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Complex Numbers: The End of the Road


Silly Druid

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I'm going back to math as promised. Let's think about how and why the different kinds of numbers were created.

First there are "natural numbers". It's a concept that is easy to understand, because it can be used to count physical objects we see around us. We can also make some simple operations on them. We can add them - no problems here, because if we add two natural numbers, the result is also a natural number. But we can also subtract them, and that's where a problem arises: if we subtract a number that is bigger than the one we subtract from, the result is NOT a natural number. So what can we do? We can invent a new kind of numbers, called negative numbers, that are the results of such operations.

The same goes with rational numbers - they were created as the extension of the set of numbers in case division doesn't give us a whole number. Irrational numbers are used to solve some equations, as well as geometrical problems, like calculating the diagonal length in a square. There are also transcendental numbers, that appear in other kinds of operations, like calculating the length of a circle.

So does this process ever end? Yes, it does. The end of it are the complex numbers. They allow all kinds of operations on all kinds of operands. We can take a square root of a negative number, a logarithm of a negative number, and so on. Almost everything is possible. (Things like dividing by zero are still impossible, but that's another story.)

Some results of these operations are multi-valued. But is this something that only the complex numbers can do? For example, everyone (with basic math knowledge) knows that the square root of 4 is 2, because 2 squared is 4. But -2 squared is also 4, so it should be another value of the square root of 4. It's just a convention that we take only the positive value, and not the negative one. So operations with multi-valued results are not exclusive to complex numbers. Another thing is that some of these operations are quite complicated (look for the formula for a+bi to the power of c+di, it's total mess).

There are further extensions to the set of numbers, like quaternions, but they are more like artificial constructs. They are not needed to make any operations possible, they are just created to serve some purpose (quaternions are used to represent rotations in 3D space, for example). So complex numbers are the end of the process of making the set of numbers complete. And they appear in many areas of physics, so it seems nature uses them a lot. So I think they are the ones that truly deserve to be called "natural" numbers, and using names like "real" and "imaginary" (for the two parts of a complex number) is just wrong, and is the result of superstitions from the time they were introduced.

Edited by PawelS

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While I have only met complex numbers in impedance diagrams for old telephones receive and transmission curves, which is something I like, finding any way to understanding how to use them and how to operate on them, is just not possible without going through the entire range of math. It is like the final step. And I don't think I got past factorials in school, so I have no chance of ever understanding this.

However, looking at how you represent complex numbers using some coordinate system, it seems like the real number is on the X and Y axis? And the imaginary is on the Z axis? Is it like, limited to 3D? Is math limited?

For me, most of complex numbers are playing with a variable that is supposed to be the root of -1, kinda like a scalable unit? I got that part, but then it forces you to use the a+bi format, instead of just bi by itself. I would understand it better if the imaginary number was its own thing, and not mixed with a real number part.

It seems to the the final end of math and numbers. Any chance to find anything past that?

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@Splashee If you just want to represent the complex numbers, it's easy - the real part is on the X axis, and the imaginary part on the Y axis (this representation is used with the Mandelbrot set, for example). But a complex function is harder to fully visualize, because it would require a 4D graph.

9 minutes ago, Splashee said:

Any chance to find anything past that?

I mentioned the quaternions, it's similar to complex numbers, but there are 3 different kinds of imaginary unit (i, j, k) instead of just i. And there are some weird things happening with them, for example multiplication is non-commutative. There are even more concepts beyond that, like numbers that have 8 parts, but I'm not really familiar with this kind of things.

Edited by PawelS
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7 minutes ago, PawelS said:

because it would require a 4D graph

That sounds difficult to represent in any visual form!

 

But yea, I guess my main problem with complex numbers is more of an impedance issue. This is electrical engineering stuff. I like to use math when needed, instead of learning about it all, to find purpose.

 

Spoiler

Kinda off topic: Square root is a big thing, right? I have heard problems about solving quadratic equations. Me and polynomials..... I don't even deserve to talk about math. Shame. I am quite better at trying to understand Finite Fields and Diffie-Hellman stuff, or even play around with Prime numbers, because I find the practical concepts interesting....

 

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@Splashee Using complex numbers really helps with complex reactance. But using polar notation is better because you can keep track of the phase of the electrical signal.

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@Splashee @Brony Number 42 I know what you're talking about, I had electrical engineering at school and we used complex number to solve AC circuits. We didn't even have the complex numbers introduced at math classes, so my first contact with them was at electrical engineering classes.

Edited by PawelS
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