Imaginary numbers always feel wrong
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.
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.
After delving into quaternions, complex numbers feel simple and intuitive.
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.
This is why a length of a vector on a complex plane is |z|=√(z×z). z is a complex conjugate of z.
I've noticed that, if an equation calls for a number squared, they usually really mean a number multiplied by its complex conjugate.
[ you may want to escape the characters in your comment... ]
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 + - i2) = √(1 + 1) = √2. I could be totally off base here and could be confusing with something else...
I think you're thinking of taking the absolute value squared, |z|^2 = z z*
Considering we're trying to find lengths, shouldn't we be doing absolute value squared?
Almost:
Lengths are usually reals, and in this case the diagram suggests we can assume that A is the origin wlog (and the sides are 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 our other length √2.
Squaring isn't multiplication by complex conjugate, that's just mapping a vector to a scalar (the complex | x | function).
It’s just dimensionally shifted. This is not only true, its truth is practical for electrical engineering purposes. Real and imaginary cartesians yay!
This is pretty much the basis behind all math around electromagnetics (and probably other areas).
Would you explain how, for a simpleton?
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.
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!
Yes, relativity for example!
Now calculate the angles
No thank you
Doesn't this also imply that
i == 1becauseCBhas zero length, forcingACandABto be coincident? That sounds like a disproving contradiction to me.
you are imagining things
Too complexe for me ;)
The length would be equal to the absolute value
What if not a Hilbert space?
Turn around...
Bright eyes.
Every now and then, do ya fall apart?
Turn around...
Stop daydreaming 😁
Seems like one can maybe work with complex metric. Interesting idea
Looks like a finite state machine or some other graph to me, which just happens to have no directed edges.
