Do imaginary numbers actually exist in our real world and in what ways can we apply them/make use of them?

>>16697664 (OP)
You could have just looked at the wikipedia article for imaginary numbers

.25 on loop in spacetime?

it's [math]i^2 \equiv -1[/math], retard

>>16697664 (OP)
does the number 0 exist? do negative numbers exist? do irrational numbers exist? no but they still have uses

>>16697713
0 and negative numbers are real too buddy. They're just incredible hard to detect you need to study single particles in a vacuum to study negative numbers.

>>16697715
can you show me a picture of -5 apples?

>>16697715
you can do quantum mechanics entirely without recourse to imaginary numbers, but you have to wrestle a 4-d manifold directly instead
which is beyond the mental powers of most engineers
i is for midwits
there I've said it

>>16697664 (OP)
The principal square root has a branch cut on Re<0. You can’t take negative square roots of real numbers even in the complex plane.

The way you actually construct them set-theoretically is by taking the quotient of R[x] by the principal ideal (x^2+1). This is a well-defined procedure.

>>16697725
Yeah I always thought that I was just a additional axis in the 3d space

>>16697729
What are you even on about?

[math]
-1 = e^{i\pi} \to \sqrt{-1} = e^{I\pi/2}
[/math]

>>16697745
Learn complex analysis. Maybe read up on what branch cuts are.

>>16697755
I did an undergrad course on complex analysis 10 or so years ago. There's a rule about generalization and abstraction that is used in applied mathematics. Keep it simple stupid. If you're finding that your generalization screws up your understanding of the simple stuff (or compromises its utility), then it's a bad generalization.

>>16697789
Functions in the complex plane are very restrictive. And the way we define the principal square root is via analytic continuation. It’s a precise procedure and, again, very restrictive. You always end up with a branch cut that is an open ray from the origin. Of course, you can rotate that branch cut away from the negative reals, but that’s not what people mean when they say “square root” (hence “principal square root”).

>>16697790
I'm still not understanding where the issue is when it comes to polar representations of complex numbers/functions. In the sense that it's a messy procedure with a multi-input multi-output ambiguity, that is true with basically any analytic function of a complex number. The whole "rotate it by 2 pi and get the same number" bit makes things trickier, but you don't need to make your life harder than it needs to be.

>>16697713
>0
Exists, demonstrably. It can't not. Consider the pigeonhole principle with one pigeon removed - just because "nothing's in this hole" doesn't tell you what's SUPPOSED to be in the hole doesn't mean the hole doesn't have nothing in it. The pigeonhole principle is applicable whenever counting is applicable, so you have to abandon counting to abandon 0.

It's sort of like saying shadows don't exist because they're an absence of something else. It's clearly, visibly false - both light and shadow are discernible regions within the same field.

0 is ironically necessary for defining what doesn't exist - if we define a "glorp" as "something that can't ever exist in this universe," there can only ever be exactly 0 "glorps" in the universe. 0 fills in the universe's pigeonholes for anything like a "glorp".
>but it doesn't have glorp pigeonholes
So it has exactly 0 glorp pigeonholes - still can't get away from the universe needing at least one "0," otherwise "possible" and "impossible" become meaningless.

>irrationals
It's trivial to create them physically.

There's always [math]\pi[/math] of course, but the easier demonstration is that you can't have a rational distance (i.e. a whole number of rational-spaced markings on an arbitrary ruler) between opposite corners of a square with rational length sides.

The measured distance is irrational to arbitrary precision no matter the resolution of your rational ruler markings. The only way to get the markings to align is to make the marking distance irrational (some whole number fraction of [math]\sqrt2[/math]; still irrational).

Of course, by making such an irrational ruler, you again prove irrationals are physically real... as it would now be impossible to use the ruler to perfectly measure the sides of a square with rational sides.

Irrationals have always been something of a problem for hard finitists.

Why stop at imaginary numbers? Do the reals exist?

>>16697813
Unit square has diameter of real. There is no square with side length of minus one

>>16697804
Branch cuts means calculus breaks down. There is no way to define the square root of a real number (transcendental, I should say) without calculus. You do Taylor series essentially. So you want your calculus to still work, but it just doesn't for negative square roots, even in the complex plane.

>>16697829
What a completely irrelevant answer

>>16697844
Imaginary numbers do not exist but real numbers appears on your eyes, blindo.

>>16697847
lmao

>>16697790
Nobody uses exotic branch cuts. It's always understood to use the standard one. Yes, you know what it is.

>>16697919
Well that’s why I said nobody does, but you could theoretically define a “non-principal” square root with a branch cut away from the negative reals. But that branch cut always stays. You cannot take square roots on the entire complex plane either way.

>>16697664 (OP)
Math is all about representation of perfectly abstract concepts. No number really exists, but the apparent constraints that constitute our real world make them useful. Complex numbers too.

>>16697938
they are worse than useless, like epicycles theory in physics

>>16697946
they are used all the time in electronics

>>16697950
>nooo pls don't make me solve the Maxwell equations in 4 dimensions! My algebra-fu is too weak!
sucks to suck

>>16697996
in simple ac circuits you fucking loon

>>16698000
did I fucking stutter?
http://kirkmcd.princeton.edu/examples/jefimenko.pdf

>>16697808
>Irrationals have always been something of a problem for hard finitists.
given the thread we are in, i think that polynomials(to which root 2 & i are solutions) really filter people

>>16697664 (OP)
No number "actually exists". Forms don't exist in the real world and arguably we only conceive of them because of our failure to properly convey what is happening.

Mathematics is very powerful, but the caveat of it is that it only works so accurately because it leaves a bunch of information out. Imaginary numbers work as symbolic language for humans, but there's no metaphysical imaginary numbers out there.

>>16698000
You can just use sin and cos, retard.

e^ix was disaster for midwits

okay here's a challenge
someone give me one problem that can be solved by continuations of sqrt past zero, and in no other way
for bonus points, give a constructive proof that x^(1/x) exists for any x in R where x!=0

>>16697664 (OP)
No numbers actually exist. It's all an abstraction. Negative numbers are argueably the least real of the bunch, at least i is technically rotation, but the concept of not having an amount of something or a value being an "opposite" is as far from reality as it gets.

>>16699207
>Negative numbers are argueably the least real of the bunch, at least i is technically rotation

>>16699207
>at least i is technically rotation
negative numbers are rotation too. Just 180 degrees. It's rotation all the way down. Except when it's reflection or translation.

>>16699237
Show me -1 of something, i will wait. There's a good reason why negatives were rejected as stupid bullshit for most of written history.

>>16699248
>Show me -1 of something
you have an apple but your friend has an apple and an apple

>>16699252
So i have 1 apple and he has 2. Try again.

-1 is the the difference between you and him

>>16699264
No that would be 1, actually.

for you it's +1 but for him it's -1

>>16699268
No, it's just 1. He has 1 more apple, i have 1 less. Why do you complicate this?

1 more apple = +1
1 less apple = -1
it's not complicated it's just logic

>>16699276
1 more apple = 1
1 less apple = 1

This is reality.

how could you make 'more' and 'less' the same thing ?
you just eliminate the '+' and '-' roles in the hole math

>>16699268
In both cases, the difference is represented by 1 apple. No one is holding a negative apple.

>>16699255
exactly. or more concisely, you have -1 apples with respect to your friend. what's so hard to understand?

>>16699324
You don't have a negative apple. You both have positive apples. You're describing a relation, not a thing.

>>16699327
yes. we both have positive apples. and i have a negative apple with respect to my friend. there's no contradiction.

>>16699330
>with respect to
Again, you're describing a relation, not a thing. You don't have a negative apple.

>>16698323
Why do you like making your life harder for no reason? Complex representations of signals and Fourier spectra are so much more usable and tractable than turning everything into a purely real series and dealing with convolutions.

>>16699336
you have a negative apple with respect to your friend

>>16699383
>with respect to

>>16699383
>the -1 represent the nonexistence of the apple you don't have , why exactly we are representing an apple that do not exist ? because its do exist as an opposite in the hand of the other person , -1 wouldn't be if there is no 2 and 1 .

The -1 represents the nonexistence of the apple you don’t have. Why are we representing an apple that doesn’t exist? Because it does exist — as the opposite, in the hand of the other person. -1 wouldn’t exist if there were no 2 and 1

>>16699435
>it does exist in the hand of the other person
Yes, it exists as a positive apple.

>>16698323
You can count by shoving penguins up your ass if you like it.

>>16699447
and you also have a negative apple

>>16699459
Where?

>>16699461
On your broken legs if you dont find a positive one before Friday.

nonexistence has no place ya fool

>>16699463
What? Lol what a retard

>>16699383
you are attempting to teach the most basic of notions related to debt to the financial illiterate

>>16697718
Far larger magnitude than -5.

>>16699857
Those are all positive apples you dumb dummy.

>>16697718
i can imagine debt of all poor people
>>16697713
when i had 5 apples and i eat 5 apples now i have 0 apples and it fact exists as information
>>16697664 (OP)
In quantum mechanics, the wavefunction is a complex number where the squared magnitude |ψ|2 gives the probability of finding a particle at a location. The imaginary part encodes the quantum phase, which determines how wavefunctions interfere with each other - a purely real effect that we observe in experiments.

reality is not only physical objects but also relation between them and numbers other than natural desribe these things

>>16699458
Why waste time on trigononetric functions if e^ix lazy notation is what you do

But let me give an example. e^ix *e^iy =e^i(x+y)

But cos(x)cos(y)=0.5(cos(x+y)+cos(x-y))

So you might end up in situation where your identity is just wrong.

>>16697664 (OP)
No numbers actually exist. It's just reference we can adhere to in order to feel semblance of stability here.

>>16699248
Do you remember yesterday ?

>>16697664 (OP)
Side niggas are real, simple as.

>>16697664 (OP)

>>16697718
I can show you a flow of 5 electrons and then show you a flow of minus 5 electrons (positrons).

Minus values exist in real life (unlike complex numbers -- yes, I know about quantum).
How can you model a positive vs negative charge without different polarities?

>>16697664 (OP)
Imaginary numbers are algebraic trickery. They're no more real than factorization. Try expanding a polynomial and making sense of it, It doesn't at all. In the end it's just means to an end.

>>16699352
>so much more usable and tractable
4U

the rational mind instinctively recoils at such absurdities as the square root of -1

>>16700622
you're not "wasting time"
you're choosing a mathematical representation that has physical meaning over a leaky abstraction

>>16697813
Nope.

https://www.youtube.com/watch?v=REeaT2mWj6Y

The everything beyond the rationals are justified to make modern mathematics work.

Yea I'm pretty sure they are just ways to represent the square root of a negative value that has no result like -1

>>16699207
Double

>>16699467
kek

>>16697664 (OP)
>what ways can we apply them/make use of them?
It's been a while, but I remember using them in my classical physics 2 course to calculate resistances in AC electrical networks.

>>16704173
Based Argand.