Euler's Equation... wtf?

[Thikr]

So I get that it is "elegant" and "advanced" and stuff. How exactly do you graph ANYTHING with "i" in it though. Isn't it by definition, and "i"maginary number?!?!? Um... doesn't that mean that this equation is basically null? I don't see any unicorns in Einstein's Relativity equation...

[ajb]

You can use the natural isomorphism between [math]\mathbb{C}^{1}[/math] and [math]\mathbb{R}^{2}[/math] to represent complex numbers. See Argand diagram.

 

You can also split any complex function up as [math]f(z) = u(x,y) + i v(x,y)[/math] and then plot the two real components. Or use them to create density plots or contour plots etc... A large part of the elementary theory is then extending real functions to complex ones.

 

What do you mean by null? Not take its values in the real numbers? Or do you mean the only purely real and simultaneously purely complex number is zero?

 

Either way there is not a problem with having imaginary numbers in a physical theory, like quantum mechanics as long as your observables are real.

[ydoaPs]

How exactly do you graph ANYTHING with "i" in it though.

Complex numbers take the form of z=a+bi where a and b are real numbers. If a is zero, you simply have an imaginary number; if b is zero, you have a real number! To graph them, use a complex plane(a co-ordinate plane where one axis is real and the other is imaginary). Then just graph each z as a point (a,bi).

[Mr Skeptic]

So I get that it is "elegant" and "advanced" and stuff. How exactly do you graph ANYTHING with "i" in it though. Isn't it by definition, and "i"maginary number?!?!? Um... doesn't that mean that this equation is basically null? I don't see any unicorns in Einstein's Relativity equation...

 

You graph it like you would anything else, with one dimension for each variable. In the case of complex numbers, you have two dimensions: one for the real component and one for the imaginary component. This setup is called the complex plane, since it is a 2D plane with a real and imaginary axis.

 

Imaginary numbers are used in quantum mechanics, so maybe there are these "unicorns" you speak of.

[ydoaPs]

[math]i\hbar\frac{\partial}{\partial{t}}\Psi(\bold{r},t)=-\frac{{\hbar}^2}{2m}{\nabla}^{2}\Psi(\bold{r},t)+V(\bold{r})\Psi(\bold{r},t)[/math]

 

Is that a unicorn? If so, there are unicorns in QM.

[ajb]

[math]i\hbar\frac{\partial}{\partial{t}}\Psi(\bold{r},t)=-\frac{{\hbar}^2}{2m}{\nabla}^{2}\Psi(\bold{r},t)+V(\bold{r})\Psi(\bold{r},t)[/math]

 

Is that a unicorn? If so, there are unicorns in QM.

 

Yes and no. The formulation requires complex numbers, but the observable outcome of any experiment is real.

 

I expect that this is what confused Thikr.

[timo]

Using anything other than integers and their fractions is irrational, at best.

[John Cuthber]

Using anything other than integers and their fractions is irrational, at best.

 

Get real.

[timo]

No way. I believe in natural selection.

 

(An explanation for this post, since I assume not everyone has studied math: A set is said to be countable if there is a mapping of every element onto one unique natural number, i.e. a labeling of every element via a unique natural number. Integers are countable but the real numbers are non-countable ("over-countable", though I am not 100% sure if that is the correct English term). IOW: There is no way to label the reals with natural numbers.)

[John Cuthber]

Presumably you never walk diagonally across a town square because the length of the diagonal isn't rational.

For those who hadn't spotted it I was refering to this joke.

http://www.google.co.uk/imgres?imgurl=http://rlv.zcache.com/get_real_be_rational_card-p137700676799757647qi0i_400.jpg&imgrefurl=http://www.zazzle.com/get_real_be_rational_card-137700676799757647&h=400&w=400&sz=17&tbnid=OIoplMqMyBQShM:&tbnh=124&tbnw=124&prev=/images%3Fq%3D%2522get%2Breal%2522%2B%2522be%2Brational%2522&usg=__1olZB2MTEM10YWelm4NCMAhMI9s=&ei=uEkhS_WEOIuL4QbA4JHfCQ&sa=X&oi=image_result&resnum=1&ct=image&ved=0CAcQ9QEwAA

[the tree]

Really, maths puns are the first sine of insanity, and I think we're getting a bit off track here anyway.

 

I'm unsure if it's worthwhile covering the derivation of Euler's Formula (though it's worth taking a look at) considering the OP clearly hasn't had a formal introduction to complex numbers so likely doesn't know much about the analysis behind Taylor Series et cetera.

[hobz]

It is perhaps worth mentioning that Euler (and his contemporaries) did not have any geomtric interpretation of the imaginary number. They treated it as a number that had the property of [math]i=\sqrt{-1}[/math].

 

Think about the real number axis. The introduction of [math]i[/math] gaves rise to numbers that did not exist on the real number axis, so these special [math]i[/math] numbers, can be given their own axis.

Placing the two axes perpendicular to eachother gives an intuitive representation, where by multiplication by a negative number changes the direction by [math]180[/math] deg, and multiplication by [math]i[/math]; [math]90[/math] deg. Thus the complex plane illustrates the property of [math]i[/math] very nicely.

 

Of course it complecates matters a bit, since you might have functions that take complex numbers as an input and produce complex number as an output. And that can be hard to graph.