You need to add some disclaimer to this diagram like “not to scale”…
This feels wrong. I love it.
Submitted 3 weeks ago by hydroptic@sopuli.xyz to science_memes@mander.xyz
https://sopuli.xyz/pictrs/image/325a81b6-9eec-4145-a4ca-65d0cd2dc425.webp
Comments
blackbrook@mander.xyz 3 weeks ago
hydroptic@sopuli.xyz 3 weeks ago
It’s to scale.
Which scale is left as an exercise to the reader.
puchaczyk@lemmy.blahaj.zone 3 weeks ago
This is why a length of a vector on a complex plane is |z|=√(z×z). z is a complex conjugate of z.
randy@lemmy.ca 3 weeks ago
I’ve noticed that, if an equation calls for a number squared, they usually really mean a number multiplied by its complex conjugate.
drbluefall@toast.ooo 3 weeks ago
[ you may want to escape the characters in your comment… ]
ornery_chemist@mander.xyz 3 weeks ago
Isn’t the squaring actually multiplication by the complex conjugate when working in the complex plane? i.e., √((1 - 0 i) (1 + 0 i) + (0 - i) (0 + i)) = √(1 + - i^2^) = √(1 + 1) = √2. I could be totally off base here and could be confusing with something else…
diaphanous@feddit.org 3 weeks ago
I think you’re thinking of taking the absolute value squared, |z|^2 = z z*
candybrie@lemmy.world 3 weeks ago
Considering we’re trying to find lengths, shouldn’t we be doing absolute value squared?
HexesofVexes@lemmy.world 3 weeks ago
Almost:
Lengths are usually reals, and in this case the diagram we can use assume that A is the origin wlog (badly drawn vectors without a direction)
Next we convert the vectors into lengths using the abs function (root of conjugate multiplication). This gives us lengths of 1 for both.
Finally, we can just use a Euclidean metric to get out other length √2.
Squaring isn’t multiplication by complex conjugate, that’s just mapping a vector to a scalar (the complex | x | function).
captainlezbian@lemmy.world 3 weeks ago
It’s just dimensionally shifted. This is not only true, its truth is practical for electrical engineering purposes. Real and imaginary cartesians yay!
owenfromcanada@lemmy.world 3 weeks ago
This is pretty much the basis behind all math around electromagnetics (and probably other areas).
A_Union_of_Kobolds@lemmy.world 3 weeks ago
Would you explain how, for a simpleton?
owenfromcanada@lemmy.world 3 weeks ago
The short version is: we use some weird abstractions (i.e., ways of representing complex things) to do math and make sense of things.
The longer version:
Electromagnetic signals are how we transmit data wirelessly. Everything from radio, to wifi, to xrays, to visible light are all made up of electromagnetic signals.
Electromagnetic waves are made up of two components: the electrical part, and the magnetic part. We model them mathematically by multiplying one part (the magnetic part, I think) by the constant
i, which is defined assqrt(-1). These are called “complex numbers”, which means there is a “real” part and a “complex” (or “imaginary”) part. They are often modeled as the diagram OP posted, in that they operate at “right angles” to each other, and this makes a lot of the math make sense. In reality, the way the waves propegate through the air doesn’t look like that exactly, but it’s how we do the math.It’s a bit like reading a description of a place, rather than seeing a photograph. Both can give you a mental image that approximates the real thing, but the description is more “abstract” in that the words themselves (i.e., squiggles on a page) don’t resemble the real thing.
L0rdMathias@sh.itjust.works 3 weeks ago
Circles are good at math, but what to do if you not have circle shape? Easy, redefine problem until you have numbers that look like the numbers the circle shape uses. Now we can use circle math on and solve problems about non-circles!
diaphanous@feddit.org 3 weeks ago
Yes, relativity for example!
BorgDrone@lemmy.one 3 weeks ago
Now calculate the angles
Rivalarrival@lemmy.today 3 weeks ago
That’s actually pretty easy. With CB being 0, C and B are the same point. Angle A, then, is 0, and the other two angles are undefined.
hydroptic@sopuli.xyz 3 weeks ago
No thank you
jerkface@lemmy.ca 3 weeks ago
Doesn’t this also imply that
i == 1becauseCBis zero, forcingACandABto be coincident? That sounds like a disproving contradiction to me.xor@lemmy.blahaj.zone 3 weeks ago
I think BAC is supposed to be defined as a right-angle, so that AB²+AC²=CB²
=> AB+1²=0²
=> AB = √-1
=> AB = i
jerkface@lemmy.ca 3 weeks ago
I mean, I see that’s how they would have had to get to i, but it’s not a right triangle.
ShinkanTrain@lemmy.ml 3 weeks ago
produnis@discuss.tchncs.de 3 weeks ago
Too complexe for me ;)
iAvicenna@lemmy.world 3 weeks ago
you are imagining things
AcesFullOfKings@feddit.uk 3 weeks ago
[deleted]Bassman1805@lemmy.world 3 weeks ago
The reason it doesn’t work is that 1 is a scalar while i is a vector (with magnitude 1). The Pythagoras theorem works with scalars, not vectors, so you’d get 1^2 +1^2 = 2.
hydroptic@sopuli.xyz 3 weeks ago
Far as I understand it (which is not very far), i is a scalar even if you take it to be the complex number 0 + i. Just by itself i is the imaginary unit that’s defined as i = sqrt(-1), and nothing in that says it’s a vector quantity.
Even though complex numbers do extend real numbers into a 2D plane doesn’t mean they’re automatically vectors, and – again, as far as I’ve understood things – they’re still treated as single entities, ie. scalars.
someacnt_@lemmy.world 3 weeks ago
I am sorry, but… to be pedantic, pythagorean theorem works on real-valued length. Complex numbers can be scalars, but one does not use it for length for some reason I forgor.
owenfromcanada@lemmy.world 3 weeks ago
If AB = i and BC = 0, then B would be in the same 2D space as C, but one of them would be “above” the other in 3D space (which doesn’t exist in this context, just as sqrt(-1) doesn’t exist in the traditional sense).
So this triangle represents a 2D object that is “standing up” on the page.
rtxn@lemmy.world 3 weeks ago
It makes sense if you represent complex numbers as
(a, b)pairs, whereais the real part andbis the imaginary part (just like the populara + birepresentation). AB’s length is(1, 0), AC’s length is(0, 1), and BC’s length will also be a complex number.
_stranger_@lemmy.world 3 weeks ago
A?= 90°
mariusafa@lemmy.sdf.org 3 weeks ago
What if not a Hilbert space?
I_am_10_squirrels@beehaw.org 3 weeks ago
It’s not wrong, just drawn on the imaginary plane
Boomkop3@reddthat.com 3 weeks ago
Turn around…
Maiq@lemy.lol 3 weeks ago
Bright eyes.
Zoop@beehaw.org 3 weeks ago
Every now and then, do ya fall apart?
Boomkop3@reddthat.com 3 weeks ago
[deleted]Rivalarrival@lemmy.today 3 weeks ago
Every now and then, I get a little bit lonely and you’re never coming 'round
hydroptic@sopuli.xyz 3 weeks ago
Boomkop3@reddthat.com 3 weeks ago
Exactly what I thought of, but then I was like… nah that’s too cheesy
Stomata@buddyverse.one 3 weeks ago
Stop daydreaming 😁
someacnt_@lemmy.world 3 weeks ago
Seems like one can maybe work with complex metric. Interesting idea
barsoap@lemm.ee 3 weeks ago
Looks like a finite state machine or some other graph to me, which just happens to have no directed edges.
kryptonianCodeMonkey@lemmy.world 3 weeks ago
Imaginary numbers always feel wrong
Enkers@sh.itjust.works 3 weeks ago
I never really appreciated them until watching a bunch of 3blue1brown videos. I really wish those had been available when I was still in HS.
driving_crooner@lemmy.eco.br 3 weeks ago
After watching a lot of Numberphile and 3B1B videos I said to myself, you know what, I’m going back to college to get a maths degree. I swim to actuarial sciences when applying, because it’s looked like a good professional move and was the best decision on my life.
Klear@lemmy.world 3 weeks ago
After delving into quaternions, complex numbers feel simple and intuitive.
affiliate@lemmy.world 3 weeks ago
after you spend enough time with complex numbers, the real numbers start to feel wrong
TeddE@lemmy.world 3 weeks ago
Can we all at least agree that counting numbers are a joke? Sometimes they start at zero … sometimes they start at one …
bitcrafter@programming.dev 3 weeks ago
If you are comfortable with negative numbers, then you are already comfortable with the idea that a number can be tagged with an extra bit of information that represents a rotation. Complex numbers just generalize the choices available to you from 0 degrees and 180 degrees to arbitrary angles.