/sci/ - Science & Math

I'll be honest I have never ever understood imaginary numbers

>>15600560
>trolling outside of /b/
why are mathtrannys so immature and desperate for attention?

>>15600788
That is common. You just need a better teacher.

>>15600797
>trolling
Lol. Lmao even.

>>15600788
it's a series of numbers on a different axis/dimension

>>15600982
Why the fuck would you say this? It does nothing to help him understand complex numbers and their applications.

>>15600992
i guess i don't understand them all that much myself. the most ive seen it applied is my second semester undergraduate physics course. how would you describe them?

>>15601009
>how would you describe them?
That is beyond the scope of this lesson you nigger. But basically, they are the most natural and convenient way of representing the frequency domain.

>>15601045
>the most natural and convenient way of representing the frequency domain
why?

>>15601058
>why?
That is beyond the scope of this lesson you NIGGER. But basically, phasors.

>>15601093
>basically, phasors
why?

>>15601045
>don't describe them as a different axis/dimension
>theyre good for representing a different domain
faggot

>>15601107
Idk, ask your favorite big forehead jew

>>15601113
no u

>>15601045
>the most
terrible science, no better than religious lunatics, since we can never find "the best" but only "the least worse".

>>15601207
I would slap you if we were in person, you dumb, annoying, pretentious faggot lmao.

>>15601226
tell me where I was wrong or shut up bad scientist.

>>15600788
No one has because they're a bullshit cope concept.

[math]\sqrt{4} = (4= 2*2) = 2^2 \\ \sqrt{x} = (x=y*y) = y^2 \\ \sqrt{-x} = (-x =y*y) = \text{NOPE} \\ -(\sqrt{x}) = -(x=y*y)=-(y^2) = \text{there ya go} \\ -1*(\sqrt{x}) = -1*(x=y*y) = -1*(y^2) = \text{big brain big boy gets an extra juice today} \\ \\ i=no[/math]

[math]\sqrt{4} = (4= 2*2) = 2 \\ \sqrt{x} = (x=y*y) = y \\ \sqrt{-x} = (-x =y*y) =y... \text{NOPE} \\ -(\sqrt{x}) = -(x=y*y)=-(y) = \text{there ya go} \\ -1*(\sqrt{x}) = -1*(x=y*y) = -1*(y) = \text{big brain big boy gets an extra juice today} \\ \\ i=no[/math]

>>15600788
They're square roots of negative numbers.

>>15602366
ya but why and what are they good for?

>>15602479
Solving and rewriting polynomial equations. They're also a vector space.

>>15602479
They have a cool property that when multiplied to eachother they'll cycle to positive numbers.
>>15601045
much like this fag was trying to say, it's good for frequencies because of it's oscillating nature. So when multiplied together eventually you'll come back to where you started in cycles.

Aren’t imaginary numbers a symptom of numbers being a broken system?

>>15602665
No, they're a symptom of real numbers not being algebraically closed. I.e. if you make a polynomial equation with real coefficients, its roots aren't necessarily real numbers.
Complex numbers are the algebraic closure of the real numbers, i.e. if you make a polynomial with complex coefficients the roots are complex numbers.

>>15601289
By definition, the least worse that we have is the best that we have. Logic isn’t your strong suit huh? No wonder you like science instead of mathematics.

>>15600788
Before the discover of imaginry numbers :
Maths are applied to direct physical objects.

After :
Maths on the first step outside the physical world...

>But Dirac equation ?
Lel

>>15602997
>we have
shut up child. you never said "we have" but you generalized.

leave your lies for your mommy child.

>>15600788
They're one big cope. See >>15603115

>>15603251
Your nigger tier semantic arguments don’t amuse me.
>”bUt YoU diDnT sAy “We hAvE”
Shut the fuck up faggot lmao

>>15603115
>>15603281
Complex numbers have many applications in circuit analysis and, in general, frequency domain analysis. For example, the reactance of a capacitor or inductor is an imaginary number, and in general impedance is complex. But why? Because complex numbers, behaving as phasors, account for both magnitude and phase shift. You cannot use a real number to account for any phase shift other than 180 degrees. Complex numbers must be used in order to account for phase shifts other than 180 degrees. The reason complex numbers are used instead of “regular” 2 dimensional vectors, is that complex numbers have a very convenient definition of multiplication and division.

>>15600982
Show me a picture of all these extra dimensions.

>>15602556
Regular negative one oscillates just fine when you multiply it, i only oscillates to nonsense that can't be represented on the number line.

>>15603649
>You cannot use a real number to account for any phase shift other than 180 degrees.
A number line only has 180 degrees, you can't do any other phase shit without inventing nonexistent geometries at the very least if you inventing other axis, you should have a way to connect x, y and z using those imaginary dimensions, but you don't, you just create excessive amounts of dimension that become impossible to graph.

>>15600560
why is this such a normie filter?

bump

who was the dingarangus dummie who thought there needed to be a solution to [math]\sqrt{-1}[/math] instead of just writing it as [math]-\sqrt{1}[/math] and moving on with life.

w'kinda dumbass turns a notation problem into a logical one

>>15604839
>A number line only has 180 degrees
Yeah…
>you can't do any other phase shift without inventing nonexistent geometries
A rotating vector in 2 dimensions is “non existent geometry”? Ok brainlet lmao.

>>15606236
On one hand I don’t blame them entirely because most teachers and even many professors (at least in engineering and physics) don’t entirely understand it either. On the other hand, if you are intelligent and read the right material, and are capable of visualizing and thinking through things on your own, then you can understand it. But simply dismissing it as an “incoherent and non existent thing” like others have done is just pure brainlet cope.

>>15606338
- sqrt(1) isn't sqrt(-1), it squares to 1
i is just sqrt(-1) so it is basically keeping that there

>>15600992
But he's not wrong.

>>15601009
[math]
a+b \, \textbf{i} \, \, \leftrightarrow \,
\begin{bmatrix}
a&-b \\
b&a
\end{bmatrix}
[/math]

>>15606446
a negative number squared is just the same positive number squared.
That's as simple as it gets.
-12 = 1
+12 = 1
to treat square root as the inverse as square suggests
[math] \sqrt{-1} = 1 \\ \sqrt{1} = 1[/math]

theres no need to make up fake objects and attribute them as being related to numbers.
theres no utility to it.
there doesn't need to be an object [math]i[/math] such that [math]i^2 = -1[/math]
it's called imaginary for an incredibly accurate reason.

it's neither real or objective.

>>15606556
The square root of -1 isn't 1, -1 squared is 1.

>>15606564
I mean 1 squared is 1, -1 squared from before is also 1. They don't square to -1 so they aren't the square root of -1.
Any positive real number will square to a greater number (e.g. 2^2 = 4), zero will square to zero, negative numbers will square to what their absolute value squares to (e.g. (-2)^2 = 4). So no real number could square to a negative number.

>>15606570
I mean all x where 0 < x <= 1 don't square to a greater number, but they don't square to a negative number either

>>15600788
ok so imagine you have a number line
when you add numbers, it shifts the line.
So if you add 2, the number line shifts to the left.
Subtract 2, it shifts to the right.
When you multiply/divide, the number line expands/contracts, and so on.
Multiplying by a complex number rotates the number line by 90 degrees.
Thats the intuition of complex numbers. They're the number lane rotating itself.
They're a very compact way of expressing rotations along an axis(rather than using angles).
So in a manner of speaking, complex numbers let you use normal numbers to represent rotations, and ignore a lot of trig calculations.
So instead of talking about sums/products of sins/cosines, its way WAY more compact to do it with complex numbers.

>>15606556
>it's neither real or objective.
imaginary numbers are real
they're a rotation.
Expressed very compactly.

>>15600560
>>15600788

The real question is why not just use 2D vectors?

>>15606738
Because “regular” 2D vectors do not have the extremely convenient property of being able to be represented by an exponential. By representing a complex number as an imaginary power of e, you obtain very useful definitions of multiplication and division that are not present with “regular” 2D vectors. This lends itself to more compact mathematical notation that is easier to work with and perfectly suited to things like AC circuit analysis and, in general, frequency domain analysis.

>>15600788
Helps a lot with geometric transformations that may involve negative numbers. Sometimes a formula needs 2 variations depending on if it's dealing with a positive or negative value. Obviously this can be problematic. i is really useful because when squared, you just get -1, which can conveniently flip the sign on a value that may fuck up the transformation when crossing the axis.

That's the simplest (and only) use case I can wrap my head around.

pic totally unrelated, it's just been cracking me up for days and I can't stop thinking about it

>>15607044
The purpose of the imaginary unit is to create a phase shift when a number is multiplied by it. Multiplying by j is the equivalent of phase shifting by +90 degrees. However, you can create an arbitrary phase shift by using a complex number, which is the sum of a real and imaginary component. Think of Ohm’s Law. Voltage is the product of current and a constant of proportionality called resistance. The resistance is a positive number used to “scale” the current and change its magnitude, giving you the voltage. Keep in mind the resistance doesn’t change the direction of the current because it’s a positive real number, it just scales it, so the voltage and current in a purely resistive circuit are in phase. In a similar way, complex impedance is a vector which acts on the current phasor to both scale it and phase shift it the desired amount, giving you the voltage with the proper magnitude and phase shift.

>>15600788
Just see how they are implemented in software. Just a pair of numbers (a, b) with the multiplication/exponentiation operations redefined. Math is so gay, when I want to understand some mathy bullshit I just look at the code

>>15602665
LMAO

>>15607361
Nothing you just said would indicate to me you understand complex numbers and why they are used.

This raises an interesting point, why do people have such a hard time accepting imaginary numbers? Is it just because of the name, or is it the late stage of introduction, glossing over similarly filtering properties of reals and rationals, not going into abstract algebra etc?

>>15607430
I think they just get way too caught up on the whole “square root of a negative number” thing and this prevents them from seeing its use when combined with Euler’s identity. Yeah, I get it, trying to think of the square root of a negative number in the traditional way doesn’t make much sense. But don’t let this hang up prevent you from seeing its usefulness.

>>15607486
I can't think of any reason why someone would assume that though.

>>15607517
The idea of a series expansion is beyond most people. They just stick with their elementary education and say “doesn’t make sense.” That’s all there is to it.

>>15604813
>make a new axis
>label it differently
>>15604829
good catch, been almost a decade since I looked at this stuff haha
>>15606496
holy shit this is awesome

>>15600560
Bump

>>15606604
Pythagorean theorem accomplishes anything you'd need to do with "a rotation of i" (whatever the fuck this schizo shit is supposed to mean).

>>15603115
now i get why Pythagoras MURDERED Hippasus

>>15601318
>∗
i hate that symbol like you would not believe

>>15608352
What he means is that multiplying any number (real or complex) by j causes a rotation/phase shift of that number by 90 degrees. This is similar to the way multiplying by -1 causes a phase shift of 180 degrees. It’s that simple.

>>15608383
scratch that, pythagorean theoreom is way more useful than "rotating with i".
way more rotation possibilities than 4 fucking cardinal directions lmao

imaginary brainlet cope to say it's useful for rotation lmao absolutely lol

>>15606341
There wouldn't be just 1 floating invisible dimension in spacetime, there must be one for each dimension of spacetime, so lets see you produce a picture in all those dimensions that looks anything like reality.

>>15607893
Ok, wordcel, now actually show a graph of a familiar object with all the dimensions of space, including the imaginary ones on the new axes represented.

>>15600560
(Surreal) Games + Urelements + Octonions = every number you will ever need.
Every number you will never need.
Every number not-you will ever need.
Every number not-you will never need.

>>15608499
>pythagorean theoreom
Congratulations you retarded motherfucker, you realize now why complex numbers are useful. Complex numbers are used to produce any arbitrary phase shift, not just 90 or 180 degrees. I was just giving an example.

>>15608541
What the fuck?

>>15608565
Step away from the bong and go lay down anon…

>>15608572
rotate 4 by 60 degrees

>>15600788
rotations my guy, think pic related, the name is due to a fucker that just did not like them, they are as imaginary as what makes pic related fun(and work in general)
https://www.youtube.com/watch?v=-dhHrg-KbJ0
https://www.youtube.com/watch?v=v0YEaeIClKY
https://www.youtube.com/watch?v=qtIsYbYdzCI
https://www.youtube.com/watch?v=2nuvHu6D-lA

>>15608572
The fuck to you, So this is x and xi, now do it for all the dimensions of space x, xi, y, yi, z, and zi then plot a common object on it.

Also you can do phase shifts with any two dimensions, you don't need imaginary dimensions to rotate, imaginary dimensions come about because there is no way to represent sqrt(-1) on the real number line, not so you can demonstrate rotations and phase changes about invisible spaces.

>>15608633
Not that anon, but you can multiply by e^i(60 deg) ~=0.5 + 0.866i to rotate 60 degrees.
So 4 rotated 60 degrees is approximately 2 + 3.464i

>>15607486
>But don’t let this hang up prevent you from seeing its usefulness.
This thing was invented because it's retarded, we only realized that it wasn't retarded much later.

>>15608633
See this guy >>15608667

>>15608641
I still have no idea what the fuck you are trying to say.

>>15608678
You are retarded and so is your mother. I bet you aren’t white lmao.

>>15608667
the numbers 0.5 and 0.866 are pythagorean identities lmao
[math]a^2+b^2=c^2 \\ 0.5^2+0.8660254^2 = 1^2[/math]
what is the purpose of converting this into a goofy fucking alternative (((imaginary))) rendition?

>>15608703
How convenient that you can claim to understand the usefulness of complex dimensions, but you can't even understand the implications of using them or produce a useful graph that actually includes them. But I am sure it is the other people who are the retarded ones.

>>15608714
Because (as stated above) the Euler identity provides a very convenient and useful definition of multiplication. What you are describing are “standard” 2D vectors which do not have such a convenient definition of multiplication.

>>15608717
The real number line is no more real than the imaginary number line. This will sound insane to you because you are a brainlet.

>>15608738
Sure, I bet you can really show a picture of i apples inside (+) or outside (-) of a container just as much as anyone can show picture of 2 apples inside (+) or outside (-) a container.

>>15608750
Show me a picture of |sqrt(2)| apples

>>15608757
Here is a picture of 1.41421356 apples now your turn to do i apples.

>>15608784
Obviously it isn't since it has an integer number of pixels

>>15608787
No, I weighed it very carefully and what is shown is exactly 41.421356237309504880168872420969807856967187537694% of the original weight, so now do i or at least get as much of an approximation to 10 significant digits as I did.

>>15608797
|sqrt(2)| has an infinite number of digits in its decimal expansion, tell me when you've done all of them

>>15608554
>make a cube
>one side is x
>one side is y
>one side is i
it doesn't exist in z space

>>15608714
Okay, so what would be your preferred method to rotate a 2d vector by 1 radian?
I'd just multiply a complex number by e^i, ez.

>>15608784
i apples is just one apple rotated 90 degrees anon. If you weren’t a fucking retard you would know this.

>>15608821
Nta, there's a matrix for this, you could probably derive it from the complex version, its like >>15606496 but with sin/cos which i suppose is e^ix. And it doesn't presuppose converting the vector to a complex number (unless you're using that matrix representation in the first place)

>>15608809
That is a measurement issue, not a numerical issue, I am withing one millionth of a percent accurate, you can't get nearly as close to a picture of i apples, you can't even begin you are 100% away from your target, compared to what I can show as sqrt(2) apples and it is only dependent on my ability to weight perfectly rather than not being able to even begin to make a picture because they are not equally real representations.

>>15608832
You can draw 1 apple and say it's at a scale of 1:i
You can draw a line on the i axis
You could probably do something with squaring and the inverse idk

>>15608818
I still don't see you being able to produce a graph, just a picture of your own retarded face when given the task and coming up with more stupid words instead of a simple graph of what you are trying to say.

Also you don't even seem to understand that there is x, xi, y, yi, z, and zi, rather than x,y,z, and i because every dimension has a 1, so every dimension has a -1 which means every single dimension has a corresponding imaginary space rather than there just being a single i in spacetime measurements.

>>15608830
No, that is 1 apple that fell over on its side.

>>15608837
You can have a vector space on complex numbers, but it's not automatically the case. E.g. complex numbers are (presumably) a 1D vector space on complex numbers but normally thought of as a 2D vector space on real numbers.

>>15608846
>normally thought of as a 2D vector space on real numbers.
Spacetime has 3 dimensions of real numbers to represent space, so 3 sets of 2 dimensions equals 6 dimensions in total, so I am still waiting for you to graph a common objects with with 3 sets of 2 dimensional axes.

>>15608856
It doesn't work that way. The spatial dimensions are real and the time dimension is imaginary.

>>15608864
No, the time line also uses real numbers, you can see if something happened 1 second ago in tim so that necessitates a corresponding imaginary domain for time as well, but you can't even account for space yet, so I wasn't even trying to burden you with units of time until you sorted space out.

>>15608884
I was thinking about this, not a physicist https://en.wikipedia.org/wiki/Imaginary_time
But I'm just saying it's not automatically case that all dimensions are complex i.e. basically 2 sub-dimensions
It might help to think of complex numbers as 2d real vectors with all the field operations to consider them numbers (which is kind of an arbitrary category)

>>15608896
>t's not automatically case that all dimensions are complex
All spatial dimensions include a directionality of + (forward, up, right) and - (backward, down, left), so they all necessitate an imaginary domain as a result.

>It might help to think of complex numbers as 2d real vectors
We already agreed on that and how it leads to each spatial dimension having a 2D complex space of its own, now graph a common object according to those rules or you clearly don't understand what you are talking about if you can't convert it to the actual geometry based on the spatial rules you are trying to defend.

>>15608908
That doesn't follow, nor are forward/up/right "positive" and backward/down/left "negative"
No it doesn't

>>15608923
Yes it does, it is the standard representation, have you never seen a 3D graph with their conventions before? Why do you think you have any valid answers to any questions that have been posed when you clearly have no idea what you are talking about?

>>15601318
[math]\sqrt{-x}\neq-(\sqrt{x})[/math]
and thank god for you remaining dumb untill the day you die, for you do not deserve knowledge of any kind

>>15604839
OH, its you, the mentally ill freak that believes that complex numbers involve x, y ,z, xi, yi and zi in some form of 6d shit, good to know that you will remain retarded for whatever its left of your life, and i must express my condolences to your parents and teachers for having to endure your mental illness, you time cube-tier aberration

>>15606361
nomenclature is, quite sadly, a great filter when it comes to a fair number of things

>>15608997
Until you can prove that negative numbers aren't applied to x, y, an z, you will have to accept that i is a consequence for all x, y, and z, otherwise you just sound like a seething coping retard who can't actually prove your counterpoint and has to attack the messenger instead.

>>15608703
>I still have no idea what the fuck you are trying to say.
yeah, he's a mentally ill fuck that's incapable of realizing that what he calls "the xi axis" is just the y axis, and so he believes that complex numbers ain't cause under his mental illness we would need 6d numbers, it ain't the first time he dares to show his broken head here

>>15609016
x=x, y=xi, z=xj and w=xk, get fucked you mental defect

>>15609027
>what he calls "the xi axis" is just the y axis
No, the y axis is a completely different spatial dimension, not the sqrt(-x).

>>15609033
What is the difference between i and j, usually j is just what electrical engineers use for i since they use i for current and if that is the case, you are just saying that y=z, also what is k?

>>15608633

>>15609069
again, this is literally just the Pythagorean identity multiplied by 4.
[math]c = \sqrt{a^2+b^2} \\ 4 = \sqrt{2^2+3.464^2}[/math]

>>15609087
>LaTeX
Derp

>>15601058
Calculate it yourself. Use the series expansion of exp(i*x) and see what happens. Think for yourself a bit

>>15609091
>Think
>Has no brain

How do you?

You're retarded. Don't tell people to think when you, yourself, have no brain.

>>15609093
Huh? What's wrong with a power series expansion?

>>15600788
>be ee fag
>love imaginary numbers so much I'm seriously considering teaching them to my daughter in parallel with the reals, because I think it makes more sense to teach addition, subtraction, multiplication, and division in the entire complex plane than on the real number line
Help me bros. I'm seriously not seeing the downside of using Euler's identity to introduce her to geometry.

>>15607430
>because of the name, or is it the late stage of introduction
Both. Numbers being an abstraction and inherently fictitious in a real world sense is never taught and consequently most people never reach that conclusion and have to much attachment to the name that they have been told: reals. In that context introducing something called "imaginary numbers" with no context for their existence in the world the reals are understood to exist is unavoidably cause colossal confusion. Students should be taught of the imaginary nature of numbers very early, potentially in kindergarten (We say we have two apples, but they're not the same; we have one apple A and one apple B; thus it is a convenient *abstraction* to map them to standard units and use numbers to quantify them.) and then when it's time their introduction should be better motivated by practical applications in problems they're familiar with.

>>15609091
>>15609087
Here I did it for you. Guess you just like being spoonfed, wittle baby :P

>>15609213
>We say we have two apples
We never say we have i apples because you can't even create an abstraction of i amount of apples, which means it is less real of a designation than 2 apples and before you say you can't have negative apples, you can say that I have one less apple than you, so we can come up with abstractions for -1 apple.

>>15609221
Wow, even better this method would filter retards in kindergarten and we could send them straight to special education.

>>15609217
oops I forgot the exponent on the (-1) in the sine expansion, oh well

>>15609226
I suspect you meant to reply to >>15609217 since expecting people to stay in special kindergarten until they draw a picture of i apples would filter everyone into one giant special kindergarten class.

>>15609238
It's just labels bro. That's why you're retarded. It's not 1 apple and i apple, it's 1(real) apple and 1(imaginary) apple. There's no difference. They're just labels. You can construct either number line independently.

>>15609245
>1(imaginary) apple
Which is sqrt(-1) (real) apple, which you can't draw like you can draw 1 apple because the label is not as closely correlated to reality when it is based in the imaginary number label compared to the real number label. The imaginary line is inherently dependent on the real line since it is a function of real operations.

>>15609263
>The real line is inherently dependent on the imaginary line since it is a function of imaginary operations.

>>15609266
Square root and negation functions are not dependent on imaginary numbers in any way, their use results in imaginary numbers, not the other way around.

>>15609274
>Square root and negation functions are not dependent on real numbers in any way, their use results in real numbers, not the other way around.

>>15609277
They are, though, as you can build functional mechanical devices that perform both those operations, but it only works for reals. Those operations don't create real numbers in any way either, so you are just being absurdly argumentative without any actual argument to back you up choosing only to twist other people's words to create absurd nonsensical arguments.

>>15609284
>They are, though, as you can build functional mechanical devices that perform both those operations, but it only works for imaginaries. Those operations don't create imaginary numbers in any way either

>>15609288
>you can build functional mechanical devices that perform both those operations, but it only works for imaginaries.
I know Toepler's method in mechanical calculators famously allowed mechanical calculation of sqrt, so what is this imaginary mechanical device method you are referring to that can square root imaginary numbers without using imaginary numbers in any way?

>>15609046
>y axis is a completely different spatial dimension
completely wrong
>What is the difference between i and j
quaternions, i already told you this before, and i will not let you ignore k

>>15609192
grand idea my guy, but do me a favor and watch the last two vids linked here >>15608639 so that you can emphasize properly the rotation and partial interference the complex numbers give+the "a rose by any name would smell as sweet" that is require to get over the "imaginary" in their name just being that, their name, and yes the geometry approach is near god-tier, the three things i emphasized should take it to the top, god speed my guy

>>15609348
You are completely wrong since this is the first time quaternions has been referenced ITT and it doesn't even answer any of the questions I asked of which, k was specifically brought up since you never defined it and still can't explain the difference between it, i, and j in your extremely confused "answer".

>>15609284
>as you can build functional mechanical devices that perform both those operations
>the mentally broken fuck has not heard of linkages
many such cases
https://www.youtube.com/watch?v=XOzhF3QoTCA

>>15609364
in this thread?, of course, the last time you dared to show your broken head?, no

>>15609378
If you can't explain yourself now, I am assuming you couldn't explain yourself then either, but it sounds like a fun cope and probably beats all the seething you were doing earlier ITT.

>>15609364
and there is nothing confused, x is the x axis, x*i is the y axis, x*j is the z axis and x*k is the w axis, and they multiply like so
[math]i^{2}=j^{2}=k^{2}=ijk=-1[/math]
[math]ij=-ji=k[/math], [math]jk=-kj=i[/math], [math]ki=-ik=j[/math]
https://en.wikipedia.org/wiki/Quaternion

>>15609382
mh, does this fix it?
[math]i^{2}=j^{2}=k^{2}=ijk=-1[/math], [math]ij=-ji=k[/math], [math]jk=-kj=i[/math], [math]ki=-ik=j[/math]

>>15609384
huh, im i the only one that see the "/math math"?, or is it broken for everyone else?, oh well, id still love to see people endure the retard's reasoning as to why quaternions can't be

>>15609238
Yes, I tagged the wrong one. Was mobile fagging

>>15609382
Now plot a recognizable common object on your graph.

>>15604813
F(x)=X2+1

>>15609418

>>15608831
yeah there is a matrix representation, which I'm not sure the anti-complex anon even knows about.
And yes, it can easily be derived from the complex product and euler's formula.
I find the complex representation more elegant, and desu it feels like the most natural way to understand why the rotation matrix works the way it does

>>15609213
>inherently fictitious
Bit of a stretch

>>15609780
If anon knew he'd be asking for the [1..2,1..2] of x or some other retarded shit, I'm surprised he hasn't gone on an infinite hunt for Xx, Xy, Xz, Xxx, ...

>>15610088
Holy SHIT anon you are right! Not only is there an x axis, but also an x1, x2, and so on to infinity! And this goes for y and z as well!

>>15601009
Then shut up. Seriously, how can you think what you said helps anyone? "What's a birble? Well, a birble is a 16th century's woman's dress that fastens with a French cotton-hook instead of an Genoese half-cantilever"? Stop being an autist.

I would explain it to OP >>15600788 like:
Imaginary numbers (complex numbers) are numbers that make dealing with vectors and rotations simpler. "Vectors" refers to changes in position on e.g. a 2D grid, like a map. Due to various reasons, so calculating position changes and rotations is even possible, one must have a solution for an operation that when squared yields -1. This is immediately evident when you consider that a movement 1 pace ahead would have regular solutions, but if you attempted to do a pace backwards, i.e. move -1, you couldn't calculate it anymore with the same tools (i.e. including this required squaring). Hypothetical non-invertibility would be VERY counterintuitive for movements in 2 dimensions.

I mean the last two sentences follow just from pure logic, no math knowledge required, right? Can anyone who doesn't know about complex numbers/linear algebra confirm me this?

>>15606496
hello, based department?

>>15610084
There's no qualitative difference between the existence of math and Harry Potter in the real world. One is just more useful. The universe operates through its own means. Math is just an idea we found that maps to it.

>>15600788
The key insight is Math is a bit like a game with rules and imaginary numbers have an unlucky bad name. It’s a game where you decide the rules, and then consequences result from those rules.

When you say
>What number multiplied by itself equals negative one?
You can’t find a natural or real number that satisfies this, ever, but because math is a game, we can say
>let’s bend the current rules a bit and see what happens - let’s pretend such a number existed. We’ll call that number “i”, so i^2 = -1
>What are the implications of this?

The answers are the reason people are replying with answers like

>>15600982
Because if we assume these “i numbers” can be meaningfully scaled (EG saying “three times i” is just as meaningful as if we said EG “three times one”), and recognize no real number scaled by “i” is ever going to be a real number, we realize that i numbers seem to exist on their own number line - so just like we can plot an x-y axis as two overlaid number lines, we therefore can plot a “real-imaginary” axis as two overlaid number lines that form a 2D space.

Once you start playing with the above, since you’ve now defined a new 2D space, you quickly find excuses to start playing with vectors, integrals, and rotation. For many interesting reasons, these numbers show up in many real world equations, particularly in physics and electricity, because the real world doesn’t seem to care how “imaginary” we think the numbers are.

>>15610650
That's just your false opinion. Many concepts really exist.

>>15610760
>and recognize no real number scaled by “i” is ever going to be a real number,
anon, you forgot about 0

>>15610825
>Imaginary numbers aren't real bro! Show me 0i apples

>>15610825
>0 is on both number lines
I did. For plotting an x-y style axis, note that this is not a bug, but a core feature

complex numbers excel in ease of use and interpretability to model ANY and I mean ANY harmonic oscillation. You need to dig a little bit into physics and materials science to understand how big of a deal that is.

>>15607893
>holy shit this is awesome
representation theory of groups and algebras, Lie-groups (and -algebras) are especially rich.

thank me later.

>>15610818
Such as?

>>15610833
indeed it is, i just found the "no real number scaled by “i” is ever going to be a real number" a bit silly

>>15610583
holy shit that was the most autistic retarded shit i've read. You don't know how to speak to normies at all, do you?
>Seriously, how can you think what you said helps anyone?
they're arguing about the practicality of it and it seems like most of what I've said is backed up in this thread. You sound like a word maths faggot

>>15610916
What's the point in me telling you, when you would deny them even though your entire life revolves around them?
The evidence of every single thing you claim is real is filtered through things you say aren't.
I'm not saying all of the things people's lives revolve around are real, it's a well known fact that things that aren't real can cause real events.

>>15611714
Yes. Real things are real, dipshit. Math isn't one. It's just an idea. A useful idea, but it exists in the real world as much as any other idea.

>>15612077
>more unsubstantiated claims
If only you had any evidence, unfortunately you don't think evidence is real. It's such a totally unnatural position. And ultimately there isn't even anything specific you can say exists, while you deny existence to everything else.

>>15612077
nta, what things would you consider real? Aren't almost all 'things' simply abstractions of incomprehensible micro-scale interactions according the laws of physics?

>>15612097
There isn't anything he can consider real, because it would be another abstraction or arbitrary collection of observations (themselves abstractions). And for all he or anyone else knows there could be yet another level of undiscovered or undiscoverable physical law (which also isn't real) thus making the previously real things also not real by his standards.

>>15608373
Imagine if mathematics was named acusmatics instead...


Also
...999.999... = e^(i.Pi)

>>15611391
If you can't follow the thread of the discussion other people are having you should refrain from butting in.

>>15602556
Why don't just use modulo?

>>15614587
the modulo function is just one projection of the exponential function, you dumb freshman CS major nigger

I got something that will blow your mind

>>15615377
ln(-1) could be i*3pi, making this -1*i^2 * 3 pi
so the expression is equal to 3 pi
Nice identity anon.
(okay you can use i*pi for the ln(-1) if you insist)

>>15613716
no, ...999.999...=...999+0.999...=(-1)+1=0

>>15600788
They're not called imaginary for no reason, they knew from the beginning it was a load of madeup bullshit.