Math experts... have you seen this before?

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e is the base of the natural logarithms
pi is the ratio of a circle's circumference to its diameter
i is the imaginary number equal to the square root of -1

Have you seen this equation before, and is it true?
Thanks!

[KitchenWitch]

1. Not seen the equation

The answer my fancy calculator gives me is (-1,0)

Meaning there are two answers to the equation (it is, in essence, a quadratic)

[ChoralScholar]

2. Google Calc says so....

Google e^(pi*i)+1

[Random_Australian]

3. YES! That's the famous Euler's Formula! That's the most amazing bit of math!

I LOVE that formula.

Seriously, the only thing that is more entertaining is the weierstrauss function, but I never use that one.

I do use Euler's formula all the damn time, though. It's so cool.

Edit: And the reason incoherent sinusoids form a single sinusoid of the same frequency.

[begin_within]

5. Thanks... I never knew it had a name. What is it useful for?

[Random_Australian]

6. Well, Euler's Formula is actually

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

And they just used the appropriate value of x.

It's useful because when I want to do something that involves a LOT of cosines, like what happens when a wave hits an aperture (light hits a hole, or wave in water hit a wall with a section missing) then instead of having to go through all those extremely tortous cosine and sine rules, you can actually just say "cos (x) is the real part of e^i*x " and work it out like that.

It's so good for those because often you have to integrate a function that changes rapidly.... inside the cosine function.

Which would be evil/impossible, but becomes simple.

We use electrical waves too, so anything that uses electricity needs this stuff too.

[begin_within]

8. Thanks

[triguy46]

31. We need way more of this stuff on DU.

I'm not a mathematician, but I appreciate knowledge and concepts and ideas. This is great stuff. Thanks.

[Rabrrrrrr]

7. Used it all the time in electrical engineering doing frequency diagrams

and other electrical wave stuff.

[rug]

9. I can think of more entertaining things than the weierstrauss function.

[Dr. Strange]

10. No you can't!

[Random_Australian]

11. Well, don't just leave us in suspense - tell us!

Yeah, there are probably more entertaining things than the weierstrauss function - I just happen to like the idea of a line without a slope. ;) (Putting it so normal people can read it)

[TZ]

12. a line without a slope? Can you explain it more?

and in terms a "normal" (ie non-mathemetician) can understand? Calculating the slope of a line is one of the ways I get my data results so this caught me eye and curiousity...

[Random_Australian]

14. Well, if you want to know the slope of a line, what you do is take two points on the line,

and compute change in y/change in x.

For a better approximation to what the slope at a point is, you bring the points closer together.

The limit as you bring them together is the derivative - the slope at a point.

However, if you bring the two points closer together on the weierstrauss function, it does NOT get closer and closer to some value. It oscillates back and forth, and the closer you bring the points together, the faster it goes.

In other words, the limit does not exist.

In other words, it does not have a slope at any point.

In other words, it is a line without a slope. (Or should I say curve?)

[pokerfan]

17. In other words it is a continuous but nondifferentiable function

[JVS]

13. Are the things you are thinking about of a carnal nature?

[rug]

15. I was thinking more along the lines of the Nowhere Monotonic Continuous Function.

I'd say more but monotonic continuous function threads are not allowed.

[krispos42]

19. (zoom)

over my head...

[pokerfan]

20. Not that difficult

It's just a polar to rectangular conversion.



You can locate a point in 2-d space by two methods: x/y coordinates (like latitude/longitude) OR by saying the point is so many units and rotated so many degrees.

When adding and subtracting, it's easier to work in rectangular coordinates.

When multiplying and dividing, it's easier to work with polar coordinates.

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

[krispos42]

22. Cartesian and polar coordinates are okay with me

It's when people begin sprouting obscure theorems named after obscure peple that my brain seeks nurishment elsewhere... :-)

[pokerfan]

24. LOL

I enjoy math for what it can do as a tool, not as and end in itself.

In electrical engineering we use two dimensional space to represent both amplitude and phase of a signal. And it beats the hell out of solving a differential equation every time you want to analyze a circuit with reactive components (inductors and capacitors).

Richard Feynman once called Euler's formula "our jewel" and "the most remarkable formula in mathematics".

[krispos42]

25. phazor diagram

I'm having flashbacks now... oy, vey...

[pokerfan]

26. which naturally leads into a discussion of Laplace transforms

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

Welcome back to sophomore electrical engineering.

[krispos42]

27. I was mechanical...

give me mass-spring-damper systems!

Actually, I did better in thermodynamics.

[pokerfan]

28. Yep, the ME analog to RLC circuits

Back then they even made us sparkies take Thermo AND Heat Transfer. They wanted us to be well-rounded. Now, it's overly specialized IMO.

[Rabrrrrrr]

4. Yes, indeedy! It was also voted the most elegant formula

I cant remember what magazine did it - Time magazine, maybe? Or the NYT? Back in 2005, maybe 2006, whoever it was had a list of the most elegant, or beautiful, equations.

That was the lead one, if memory serves.

[pokerfan]

16. Used in electrical engineering and other fields

It's a phasor.
http://en.wikipedia.org/wiki/Phasor_(electronics)

The complete form is:


Since we sparkies use i to represent current, j is used for the square root of -1.

And yes, the equation as shown is true.

[MikeH]

18. Euler's formula in wikipedia

http://en.wikipedia.org/wiki/Euler%27s_formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. (http://en.wikipedia.org/wiki/Euler%27s_identity">Euler's identity, i.e. the formula referred to in the OP, is a special case of the Euler formula.)

Euler's formula states that, for any real number x,

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

where

e (roughly 2.71828) is the base of the natural logarithm

i is the imaginary unit (the square root of -1)

cos and sin are trigonometric functions.

* is the symbol used above for multiplication

^ is the symbol used above for exponentiation


The independent variable in a trigonometric function is expressed in http://en.wikipedia.org/wiki/Radian">radians. One radian is the angle subtended at the center of a circle by an arc of circumference that is equal in length to the radius of the circle, roughly 57.2958 degrees. Pi radians is equal to 180 degrees.

It turns out that the base of e for the exponential function, and the unit of one radian for trigonometric functions, are by far the best units for these functions; any other units would make formulas such as the http://en.wikipedia.org/wiki/Taylor_series">Taylor series expansions for these functions much more complicated.

I worked with these functions in advanced mathematics classes in college (such as advanced calculus, and mathematical methods for physics); I majored in math, and always liked math and was good with numbers and math since I was a kid.

[smedwed]

21. Euler

Euler was also the first mathematician to use the notation pi, e, and i/j.
Or so i seem to remember from maths history :)
So he was the first person to be able to write it as exactly that, which must have made him smile.

[Dr. Strange]

23. A Tercentenary Celebration

For all you Euler fans: this year is the tercentenary celebration of Euler.

Feel free to party, but please, don't drink and derive. (Oh, that was bad!)

[wildhorses]

29. this thread is oddly fascinating and strangely arousing

or as paris would say: that's hot.

[TZ]

30. In a math nerdy kind of way....

:)