Imaginary numbers are as real as negative numbers.
Are they as real as real numbers though?
I imagine
Damn, this is more complex than I thought.
Me, a nerd: its just a joke don’t go off on a tangent, dont go off on a tangent, dont go - just let it be… I just… I just…
I NEEEED IT
So achtbually, nature works with transcendental real numbers on the complex plane with uncountably infinite precision.
Lets break this math nerd statement down in a way normal people might understand. First, most numbers are multi-dimensional and live on a plane instead of a line. The straight integer number line like 0, 1, 2 is just a small slice of the plane. In this plane, imaginary numbers occupy their own dimension. Complex numbers which are made of both real and imaginary parts occupy another dimension.
Moreover, most numbers are also infinitely precise thus being uncalculatable. Their decimal places go on forever and ever without repeating or being representable with a ratio of integers. Its why we only have good approximations for pi instead of an exact pinpoint knowledge of it. There are methods to get closer and closer approximations but you need an infinite time frame to complete that unending process.
Theres actually somehow more uncountably real numbers than countable integers and ratios, even though they are both infinite. There are more decimal numbers between 0-1 than there are integers between 0-infinity. One form of infinity is literally bigger than another, and that bigger infinity is the one nature likes working with.
Moreover, most of our universal physical and mathematical constants are transcendental reals because nature gets a hard-on for baking multidimensional fractal holomorphic topology and complex nonlinear equations into its magical abstraction bullshit logic.
Theoretical physicist during the 20th century were VERY salty about finding complex and imaginary real numbers in their physical equations. Since it implies that complex numbers arent just imaginary tools of abstrction but somehow “real” and affects the universes physical machinery. Nonlinear dynamic equations put a bullet through the brains of classical scientific determinism. Thank you very much, chaos theory and entropy.
It’s not that we invented imaginary numbers, its that they were the missing piece to fully complete our understanding of algebra. With them, we finally graduated from cave man linear algebra, to discovering holomorphic dynamics which model the way natural systems actually work. After 2000 years of banging basic logical abstractions together to make a enough decent sparks of discovery for a real smoldering fire.
Computer processing power sure helped to visualize these higher dimensional topologies for our little monkey brains to process with our eyeballs in real time instead of just thinking about this stuff in the minds eye. I sure cant visualize a 4D hypercube let alone a 20ishD hyperstructure that AI image network picture forms brought down to three dimensions.
Really its a miracle that we have even a thin narrow portion of numbers we can compute, all our regular integers and ratios are islands distanced apart by an infinitely deep ocean.
In case you were wondering about the stuff in the image: Multidimensional AI activation map showing how and image AI organizes its knowledge on a neural network. Similar concepts or images are closer together.
3D mandelbrot set with the logistic map highlighted along its real number line axis. https://github.com/jonnyhyman/Chaos
minibrot zoom in
algae colony arranging itself into conjoined 2nd iteration sierpinski triangle, screenshot from a journey to the microcosmos video.
pascals algebraic triangle encoding the sierpinski triangle by if the number is even or odd (base/mod 2)
the dynamic map of where a pendulum will land if pulled upon by three magnets equally spaced given its initial starting spot. https://youtu.be/C5Jkgvw-Z6E
Imaginary numbers are the proof that even in mathematics you can discover stuff even though you don’t understand what you have found. Complex numbers encode rotation.
Yup. When you have a circuit that is not purely resistive the inductive or capacitive load causes the voltage and current to not be in phase. It looks like ohms law is being violated. However the missing part of the energy is in the imaginary component to be returned latter.
Ever since I went down a particularly nasty rabbit hole and came out with a tenuous grasp on quaternions, imaginary numbers started feeling very simple, familiar and logical.
Yeah. The thing that made me “get” quaternions was thinking about clocks. The hands move around in a 2d plane. You can represent the tips position with just x,y. However the axis that they rotate around is the z axis.
To do a n dimensional rotation you need a n+1 dimensional axis. So to do a 3D rotation you need a 4D axis. This is bassicly a quat.
You can use trig to get there in parts but it requires you to be careful to keep your planes distinct. If your planes get parallel you get gimbal lock. This never happens when working with quats.
I still maintain that quats are the closest you can get to an actual lovecraftian horror in real life. I mean, they were carved into a stone bridge by a crazy mathematician in a fit of madness. How more lovecraftian can you get?
And THEN the imaginary numbers started EATING ALL THE CATS AND DOGS
Math: Imaginary numbers!
Me: Fuck that, imaginary dragons.
