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I've always thought "imaginary" vs "real" was an unfortunate naming convention.
I don't know about other fields, but electrical engineering uses imaginary numbers with AC circuits and changing electrical fields. Since electricity moves as waves, imaginary numbers let you represent what's coming 90 degrees later in a compact way.
Yeah, I think it causes unnecessary difficulties. I actually think they're introduced at a time when you could instead teach them as two-dimensional vectors with pointwise addition and a special multiplication and division rule, and prove that (0, 1)×(0, 1) = (-1, 0) using that rule, so that sqrt(-1, 0) = (0, 1).
Then you can establish a convention that you write (a, b) as a + bi (and i = (0, 1)).
This is too abstract for younger students, but nowadays I don't think they learn complex numbers anyway, and I think it would be less spooky for the older students.
Also tbf, rational numbers are just the lattice modded out by the equivalence relations (a, b) ~ (c, d) iff ad = bc, and the equivalence classes just happen to form an ordered field. If you show an undergrad math/science student this esoteric definition of rationals, the motivation of "2D numbers" makes a lot more sense. Of course, please don't show this to an elementary or middle schooler, a high schooler might be able to handle this if they're passionate about math.
Not just electricians.
3d graphics are all about about quaternions (4d version of imaginary numbers).
Quaternions are useful, but no means necessary or intrinsic to 3d graphics.
They are useful for describing rotations in 3d space, but the same can be done with a matrix, or even Euler angles. It’s just a more calculations.
That said, basically all the major game engines use quaternions, so unless you are creating your own 3d renderer, you have to learn at least the basics about them.
Quaternions really shine at spherical interpolation (Slerp - yes thats the real terminology) between two rotations.
I'd say Quaternions are necessary purely because they avoid gimbal locking common with 3 axis Euler rotations.
I know in Blender all other rotation options are really just quaterinions under the hood. But an advantage Euler rotations have over quaternions is continuous rotation. I can set 2 key frames with linear extrapolation and have them just continue along each axis forever. Everything quaternion ideally should be bounded to the 4d unit circle.
Electrical engineers also use them as a simple way of solving differential equations with laplace transforms. Much easier as long as you don't care about what happens at infinity.
Yep, it is bad naming. I took it in highschool but it didn't really make sense till I saw how it described electric fields with the wave 90 degrees into the flat page.
Just one more axiom bro and it will all make sense... just one more I swear it's consistent... BRO I just NEED one more conjecture and it will BE complete... tHe pRincIPles ARe sOLid BrO jUST TruST mE!
It's more complex than that...
cmon, get real
Think rationally
But it's plain to see...
It's not like the regular numbers behave themselves. Have you ever heard a coherent, non-self referential definition of the number 3?
3 is a sideways nutsack
W is a nutsack getting elaborately crushed
3 = . . .
It's 2 plus 1
Oh yeah? Prove it
{{},{{}}} (I think)
That would be 2. 3 would be {{},{{}},{{},{{}}}}.
the only numbers that actually exist are 0 and 1. everything else is a reference to those
I can define zero as the absence of 1. But also 1 itself is a reference to the infinitely fractional 0.00...1 and I'd consider that a different number to the whole number 1
that doesn't actually exist in the standard number system; there is no single real number that's immediately next to 0, you can always get closer. some number systems have infinitesimals, but real numbers don't. and any very small real number with a terminating decimal expression is simply the inverse of a very large number, which is simply a long string of 1+1+1+1+1...
λ f. λ x. f (f (f x))
alright wise guy, what IS the square root of -1 then?
alright wise guy, what IS -1divided by 0 then?
-1/0 = Boom
But more literally, the answer is undefined.
Instead of thinking of division as a "proper" operator, think of it as shorthand for "x0 = -1". Since the multiplication operator already defines "x0 => 0" and "-1 != 0", there literally is no answer by definition. So "undefined".
For "0/0", it's a little different. Since "x*0 => 0", literally every number satisfies "0/0". Since we're expecting a single number as the answer, we call the answer to this "undefined" too.
So without i, sqrt(-1) would be.... Undefined?
Why should I care?
All numbers are imaginary. So are words.
for real, which is imaginary
Especially for solipists.
They're just closure-minded.
Physicists (only because I watch Sabine, and this is what she says they do)
That's the face I make when someone mentions she is watching Sabine (only because I watch prof Dave)
"Quārum hī vitiō moriānis?"
"Moriānis‽ Moriānis vestris incipivit rem tōtam!"
"Et herem facĕre debēre īre stultum."