Complex numbers

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I'm trying to get into control systems a bit and all the transforms seems to be happening with complex numbers so i figured it'd be a good idea to get a good understanding of these. What differentiates a complex number from a tupple<2> ? It seems to me that a complex number is almost if not exactly synonymous with a vector that we use every day in graphics programming. Or perhaps its a bit more like a 2D matrix, where the complex number is some transform from the real coordinate space to a value in this arbitarily defined 2d coordinate space. Where i could easily be a real, measureable value. Is it safe to assume that complex numbers and vector math are the same? If so, or even remotely, then how come math teaches never make this connection? I always see complex numbers plotted on cartesian graphs, but they're never linked to cartesion coordinates. Any tips would be helpful, thanks.

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Complex numbers are scalars. While visualizing them as vectors may help understand certain operations, at the end of the day they're still scalars, and if you spend too much time thinking of them as being similar to vectors then you may end up confusing yourself more in the long run than if you just accepted them as scalars wholeheartedly from the start. That's probably why you never see a math teacher make the connection between the two, because while Cartesian graphs are great for visualization, they don't want to give their students the wrong idea about complex numbers. As it turns out, complex numbers were used by mathematicians to perform 2D analysis long before vectors were even invented.

I think I have a good PDF at home that goes over complex numbers in more depth that I can try to post later.

EDIT: Here we go. It's mostly about quadrature signals but it assumes you're (mostly) a beginner so there's a good complex number introduction. Plus signal processing is cool is general so feel free to continue reading through the whole document :)

[Edited by - Zipster on July 23, 2009 12:06:22 AM]

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No, it is correct to see complex numbers as vectors in a 2-dimensional space. However, it is important to remember that the set of complex numbers has some additional structure that vectors don't possess in general. Namely, there is a multiplication operation defined for every pair of complex numbers. This multiplication also has the property that every non-zero complex number has a multiplicative inverse.

Beware, do not confuse this with the dot product on vector spaces. A dot product takes two vectors and produces a number (a scalar) as a result, rather than another vector. The complex multiplication takes as input two complex numbers and produces as output a complex number again.

If you see a complex number as just a pair (a, b), then complex multiplication is defined as follows
(a, b)*(c, d) := (ac - bd, ad + bc)

This is coherent given the equivalent representations of complex numbers as either a pair (a, b) or as a sum of a real and an imaginary part : a + bi. Indeed, using the axiom i^2 = -1, you can see that
(a + bi)*(c + di)= ac + adi + bci + bdi^2= ac + (ad + bc)i - bd = (ac - bd) + (ad + bc)i

Note that Zipster is correct in saying that we must think of the complex numbers as scalars rather than vectors, because this additional multiplicative structure really make complex numbers much more similar to rational numbers or real numbers than they are to vectors.

Formally, we define the set of complex numbers as the vector space R^2 along with a multiplication defined on pairs (a, b) and (c, d) as given above. This makes the set of complex numbers into a structure called a field. Real numbers and complex numbers are both example of fields, but in general vector spaces are not fields.

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Let me add a little geometric interpretation.

First, consider the real numbers:
- Multiplication by a positive number scales points on the number line.
- Multiplication by -1 "flips" the number line.
- Multiplication by any real number, positive or negative, does some combination of scaling and flipping.

What if you want to do "half a flip?" That's a 90-degree rotation. Multiplication can be generalized to mean, geometrically, "rotation and uniform scaling."

That's what the complex numbers do.

The imaginary number, i, represents a 90-degree rotation. Take any complex number p and multiply it by i, and you get the point p in the complex plane rotated by 90 degrees about the origin. Two 90-degree rotations gives a 180-degree rotation, a flip: In other words, i*i=-1.

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Quote:
 Original post by HedosNo, it is correct to see complex numbers as vectors in a 2-dimensional space. However, it is important to remember that the set of complex numbers has some additional structure that vectors don't possess in general. Namely, there is a multiplication operation defined for every pair of complex numbers. This multiplication also has the property that every non-zero complex number has a multiplicative inverse.

I never said you couldn't visualize them as vectors, but if you need to be careful if you do because it's easy to confuse yourself when you try to do something like multiplication, and you're thinking in terms of vectors. Or if you're trying to use exponentiation, which doesn't make any sense at all for vectors.

As far as those operations are concerned, I personally prefer the magnitude-angle representation because you reduce the problem down to operations on real numbers. Exponentiation in particular is mind-boggling in rectangular form. As an added bonus, you gain some particularly interesting insight into polynomials.

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I too have found the angle magnitude representation easiest to understand. But then again i'm a complete noob at this stuff so that doesnt really mean much.

Thanks for all the info guys.