this post was submitted on 30 Oct 2024

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384

This feels wrong. I love it. (sopuli.xyz)

submitted 1 month ago by [email protected] to c/[email protected]

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[–] kryptonianCodeMonkey 45 points 1 month ago (4 children)

Imaginary numbers always feel wrong

[–] [email protected] 30 points 1 month ago (1 children)

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.

[–] [email protected] 23 points 1 month ago* (last edited 1 month 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 switched at last moment to actuarial sciences when applying, because it's looked like a good professional move and was the best decision on my life.

[–] Klear 20 points 1 month ago

After delving into quaternions, complex numbers feel simple and intuitive.

[–] affiliate 17 points 1 month ago (1 children)

after you spend enough time with complex numbers, the real numbers start to feel wrong

[–] TeddE 5 points 1 month ago (1 children)

Can we all at least agree that counting numbers are a joke? Sometimes they start at zero … sometimes they start at one …

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[–] [email protected] 7 points 1 month 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.

[–] [email protected] 44 points 1 month ago (1 children)

You need to add some disclaimer to this diagram like "not to scale"...

[–] [email protected] 54 points 1 month ago (1 children)

It's to scale.

Which scale is left as an exercise to the reader.

[–] [email protected] 10 points 1 month ago (2 children)

I really don't think it is.

[–] captainlezbian 14 points 1 month ago

Yeah, 1 and i should be the same size. It’s 1 in the real dimension and 1 in the imaginary dimension creating a 0 but anywhere you see this outside pure math it’s probably a sinusoid

[–] [email protected] 5 points 1 month ago

I may not have been entirely serious

[–] [email protected] 40 points 1 month ago (2 children)

This is why a length of a vector on a complex plane is |z|=√(z×z). z is a complex conjugate of z.

[–] [email protected] 18 points 1 month ago

I've noticed that, if an equation calls for a number squared, they usually really mean a number multiplied by its complex conjugate.

[–] [email protected] 7 points 1 month ago

[ you may want to escape the characters in your comment... ]

[–] [email protected] 34 points 1 month ago (2 children)

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...

[–] [email protected] 15 points 1 month ago (1 children)

I think you're thinking of taking the absolute value squared, |z|^2 = z z*

[–] candybrie 6 points 1 month ago

Considering we're trying to find lengths, shouldn't we be doing absolute value squared?

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[–] captainlezbian 29 points 1 month 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 27 points 1 month ago (2 children)

This is pretty much the basis behind all math around electromagnetics (and probably other areas).

[–] A_Union_of_Kobolds 13 points 1 month ago (3 children)

Would you explain how, for a simpleton?

[–] owenfromcanada 32 points 1 month ago (2 children)

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 as sqrt(-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.

[–] A_Union_of_Kobolds 5 points 1 month ago (1 children)

Makes sense, thanks. More of a data transmission than an electrical power thing.

[–] owenfromcanada 4 points 1 month ago (1 children)

Yeah, it's about how electromagnetic energy travels through space.

[–] A_Union_of_Kobolds 3 points 1 month ago

Thanks!

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[–] [email protected] 24 points 1 month 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!

[–] [email protected] 4 points 1 month ago (1 children)

Use the Pythagorean theorem: the length of the hypotenuse is equal to the sum of the squares for the other two sides. 1x1=1. ixi=-1. 1+(-1)=0.

[–] A_Union_of_Kobolds 3 points 1 month ago (3 children)

Yeah I get that, but what's the application to electromagnetism? I'm an electrician, it's been a few years since I had to think about induction and capacitance calculations, but I do recall them being based mostly on trigonometry. Where does i come into play, I guess is what I'm asking.

[–] marcos 3 points 1 month ago

I think the GP is going with complex numbers representing a magnitude and a rotation angle, so that side with length "i" is rotated, and A is an angle of 0°.

But this image is out of order for that. This one would lead to A = 180°. Either way, you can't use Pythagoras theorem anyway.

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[–] [email protected] 3 points 1 month ago

Yes, relativity for example!

[–] [email protected] 17 points 1 month ago (2 children)

Now calculate the angles

[–] [email protected] 14 points 1 month ago (4 children)

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.

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[–] [email protected] 5 points 1 month ago

No thank you

[–] [email protected] 14 points 1 month ago

[–] [email protected] 13 points 1 month ago* (last edited 1 month ago) (1 children)

Doesn't this also imply that i == 1 because CB has zero length, forcing AC and AB to be coincident? That sounds like a disproving contradiction to me.

[–] [email protected] 6 points 1 month ago (2 children)

I think BAC is supposed to be defined as a right-angle, so that AB²+AC²=CB²

=> AB+1²=0²

=> AB = √-1

=> AB = i

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[–] [email protected] 12 points 1 month ago

Too complexe for me ;)

[–] iAvicenna 11 points 1 month ago

you are imagining things

[+] [email protected] 10 points 1 month ago (3 children)

[deleted]

[–] Bassman1805 31 points 1 month ago (2 children)

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.

[–] [email protected] 7 points 1 month ago* (last edited 1 month ago) (1 children)

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) (edit: or, well, the solution to x² + 1 = 0, but same difference), and nothing in that definition 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. i by itself isn't a complex number I think, though.

The joke is that i² = -1 by definition, so i² + 1² = 0²

Edit: eg. nothing on the imaginary number wiki page implies that the imaginary unit is not a scalar value

[–] affiliate 6 points 1 month ago (2 children)

whether or not i is a scalar depends entirely on the context.

every vector space has an associated field of coefficients. in practice, this field is typically the real numbers. but you can have lots of other kinds of vector spaces as well, and they can be useful for certain things.

anyways, if you have a vector space over the complex numbers, then i is a scalar, because it is a complex number. if you have a vector space over the real numbers, then i is not a scalar, because it’s not a real number.

its worth mentioning that you can view the complex numbers as a vector space over itself. this is just a fancy way of saying that you can add complex numbers together, and you can multiply a complex number by a complex number. (one of those numbers is playing the role of scalar, and the other is playing the role of vector.) but you can also view the complex numbers as a vector space over the real numbers. and this is just a fancy way of saying that you can add complex numbers, and you can multiply a complex number by a real number.

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[–] someacnt_ 6 points 1 month 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 11 points 1 month ago (1 children)

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 7 points 1 month ago* (last edited 1 month ago) (1 children)

It makes sense if you represent complex numbers as (a, b) pairs, where a is the real part and b is the imaginary part (just like the popular a + bi representation that can be expanded to a * (1, 0) + b * (0, 1)). AB's length is (1, 0), AC's length is (0, 1), and BC's length will also be a complex number.