Complex Math

Depends on where you put your brackets.

2i^2 will normally be handled as 2*(i^2), which means 2i^2 = -2.
If you meant to write (2i)^2 then yes, that'd be -4.

Graph the real part of [math]i^x[/math]. What does it look like?

It's not so much useful as a necessity, as practically all wave functions are complex. When doing classical physics, you can mostly ignore the complex part of a wave (although you should be aware of its existence) but that's not possible anymore when you go smaller.

like a wave?

It's
[math] \cos( \frac{ x \pi }[2} )[/math]

Exactly.

Wavefunctions have phase to them which is very nicely modeled by complex numbers

> Essentially it's just a 2D vector of 2 real numbers,
Another way to view a complex number is as a compressed representation of a linear transformation consisting of rotation and scale:

Given complex numbers z=a+b*i, z1=a1+b1*i and z2=a2+b2*i:
z = z1*z2
=> a+b*i = (a1+b1*i)*(a2+b2*i)
= a1*a2+a1*b2*i+a2*b1*i+b1*b2*i*i
= (a1*a2-b1*b2)+(a1*b2+a2*b1)*i
=> a = a1*a2-b1*b2, b = a1*b2+a2*b1

This can be expressed in matrix form as:
[ a b] = [ a1 b1][ a2 b2]
[-b a] [-b1 a1][-b2 a2]

Similarly, each matrix can be factored as
[ a b] = [r 0][ cos(t) sin(t)]
[-b a] [0 r][-sin(t) cos(t)]
where r=sqrt(a^2+b^2) and t=atan2(b,a)

I.e. a=r*cos(t), b=r*sin(t).

Essentially, given a 2D vector, you can construct a rotation-and-scale transformation; one of the axes is the original vector, the other is a 90-degree rotation of it.

And if m(t) is a matrix corresponding to a rotation by angle t and unit scale, then m(t1)*m(t2)=m(t1+t2) and m(t)^n = m(n*t), i.e. multiplying unit-magnitude complex numbers adds their angle, while raising a unit-magnitude complex number to a power multiplies its angle by that power.

youtube.com/watch?v=F_0yfvm0UoU

Complex numbers are very natural for the following reason.

Polynomials are very important and the most important thing about them is their roots or solutions. It is very natural to want to work in a system of numbers such that you can always find the roots to these polynomials, but in the real numbers we have things like x^2+1=0 that have no solutions.

The fix is to add in all of these roots for all of these non-solvable polynomials, such as x^2+1 or x^3+x+1 ect. This construction is called the algebraic closure of your system of numbers (for us the real numbers).

Now the big result is the fundemental theorem of algebra, which states that this new system of numbers we have created is actually the same as the complex numbers. That is the numbers we get by only extending the real numbers by only the root of x^2+1 (or the imaginary number i). This is a very big deal. Any root to any crazy polynomial you can write down can be written as a+bi for a,b real. HUGE.

Kind of what he was getting at. The advantage is well defined orthogonality.

I don't understand how you could use it in applied science, it's not like the distance of away from the origin is different at -x than at x

Math is not science. The application of math to a real life is. We found these fancy numbers which have orthogonality, hoopla lets use them to describe systems with an orthogonal likeness.

Youre thinking too hard. Math is not a science.

>distance of away from the origin is different at -x than at x

Vectors yes, scalars no. We use imaginary numbers in vector descriptions, thus direction is very pertinent.

Also note user, that i should be easier to understand than pi, and both are abstract numbers that don't actually exist in the real world.

i is just the solution to x^2+1.

Yeah sure pi has an interpretation as the ratio of the diameter of a circle to it's circumference, but it's a lot stranger than that because you can't write this number down. You also can't write down pi as the solution to any polynomial equation like you can something like sqrt(2) (solution to x^2-2).

To really talk about pi you need to talk about convergence of cauchy sequences. That is, the proper definition of pi is the abstract number that we assign to the limit of the cauchy sequence of fractions that approach the circle constant.

i is ALOT nicer than this.

Complex numbers have more structure then just as a vector space. It's an algebra, you can also multiply the vectors together.

betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

>"I like to imagine a wiseguy"
Mafia invented complex numbers confirmed

Just study differential equations and it will all become clearer.

> You also can't write down pi as the solution to any polynomial equation

x-pi

don't get cute with me kid, If we are starting with the rationals and can only define things with polynomials, there is no way to write down pi.

From what I understand, the usefullness of [math]i[/math] comes from the fact that it "rotates".

[math]i^0 = 1[/math]
[math]i^1 = i[/math]
[math]i^2 = -1[/math]
[math]i^3 = (i^2)i = -1i[/math]
[math]i^4 = (i^3)i = (-1i)i = -1(i^2) = -1(-1) = 1 [/math]
[math]i^5 = (i^4)i = (1)i = i [/math]
[math]i^6 = (i^5)i = (i)i = -1 [/math]

and as you keep raising it, it goes [math](1,i,-1,-1i,1,i,-1,-1i,1,i,-1,-1i...)[/math] forever. This property is neat because you can rotate 2d vectors without the trig functions. On an (x,y) plane you can have a vector pointing at point (2,3), and you want to rotate it 45 degrees clockwise. You just multiply the vector (2,3) with another vector pointing at -45 degrees (1,-1) like so...

The x value it the real part, and the y value it the imaginary part
[math](2+3i)*(1+-1i) = [/math]
factorize
[math]2*1 + 2*-1i + 3i*1+3i*-1i = [/math]
add like terms
[math]2 + -2i + 3i + -3i^2 = [/math]
This part is what makes it work. [math]i^2 = -1[/math] so therefore [math]-3i^2 = 3[/math]
[math]2 + 1i + 3 = [/math]
[math]5 + 1i[/math]

The new vector is (5,1) which is a -45 degree difference between (2,3). Didn't have to use sin or cos, or a rotation matrix.

This doesn't work in 3D, but they found a way to make it work in 4D then apply it to 3D (quaternions). Apparently, quaternion transformations are just better than traditional transforming for a number of reasons, so you can see why it would be useful in physics.

That's all I know about complex numbers. There might be more applications or better abstract definitions, but this makes the most sense to me

JUST LOOK UP Quaternions IF YOU ARE STILL CONFUCIOUS

I'm pretty certain that quaternions only work for 3D, and you need octonions to move up to 4D. The number of "variables" needed to define rotations in space (also known as degrees of freedom) is NOT equal to the number of dimensions, instead it's equal to 2^(# of dimensions-1). So for example in 2D you need 2^1=2 (complex), in 3D you need 2^2=4 (quaternions), and in 4D you need 2^3=8 (octonions). Correct me if I'm wrong.

I probably shouldn't have used the phrase "but they found a way to make it work in 4D then apply it to 3D" so confidently when I wasn't sure.

But yeah, the point is, un-intuitively, you can't use a 3 part complex number for 3D rotation.

>quantum physics is based on numbers that literally don't exist

really makes you think...

>I don't understand how you could use it in applied science
I'm not in applied science so I don't give a single fuck and neither should you

To add, you use +. To multiply, you use ×. To subtract, you use -. To divide, you use ÷. Finally, to rotate real numbers, you use i.