Complex Numbers

So
What are the complex numbers? why were they created and how did they came with the ideia of add a real number with ''something'' to get to a point? what and where is this point?

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Defining the square root of a negative number yields i. From there you can form the complex shit.

Real numbers sit on a one-dimensional real line.
Extend this line with one extra dimension and you get a two-dimensional complex space. You still have all the real numbers but now you can move along the complex dimension as well.

Let's say you want to find the roots of the polynomial x^2 + 1.

What are they?

That kind of question led to the invention of the complex numbers as a way to give every polynomial at least one root.

It happened, IIRC, in the 1500s with people like Cardano and root finding competitions that people used to show off.

this shit is making more sense now, although math will always be for the crazy ones

>What are the complex numbers?

Take the cartesian product of the set of real numbers with itself and then take the ordered pairs this results in and while no one is looking paint a cute little 'i' next to the second component.

That is your geometric definition of the complex plane.

While every mathematician is still not looking you have to figure out what will this little i you painted be for.

You could decide for anything but you just randomly decide that you want to have your new set be closed under the root operation so you make your i equal a root that is not real. You could say that i is the root of -2, or -500000000 and it would all be the same, but for simplicity you choose i to be the root of -1

Now you can literally perform any standard operation on your set and you can also solve every polynomial equation.

That said, there are probably a billion ways to make a system in which we can solve all polynomials but this is probably the most intuitive.

The idea was to use them for temporary calculations and then reduce to get a real solutions.

They weren't used for quadratic equations originally but 4th degree equations

My math professor explained it once he introduced the complex numbers. Look up the history.

Cardano, Gauß, Cauchy and even Kant will help you out on that.

I know this will take your motivation to learn math, but I have to tell you what I think is the truth, since you asked. I do recommend that you learn math, that you call yourself a mathnoob and all that.

Complex Numbers, and more specifically, negative roots are Bad math. Bad math is accepted on a rush for rhetorical reasons, like the artificial complexity involved in it. Why this happens? The complexity is what attracts smart people around this part of math and results in useful things coming out of it. I say this believing that you already accept that smart people are attracted to complexity.

I wish I am wrong though, that my knowledge about this is plain superficial and emotional, because I rather live in a world where people don't fuck mathematics up.

But as far as I know, modern mathematics is a mess, taught by people without interest in proofs and good definitions. It ends up being a brainwashing that makes dependent slaves instead of conscious workers, your general problem solving ability is diminished by it afterwards instead of increased. I can go on and on about this, consider the terms that you use in math: reals, complex, integrals, calculus... etc these are terms designed to castrate you mentally.

Algebraic closure of the reals.

Math prof told us almost anything you could do in EE, physics and wherever you need complex numbers you could do it "manually". But they are just as handy as they come to simplify alot. It's in the end just a tool.

Though pure imaginary roots of negative numbers were conceptualized long before, many thought they were useless curiosities of allowing a broken number to exist. Something I read recently in Ian Stewart's In Pursuit of the Unknown suggests that one of the factors that started to lead to their acceptance was that they could appear as intermediates in real solutions. An example from the book uses Cardano's solution of the cubic equation:
[eqn]x = \sqrt[3]{-\frac{b}{2}+\sqrt{\frac{b^2}{4}+\frac{a^3}{27}}} + \sqrt[3]{-\frac{b}{2}-\sqrt{\frac{b^2}{4}+\frac{a^3}{27}}}[/eqn]

The example given is: [math]x^3 - 15x - 4 = 0[/math] which through the above equation becomes [math]x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}[/math]. These problems were treated as cases that could not be solved by the equation, until others, particularly Bombelli as noted by the book, noticed that if you ignore the "meaninglessness" of [math]\sqrt{-1}[/math] you could let [math]2 + \sqrt{-121} = (2 + \sqrt{-1})^3[/math] and [math]2 - \sqrt{-121} = (2 - \sqrt{-1})^3[/math], which would allow you to rewrite the equation [math](2 + \sqrt{-1}) + (2 - \sqrt{-1})[/math] which simplifies to a nice clean real number 4 - and a valid solution.

It's a two-dimensional number system. One that works for equations. So think what happens with regular math, and now all of a sudden your equations are 2-d because the numbers your are using account for two movements of the number line.

How do you visualize in Analysis the complex numbers in terms of (complex) polynoms and thus the complex zero(s) of a function?

> is looking paint a cute little 'i' next to the second component.

i could have said paint a horse and create a math based on the square root a horse. I think OP is asking why square root of -1 is a deal at all.

This. A field F is called algebraically closed if every non-constant polynomial with coefficients in F has at least one root in F.


So consider the field of real numbers R. And consider the polynomial p(x) = 1 +x^2.

This polynomials has coefficients in R, yet there does not exist any real number n such that p(n)=0. So we define a number n = i that does satisfy this property. This leads to the field of complex numbers being defined as the algebraic closure of the field of real numbers.

You compromise and visualize features of the system. A full map from C to C would be 4d. A common approach is the color-wheel system, which shows the direction and much less effectively the magnitude of the solution of each point. An alternative 3d visualization in pic related is to concentrate solely on the magnitude of the solution at each point. If you wanted to, you could graph the real or imaginary part as well.

Algebraic closure seems to be more of a side-effect of accepting them that was appreciated much later. Imaginary numbers were discovered long before anyone accepted them, and the path to acceptance involved first showing that they were useful for real solutions.

The complex numbers are algebraically complete. That's pretty important, every polynomial has its roots.

conspiracy theories aren't rigorous math either.

>even Kant

Really? I'm curious if you have a reference relating to this on hand.

low quality bait my friend

>How do you visualize in Analysis the complex numbers

usually as a grid with real component as the horizontal axis and imaginary component as the vertical

>terms of (complex) polynoms and thus the complex zero(s) of a function?

You don't really do this? The hard thing is to graph complex valued functions, which is usually accomplished with two 3-d graphs, one for the graph of the real part of the function and another for the imaginary part.

I still have some old shit-tier python code that runs to generate a folder with 111 images of mandelbrot sets from z0+cz10+c in increments of zn + 0.1+c

Notation didn't come up correctly.
z^0 + c to z^10 + c
in increments of
z^(n + 0.1) + c

Maybe think of real numbers as vectors with 1 as a basis. Now add a second basis "i" except that every time you multiply numbers by "i" they get rotated by another 90 degrees.

ordered pairs of real numbers with certain properties
[math](a,0)=a\\
(a,b)=(c,d) \implies a=c\ \text{and}\ b=d\\
(a,b)+(c,d)=(a+c,b+d)\\
\lambda(a,b)=(\lambda,0)(a,b)=(\lambda a,\lambda b)\\
(a,b)=a(1,0)+b(0,1)\\
(0,1)(0,1)=(-1,0)\\
(a,b)(c,d)=(ac-bd,ad+bc)
[/math]
adding the term (0,1) makes this set of "numbers" closed under pretty much every operation. No one wants to write all those parentheses so we identify (0,1) as i and (a,b) as a+bi

Historically it comes from finding roots of polinomials of degree 3.
Here is the general formula:
>en.wikipedia.org/wiki/Cubic_function#General_formula

The point is that even if the root are real the intermediate steps may involve squares of negatives, something that makes no sense for the reals, but it gave the correct answer.

Very important indeed. It's the whole point of complex numbers really.