/sci/ - Science & Math

>>11101083
can't solve
x^3-5x+1=0
without imaginary numbers
even though
all three solutions are
normal real numbers

https://www.google.com/search?q=y=x^3-5x%2B1

https://youtu.be/_qvp9a1x2UM?t=3m

>>11101083
>Math is all made up
Stop the press!

>>11101083
>Lets sort all things with "x" property together and name it
>>But what about those things who do not have that property?
>Lets call them something else
>>OMG YOU'RE JUST MAKING SHIT UP
Yeah but who cares?

>>11101083
Complex numbers are just an elegant way of extending the 1D real number line to the 2D Argand Plane

You could achieve the same results with 2D vectors, but it wouldn't be as aesthetically pleasing

>>11101083
x^2 = -1
sqrt(y)^2 = -1
y^2/2 = -1
sqrt(-1)^2 = -1
what's wrong with this?

>dudes what if square root, right..?
>but... square root... of negative one..!
>>"Can't do that little nigga. Negative numbers don't have a square root."
>yea bu-
>>"nigga... no."

imagine how much simpler things would be if there was always a real nigga around to keep things real.

>>11101187
A lot harder. Try writing sin(17x) in terms of only sin(x) and cos(x) without complex numbers.

>>11101194
There are no complex parts about it. Sin and cos are used relative to geometry and angles, which necessitates a 2d graphing plane. imaginary-i is functionally just the y plane on an xy graph.

the y-axis would remain to exist without i

>>11101208
Then fucking do it, retard.

[math] \displaystyle
\\ z_1 = x_1+y_1i \; \; \; \; \; z_2 = x_2+y_2i
\\ z_1^* = x_1-y_1i \; \; \; \; \; z_2^* = x_2-y_2i
\\ | z_1 | = \sqrt{x_1^2+y_1^2} \; \; \; \; \; | z_2 | = \sqrt{x_2^2+y_2^2}
\\ z_1+z_2 = x_1+x_2 + (y_1+y_2)i
\\ \left | z_1+z_2 \right |^2 = \left ( \sqrt{(x_1+x_2)^2+(y_1+y_2)^2} \right ) ^2 = (x_1+x_2)^2 +(y_1+y_2)^2
\\ z_1z_2^* = x_1x_2 -x_1y_2i +y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_2y_1-x_1y_2)i
\\ z_1^*z_2 = x_1x_2 +x_1y_2i -y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_1y_2-x_2y_1)i
\\ z_1z_2^* + z_1^*z_2 = 2(x_1x_2+y_1y_2) = \text{2Re}(z_1z_2^*) = \text{2Re}(z_1^*z_2)
\\ |z_1|^2+|z_2|^2 + z_1z_2^* + z_1^*z_2 = x_1^2+y_1^2 + x_2^2 + y_2^2 + 2(x_1x_2+y_1y_2)
\\ = (x_1^2 + 2x_1x_2 + x_2^2) + (y_1^2 + 2y_1y_2 + y_2^2) = (x_1+x_2)^2 +(y_1+y_2)^2
[/math]

>>11101083
Why does this thread exist? sage, reported, cringe and bluepilled, kys, dilate, etc.

nooo bros nonononono the 300k noooooooo

>>11101375
based

>>11101375
|1+i|=sqrt(2)

>>11101375
My favourite part of this is the fact that he placed an image in the speech bubble.

>observe something, make massive assumption that observation even occurred as remembered in the first place
>ignore that same set of conditions that were first observed can never exist again under any circumstances
>claim to see "same" set of condition again
>enough people make the same claim, now its Fact™
Wow, the sciences are so rigorous and grounded in reality, not like that phony math...

yeah

>>11101100
Its called solving by substitution and it works perfectly without complex numbers. You can't get a closed nice formula.

>>11101083
Just curious. Can a quaternion be a complex root to a certain polynomial? Or even an octonion?

>>11101156
No, you're missing the point. You don't multiply and divide 2D vectors

>>11101469
If the polynomial has only complex coefficients, no.

>>11101444
>Rube Goldberg machine math
found the engineer

>>11101436
it follows directly from inf's definition:
-unbounded
-larger than any number

>>11101083
"imaginary" is the most unfortunate misnomer in math. calculations based on these "imaginary" quantities reliably predict real-world phenomena

>>11101565
When you spend years formulating the mathematics to explain 3d physics that fully capture the mathematical elegance of what is going on in the world. But everyone is a brainlet so they go with vectors instead and all they give you is your I j k notation and make fun of imaginary numbers

>>11101651

>>11101651
What are you even trying to make fun of?

>>11101660
The picture of Hamilton didnt upload for some reason even though I included it. Its Hamilton. Learn your math history.

>>11101565
Phasor diagrams actually make the analysis really easy

>>11101083
Pic is wrong. i^2 = -1, yes, but there's no such thing as square roots of negative numbers.

>>11101083
Nothing wrong with imaginary numbers, they're well defined and self consistent.
Reals on the other hand...

>>11101723
At this point I can't tell if this is a joke or not

>>11101565
can someone explain how this works, i mean how do imaginary numbers predict real world stuff

>>11101742
Rotations. Instead of saying "90 degrees" or "120 degrees," you say [math]e^{i\frac{\pi}{2}}[/math] and [math]e^{i\frac{2\pi}{3}}[/math]. Calculations are then done with exponentials, in a way that's unambiguous** and mathematically rigorous
>**mathfags don't seem to get this

>>11101742
Real world stuff involves differential equations with time derivatives. If we do a Fourier transform that derivative becomes multiplication by a frequency and i. So rather than thinking about differential equations we can just do algebra with complex numbers

>>11101740
not. it's a popsci sloppy meme

>>11101083
If imaginary numbers are 2 dimensional, what would be 3 dimensional numbers be? How do you even get into 3rd axis?
Because square root of i is still on complex 2d plane

>>11101783
4D, 8D, 16D, 32D etc.
https://www.youtube.com/watch?v=3BZyds_KFWM&list=PLNxhIPHaOTRZMO1VjJcs7_3dgyJ2qU1yZ

>>11101783
>Hasn't heard of sedenions.
>Hasn't heard of quaternions.

>>11101083
You're actually retarded

>>11101083
So all of Euler is essentially a mathematical sort that says: "Irrationals+Negatives belong on the left, Variables/Exponents/Negatives/Iterants/Directions belong on the right."

>>11101779
>>11101740
You both don't understand imaginary numbers. Roots of imaginary numbers have infinite solutions. In order to make them unique, their domain is restricted to the cut imaginary plane, the cut being exactly on the negative real numbers.

https://math.stackexchange.com/questions/44406/how-do-i-get-the-square-root-of-a-complex-number

The absolute state of /sci/...

>>11101083
complex numbers represent rotations and uniform scalings. -1 is the rotation by 180 degrees (or a point reflection through the origin). the equation x^2 = -1 literally says "what do you need to do twice to get rotation by 180 degrees?" that's right motherfuckes, rotation by 90 or 270 degrees. that's i and -i.

>>11101833
Hey! My right is YOUR left! FANCY THAT!

>>11101826
>In order to make them unique, their domain is restricted to the cut imaginary plane
Arbitrary bullshit.
And from your link,
>What's the rational basis for having positive square roots as the principal square roots instead of negative square roots?
I still haven't heard a good answer for this

>>11101833
Hot take: negative numbers are undefined in the complex plane

>>11101853
>arbitrary bullshit
So are your ramblings. You seem to have never heard a math lecture before.

What's the third root of -8?

>>11101855
>What's the third root of -8?
It depends.

>>11101869
>On how much we get paid, fellow Jew. Kekek number manipulation.

>>11101758
>>11101761
so imaginary numbers are just a trick to (ironically) avoid complexity and make our calculations easier e.g. instead of degrees we use exponentials like anon said
as far as real world stuff cares its all same from their perspective

>>11101742
Do you know about Power in circuits? How P = VI?
Now, a motor, for example, has an apparent power, which in mathematics is given by:
S = VI*, I* being the complex conjugate of the phasor I.
Now, S, the apparent power, is:
S = P + iQ
P = True Power(what your motor gives or drains in reality).
Q = Reactive power(not real, it just to account for things like Capacitor and inductor that DOESN'T dissipate power, but still drops voltage and draw current)
Yet, you can use things in real world to calculate the Reactive power and you want it to be low as possible(because if you fix S and you low Q, that means your P, the true power, increase and that is ALWAYS good). Reactive power is an example of what is a complex number that can be calculated in the real world.
Reactive power keeps circulating around your circuit from machine to machine and so on.

>>11101915
thank you anon
yes im familiar with those things back from high school of electrical engineering
except nobody ever explained us the real world correlation
it was always like, this is how things go, learn it and gtfo

>>11101758
seriously

try doing AC circuit analysis, literally most of quantum mechanics without complex numbers...

The amount of trig functions would make you go insane.

>>11101933
so is this correct thinking >>11101894

>>11101945
well sure but I think many academics also see complex numbers and the complex plane as an actual mathematical/physical structure that nature is taking advantage of.

It's simply wonderful that you can add arguments of multiplying exponents. Doing calculations in the complex plane enough and rotations become second nature. admittedly it's been a few years for me.

think of 'i' as a placeholder
we'll fill our equation with i's, shuffle things around, and then at the end we multiply them all out again.

>>11101083
Holy shit are you underage or just trolling? Also, the image is wrong, but I won't even bother telling you why.

>>11101869
Not really, no.

>>11101853
>>What's the rational basis for having positive square roots as the principal square roots instead of negative square roots?
there's none

>Hot take: negative numbers are undefined in the complex plane
they are

>>11102218
The rational basis is that it doesn't work for real numbers, so in order not to confuse newfags and brainlets it would be better to shun roots of negative numbers. But sure, any cut does the trick.
Whelp, here it happened anyway.

>>11101083
>>just starting making up random shit
That's actually exactly how math works. You make up random shit and figure out what properties it has.

Then when you know its properties you may realize wow, this I can use this random shit to describe some real world stuffs.

>>11102224
Actually I misread a I thought we were talking about complex square roots. I guess with the real square root it makes *some* sense. The map [math]\mathbb{R}\setminus 0 \to (0,\infty), x \mapsto x^2[/math] is a homeomorphism on both connected components, so in the first place it's actually possible to choose an inverse defined on the whole codomain (not possible with the complex sqrt). You choose the component including [math]1[/math], because it's a distinguished element.

>>11101785
It's almost split up like musical time. Quarter notes, eighth notes, sixteenth, and thirty second notes.

>>11102284
>dude powers of 2

>>11101083
[math] \mathbb C = \mathbb R[x]/(x^2+1) [/math]
[math] i = x + (x^2+1) [/math]

>>11101470
Sure complex numbers are equipped with an additional bilinear map.
Indeed, this bilinear map is typically interpreted geometrically: as product of the magnitudes rotated to the sum of the arguments.
If you want to be pedentic about using linear algebra formalism though, its pretty easy to represent it using ordinary matrix multiplication. All you need is a basis that respects the negative identity:
[math] J^2 = -J [/math]

>>11101651
>>11101718
>misnomer - a wrong or inaccurate use of a name or term.
that post isn't saying imaginary numbers are bad, quite the opposite
it's that they were given a really bad name that makes brainlets think of i like it's a magical fantasy unicorn

the term "complex" is much better and it would be really nice if people would just stop using "imaginary" altogether

>>11101894
>as far as real world stuff cares its all same from their perspective
philosophically speaking, all numbers are in your imagination
the "real world" doesn't "care" about numbers
if you think complex numbers are a "trick" because they help us more easily design real-world systems, then you have to also think of real numbers as a "trick"

>>11101792
He's asking about 3 dimensional, not 4 or 16 dimensional.
>>11101783
You can invent a 3 dimensional algebra, but it will not be useful, because it must lack some certain properties. It can be proved that the only real finite dimensional normed division algebras are octonions, quaternions, complex numbers, and real numbers.
You can have a 3 dimensional algebra, for example just taking [math](\mathbb{R}^3,+,\times),[/math] where [math]+[/math] is regular vector addition, and [math]\times[/math] is the cross product. Then you have an anti-commutative, non-associative algebra. It is not a division algebra, since any vectors that a parallel give a cross product of zero.
The thing to take away here is that you can invent whatever retarded algebra you want with relative ease, but it is not guaranteed to satisfy certain properties that will actually make it useful.

>>11102514
Imaginary numbers are [math]i\mathbb{R},[/math] complex numbers are [math]\mathbb{C}.[/math] They are two different sets, and are not interchangeable.

>>11101651
What’s wrong with vectors to describe the 3D. ME student so honestly curious.

>>11102826
Confirmed for brainlet

imaginary numbers are just complex numbers with [math]\Re{z} = 0[/math]
and thus belong to the field [math]\mathbb{C}[/math]

[math]i\mathbb{R}[/math] isn't a field because multiplication of two imaginary numbers doesn't result in another imaginary number. Your construction doesn't make sense

>>11102845
are you literally retarded ? he's completely right

>>11102411
based and algebrapilled

>>11102859
Are you?
Please tell me: what FIELD is the number 4i a member of.

>>11101436
Enough! Go make your own website to spread your crackpot theories and wrong proofs. You suck! You're a no-talent piece of shit! Get out! How dare you come down here and shitpost. I've been on this board for 14 years. You're a disgrace! You suck! Who the hell do you think you are? You're any kind of mathematician? Anybody know who you are? Maybe everyone else wants to enjoy actual mathematical discussion. This is one of the most important places in all of internet for maths, who are you? WHO ARE YOU? You miserable, presumptuous, no-talent. You're no mathematician. A mathematician respects the rigour and precision that serves as the foundation for intellectual achievement. You obviously don't have the talent. You don't have enough respect for yourself or other people, or what it is to express yourself in maths or any other form of study. And I'm a Harvard graduate, sucker. You suck. You're a no-talent. If you really had talent, go practice, and then get yourself published, instead of ruining the day for everybody down here. You're a disgrace! You are everything that's gone wrong with this board. You're a self-consumed, no-talent, mediocre piece of shit. And i've earned the right to say it. Okay?
In 1975, I collaborated with Erdos on a paper. Who the fuck are you? You're nothing. You are nothing. And you will never be anything. NEVER. How dare you? You miserable mediocre nothing. Shame on you. Your crat stupid little smile. You're wrong. Go practice more. You can't even put together a simple proof. I don't care about your little alternate definitions of infinity which don't prove anything. It doesn't mean you know how to do maths.You're a crank. I'm trained clasically, I'm trained contemporaneously, and you suck.

>>11102845
Imaginary numbers are not a field, and our definitions for imaginary numbers are identical.

>>11102865
Yikes!

>>11102863
[math]\mathbb{C}[/math]

>>11102866
And yet i is a member of a field called C

>>11101651
Hamilton has entered the thread.

>>11101783
>not knowing i, j , k

>>11102876
Glad you admit it. What what set 4i is also a member of? Guess what set all complex numbers with Re(z)=0 are members of? Guess what set intersects C?

>>11102877
Your point?

>>11102886
iR is a subset of C

>>11102885
I have legit no idea what's your point. tell me exactly what part of
>Imaginary numbers are iR,iR, complex numbers are C.C. They are two different sets, and are not interchangeable.
seems so wrong to you.

>>11102885
None of that contradicts >>11102826
Seriously, how are you this retarted?

>>11102884
>Look at me. I learnt something that someone else didnt learnt. Im so smart.

>>11102898
Saying they "are different sets, not interchangeable" is pedantic and misleading considering one is a subset of the other.

>>11102912
>It is pedantic and misleading to say that the empty set is different from a non-empty set considering one is a subset of the other.

>>11102912
>Saying they "are different sets, not interchangeable" is pedantic and misleading considering one is a subset of the other.
LMAO are you for real ? either you're a CS student or an undergrad trying to appear smart. there's literally nothing pedantic or misleading on saying that the set of imaginary numbers is distinct from the set of complex numbers. mainly because it's completely correct, and also because it was actually a valid point in the discussion.

>>11102826
>Imaginary numbers are iR, complex numbers are C. They are two different sets, and are not interchangeable.
that's true, and i didn't mean to imply that they are.

maybe it's not so for other languages, but in english, "imaginary" has a certain connotation.
it'd be nice to see a new name for the imaginary part of the number.
i don't know...
anti-real numbers? implied numbers?

of course i know this is about as useful as "power triangle" or "tau vs pi" discussion.

>>11101083
>imaginary
Lateral numbers plz.

>>11102898
sure [math]i\mathbb{R} \subset \mathbb{C}[/math]

My point was that there's no point in distinguishing the concepts of "imaginary" v.s. "complex" numbers because [math]i\mathbb{R}[/math] is not closed under multiplication, and is thus not a useful construct in everyday parlance. Any time you would be plausibly working with imaginary number, you're going to be working with complex numbers, ergo its much more useful to conceptually think of them as an element of the field [math]\mathbb{C}[/math] unless you're an annoying pedant which is >>11102912
point if I understood him correctly

>>11103057
The best mathematical maniacs wear their pjs and are annoyed by photography. Imagine how pissed this guy was to be conscious and not doing math.

>>11102423
And that complex structure J is additional structure a generic 2d vector space does not have.

I was trying to teach the person I was replying to that the algebraic structure (i.e. your bilinear map) is the entire point. Complex analysis is not just 2D vector calculus

>>11101375
Incredibly retarded, jesus christ

>>11101758
If you write complex numbers in exponential form instead of polar form you're a huge faggot and your circuit analysis 101 prof shouldve fucking slapped you

[eqn] Z = |r| \angle \theta [/eqn]

Its just so goddamn elegant and aesthetic, makes arithmetic absolutely trivial, nothing can compare.

>>11103789
They are literally two forms of notation to present the exact same two numbers

>>11103805
I know, but my personally preferred notation is obviously superior to all other possible notations.

>>11103789
meh, adding two complex numbers using polar is clunky and inelegant

>>11102884
i and j are the same, j is just i used in electronics instead to make it better distinguishable between I as current and j also known as i as imaginary number.
k is wave number.
Nothing 3 dimensional

>>11102817
Thanks for the explanation, anon.

>>11101444
>>11101545
OP is a limit

>>11104136
I think you meant dimwit

>>11101156
>muh 2D vectors
That's a retarded way to think about complex numbers. The right way to think about complex numbers is as the algebraic closure of the real numbers.

>>11103858
wtf are you talking about you fucking engi
learn what quaternions are, pleb

>>11101083
Actually that first one is not true [math] i =/= sqrt{-1}[/eqn]

>>11104367
you ducked up your latex
is the problem that it isn't +/- i in OP?

>>11104367
>>11104376
√ simpliciter denotes the principal square root

>>11104383
can you say that in plain English please. not my first language

>>11104391
By "simpliciter" I mean "without qualification"
By principal square root I mean this https://en.wikipedia.org/wiki/Square_root#Principal_square_root_of_a_complex_number
There are two complex numbers that square to -1, but only one of them is the principal square root of -1

>>11104383
>√ simpliciter
what's the regular square root look like then
>>11104398
so is it just that it should have been +/- i?

>>11104406
We are not google or wikipedia.

>>11103057
Mathematicians dont love complex numbers. They just love gaussian integers. When you see a irrational complex number you leave. Engineers ftw.

>>11101100
I think the issue here is not the modality of finding 3 separate solutions to a problem. I think the issue is that -1 doesn't exist in nature.

>>11103534
have you actually read the post he was replying to ? it's you who's being an annoying pedant

>>11102209
Yes, actually. There are three possible answers, and
>it depends
on which exact phasor represents the "-1."

If [math]-1=e^{i\pi}[/math], then the cube root of -8 is [math]2(e^{i\frac{\pi}{3}})[/math]
If [math]-1=e^{i3\pi}[/math], then the cube root of -8 is [math]2(e^{i\pi})[/math]
If [math]-1=e^{i5\pi}[/math], then the cube root of -8 is [math]2(e^{i\frac{5\pi}{3}})[/math]

This is basic fucking arithmetic, and you should be ashamed to accept something as vague and arbitrary as principle roots.

>>11101375
So what's wrong with this image?

>>11101375
hm... what job in maths pays 300k?

>>11104612
in Zimbabwe an apple costs 300k

>>11104616
they could afford it, they literally have diamonds and gold just lying around everywhere on top of the ground

>>11102231
yeah it's literally how fucking from language to technology

it's all made up shit

it's the fucking applications that matter

>>11102231
yeah it's literally how everything from language to technology works

it's all made up

it's the fucking applications that matter

>>11101792
What's above sedenions?

>>11101083
naming it imaginary should've nullified the recognition for conceptualizing it.
There's nothing imaginary about it.
people invented 0 as well

>>11104714
>>>google.com
>>>wikipedia.com

>>11104594
you need to use the norms to recover Pythagoras. They did not

>cant solve an equation
>just make up random numbers that solve it
>wow im such a genius
math is a liberal art

>>11105229
>math is a liberal art
So you've to be liberal to be good at math ?
I'm ok with that.

>>11105229
So when faced with a problem you just give up?

this is the only i I need to know about:

for (int i = 0; i < 10; i-=-1) {
System.out.println("Math is for faggots");
}

>java
lol

>>11105254
I turn to God

>>11105266
checked and blessed
>>11105254
If you just invent random shit because you can't figure out a problem, and that invention (which you call IMAGINARY for Christ's sake) miraculously has the properties needed to solve the equation, then that's just laziness.

>>11101375
>Hypothenuse = i2 + 1 = 0
I'm [e^(i.Pi) - 1]K with that.

>>11102865
lmao this should become a new pasta, using the angry guys speech.

>>11105276
So what, we throw out all of the theory of DE's, etc. just because you don't like the name? Grow up. The math works. Maybe come back after taking a course in analysis and realize how fucking stupid you sound.

>>11105311
>analyzing the dreams and lsd trips of 1600s mathemagicians.

No thanks. My time is far too valuable.

>>11104620
Fourier transforms

>>11105320
>My time is far too valuable.
>spends embarrassing himself on /sci/
uh huh

>>11105320
you've done nothing with it up to now

Mathematics isn't invented you stupid fucking morons.
That the imaginary numbers close the Reals is an intrinsic and beautiful aspect of reality. No one "made this up" the Italian realized that this works and wrote a paper on it. Just like no one "made up" zero or negative numbers or any other mathematical idea.

>>11105276
>If you just invent random shit because you can't figure out a problem, and that invention (which you call IMAGINARY for Christ's sake) miraculously has the properties needed to solve the equation, then that's just laziness.
imagine that some people unironically believe this

>>11105329
>Mathematics isn't invented
yes it is

>>11104587
nigga learn something about covering spaces

>>11105334
No it isn't
Whatever tautologies you find from a set of WFF is intrinsic to the set and isn't invented, you can only discover them.
You can not invent Euclidian geometry, once you have the axioms the set of true statements are always true and uninvented.

>people still arguing about imaginary numbers
>Nobody arguing about the powerset axiom

>>11105345
you always invent the axioms though

>>11105354
You, you discover axioms a posteriori using your senses, mathematics is both empirical and rational.
The foundations of mathematics are in a posteriori counting, which is not invented. The axioms built on top of that are not invented either, they are discovered based on the observations of how counting works.
Then LITERALLY ALL OF MATHEMATICS THAT WE USE NOW are rationalized from the set of WFF that we built off of the discovery of counting.
All of math is discovered, none of it is invented. Imaginary numebrs are not invented, made up, or imaginary. They are real, there exists a number that you square and get negative one, and in fact it forms an algebraically closed field over the reals that is unique up to isomorophism.
None of this shit is invented and I'm tired of engineer fucking retards saying it is. When a mathematician writes a new theorem of all those crazy symbols he is discovering a new intrinsic aspect of reality no matter how abstract it is.

>>11105349
do you honestly think most people on /sci/ have taken a course in set theory?

>>11105345
>Whatever tautologies you find from a set of WFF is intrinsic to the set and isn't invented
Maths are based on the cultures it evolved from.

>You can not invent Euclidian geometry
You can imagine a world where those E.T. Mathematicians :
- Have no fingers : So no base 10 or 12 or 20 or whatever.
- They live in water : No 2D plan, then no Euclydian in a plan, but perhaps 3D mat based on volumetry.
- Another one if they live in water : Base 2 could be a math model, "closed palm" = 1 and "opened palm" = 0.
- Etc.

>>11105364
fundamental theorem of calculus is a discovery, but riemann integral is an invention

>>11101375
literally rekt lmao