he developed the Cartesian coordinate system (the x,y plane for graphing).

I'm not familiar with the particular's of Godel's proof. He proved that a symbolic system 'capable of doing something like math' cannot be internally proven, among other things.

You'd have to look that one up.

meshpig

09-28-2009, 03:31 AM

Whether Descartes ideas are discredited today or not (I don't know that they are), he still made significant contributions to philosophy, and should be counted as a philosopher.

Indeed he did. Definitely not one to argue against the rock of achievement:)

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