air to be sucked
There really is no such thing as
suction. Moving on...
I like math(s?)... If I had enough time in my life, I would probably make a thread about cool math things. But I don't, so I guess I'll just post on here.
So, this is something I learned this year which is pretty nifty. As many of you probably know, we run into problems when we try to take square roots of negative numbers. So, as a nice little workaround, we just define the square root of -1 as the number "i"*, an imaginary number. This gives us a whole domain of complex numbers (we call them z's, by convention) of the form z = x + iy, where both x and y are real numbers (so iy is the imaginary part of z, and x is the real part).
This is where things get interesting: in order to keep track of complex numbers, we don't use a number line like we do for real numbers, but a plane (called the complex plane). We represent each point z=x+iy on the plane by its x and y coordinates, just like we would if we were graphing an equation of one variable onto a real graph.
However, something cool we can do with this complex plane is "map" it onto a sphere (in this case called a Riemann sphere). We create a basic bijective, conformal function that takes each point on the complex plane, and relates it to a 3D point on the sphere. Similarly, we take the inverse of the function to map every point on the sphere back onto the plane.
When we do this, we see that all lines through the origin (the point 0,0) on the plane are mapped to circles on the sphere that intersect the north and south poles, IE longitudes.
THIS is the interesting part (I SWEAR). So, the circles intersect each other twice (once at each pole) but the lines only appear to intersect each other once. Because the function we are using is conformal, we know that every intersection on the sphere must be mapped from an intersection on the plane. So, where is the missing intersection on the plane? I'll tell you! It is at the POINT at infinity! You see, in the complex plane, there is no difference between "negative" infinity and positive infinity. I mean, what would infinitely imaginary mean, anyway? So, we consider all the different kinds of complex infinity to just be one point on the plane that all lines must pass through.
So, all non-parallel lines in the complex plane intersect twice, and all parallel lines intersect once. Crazy!*The interesting this about this definition is that imaginary numbers have no inherit concept of size or polarity (like positive versus negative) the way real numbers do. Think about it: what is i squared -1. But, What is negative i squared? -1. (just like how -2 x -2=4). So, what is the square root of -1? both i and -i. So, every time we define i, we don't know if we are getting negative i or positive i. So we can't say things like -i < i, since, we don't actually know which is which!
