Understanding the Complex Plane and Real Plane Relationship

[chaoseverlasting]

I may be asking a stupid question, but what is the co-relation between the complex plane and the real plane? I know Euler's equation e^i\pi+1=0 relates them, but graphically, how are they related?

 

[CompuChip]

If you know that $e^{i \phi} = \cos\phi + i \sin\phi$ then you can show for yourself that it is just a rotation (e.g. take two basis vectors 1 and i, then you can write it out in matrices and you will see that the matrix $$\begin{pmatrix} \cos\phi & 0 \\ 0 & \sin\phi\end{pmatrix}$$ corresponding to $e^{i \phi}$ is just a 2d rotation matrix. So if you apply it to a complex number $$(a + b i) = \begin{pmatrix} a \\ b \end{pmatrix}$$ it will perform a rotation.

Now take a = 1, b = 0 and $\phi = \pi$ and see what happens to the number 1 when multiplied by this matrix.

 

[chaoseverlasting]

I know that. That isn't the question. What I mean is, graphically what is the meaning of complex numbers? I know a fair bit of complex analysis and its applications in engineering, but complex numbers themselves, what do they mean? Do they have any analogy to the real world?

 

[Gib Z]

What do you mean by "any analogy to the real world?". Usually I would think you meant application instead of analogy, but seeing as you know the applications of the complex numbers in engineering, you obviously meant something else. Perhaps to clear it up for us, tell us what analogy the reals have to the real world?

 

[chaoseverlasting]

For example, the numbers 1, 2, 3, 4, 5 on the real line can denote a box, two boxes, three boxes, etc. The equation $$y=a sin(\omega t+\phi)$$ can be used to represent the simple harmonic motion of a particle. Similarly, with respect to the world around us (real world), what do complex numbers represent?

For example, there are countless places where I've used the equation $$i(t)=Asin(\omega t+\phi) +iBcos(\omega t+\phi)$$, but what does the complex current denote? In any such equation which is used to define some aspect of the world around us, what do complex quantities denote? What do they mean?

 

[olliemath]

Maybe you should ask if they need one?

Consider the negative numbers. I think these were origionally invented (is this the right word?) by the Ancient Chinese in the form of white tablets - which signified debts. These were still treated with suspicion by some mathematicians as late as the middle ages! For instance, positive numbers can be used to represent a volume, or the number of dots on a sheet of paper - how can you have negative volume or a negative number of dots?! Of course today we are perfectly happy and we know they are good for representing other quantities.

Some schools of Ancient Greek mathematicians believed that the only possible numbers were integers and fractions. Then they proved that root 2 was not. This seemed to them to be quite literaly 'irrational'! (See http://en.wikipedia.org/wiki/Number" for an interesting history of numbers).

We are fine with both of these. We know that the algebraic structure of the natural numbers and positive fractions is good for counting things or bits of things; the algebraic structure of the positive reals is good for lengths, areas and volumes; the algebraic structure of all real numbers seems to correspond well with points in space or the money you own.

To move the discussion on a bit, consider a circle. Can you ever draw a perfect circle? The answer is 'No', simply because of the discrete nature of matter. But we use the idea of a circle all the time.
Hilbert put forward an argument that all mathematics is just a game played on paper according to certain rules. It has nothing to do with real life, but we seem to think it does, and we picks the rules (e.g. 1+1=2) accordingly.

So the complex numbers don't have to correspond to any sort of length, area, quantity. On the other hand their algebraic structure is the same as that of many hard to visualise things - the applications you just mentioned use it. The wave function of a quantum particle uses it. Of course all this doesn't mean that imaginary numbers don't 'exist'. http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism" would certainly tell us that they do. :)

All this is very philisophical.. perhaps I've confused you more than I've enlightened you? Some mathematicians (see Cantor, Dedikind etc.) were very unsatisfied with the 'common sense' definition of numbers so tried to define them more axiomatically with set theory

 

[berkeman]

For example, the numbers 1, 2, 3, 4, 5 on the real line can denote a box, two boxes, three boxes, etc. The equation $$y=a sin(\omega t+\phi)$$ can be used to represent the simple harmonic motion of a particle. Similarly, with respect to the world around us (real world), what do complex numbers represent?

For example, there are countless places where I've used the equation $$i(t)=Asin(\omega t+\phi) +iBcos(\omega t+\phi)$$, but what does the complex current denote? In any such equation which is used to define some aspect of the world around us, what do complex quantities denote? What do they mean?

I think this will help.

The equation you wrote (I changed i --> V to avoid confusion): $$V(t)=Asin(\omega t+\phi) +iBcos(\omega t+\phi)$$

represents a sinusoidal voltage in the real plane that has a phase shift at time t=0. Take the complex plane and hold it out in front of you like a sheet of paper, that you are looking at so that the sheet is vertical, and you only see the edge nearest you. That complex plane has the + real axis aimed vertically, and the + complex axis pointing at you. Now, the real plane is perpendicular to that, running left and right, oriented vertically, so that the + time axis is to the right, and the + real axis is still vertical. Now, picture the spinning complex plane vector that represents the V(t) signal, and move the complex plane slice to the right in time, as the complex V(t) vector rotates in the complex plane. You only see the real up/down motion of the tip of the complex vector (picture it as a point) as the complex plane moves to the right in time, and that tip traces out the real V(t) plot that you see traditionally written.

So if there is a phase shift in the complex plane at time t=0, that correlates to a phase shift in the real V(t) signal that you see traced out as you sweep the complex plane from t=0 on to the right to plot out the signal.

Does that help? It sure helped me to visualize this stuff.

 

[berkeman]

BTW, there are other useful mixes of the complex plane and real plane. The one I mention above is for the complex and real amplitude planes. But another use is in Bode plots, where you combine the sigma + j Omega plane with the frequency response amplitude plane. Check out figure 4a:

http://pdfserv.maxim-ic.com/en/an/AN733.pdf

Visualizations like this really help to get a mental handle on the pole-zero plot for a transfer function, and what that does to the Bode plot... Cool stuff.

 

[chaoseverlasting]

Thats how I used to imagine the complex and real planes (post #7) and I thought that perhaps euler's equation $$e^{i\pi}+1=0$$ could somehow relate the complex and real planes, just thought something else already existed. Pretty cool though.

EDIT: That is one cool figure! And this pdf is especially helpful cause I am going to be studying analog circuits this year, college starts on 4th Aug, so I won't know what exactly we're studying, but cool nonetheless.

 

[gmax137]

Until recently I always followed these conversations about the greeks and irrational numbers, and felt smugly 'superior' in that we now 'accept' the irrationals easily... But I have just started to find out about Dedekind (mentioned by olliemath in post 6). I don't have the words worked out yet, but irrational numbers are pretty weird. If you take sqrt(2) as the length of the hypotenuse of a unit right triangle, it seems pretty well defined. But if you are crawling along a number line, headed to root2, it's hard to see when you would stop. It has to do with the idea that the numbers are continuous. This sounds 'obvious' but when you think about it, it gets weird. Sorry for being vague, and I guess this is diverging from the OP (sorrry to hijack the thread). Anyone have pointers to these issues?