/sci/ - Science & Math

[Anonymous No.12598199]

I get extremely infuriated every time I encounter imaginary numbers.
How is that allowed? It’s literally introduced a tainted, absurd object as a building block and then building logic on top of it, of course you will come up with weird shit like Riemann hypothesis.

Any anon can change my mind? Why imaginary numbers have a right to exist? Why not also create 2+2=4 mathematical objects, mix them into real math to create ((complex numbers))? I just don’t see how that’s not absurd, this absolutely infuriates me that math cucksois literally take this at face value.

Also I’m particularly interested in foundations of number theory, eg why people think the initial distinction between 0 and 1 makes the solid foundation for all the modern math we have right now, why we even make this distinction and never question the foundation of real math as well (besides it being useful in our everyday life to operate on our primitive world concepts).

[Anonymous No.12598223]

>>12598199
It started as a game: what are the roots of the polynomial [math] x^{2} + 1 [/math] ? Then interesting properties were found, and it led to pretty much the whole field of modern algebra. You say it has to right to exist, but it has many legitimate uses. You like number theory? Without complex numbers many problems couldn't even hope to be solved. You like physics? Quantum theory can't be formalized in the real domain. Trigonometry? Requires complex numbers.
Just because you don't like it doesnt mean it's pointless. Why can't you just get it through your head?

[Anonymous No.12598233]

>>12598223
>It started as a game: what are the roots of the polynomial x2+1
wrong
Can't solve x^3-5x+1=0 without i even though all three solutions are real numbers
https://www.google.com/search?q=y=x^3-5x%2B1
https://youtu.be/_qvp9a1x2UM?t=3m

[Anonymous No.12598255]

>>12598233
I guarantee someone though of the quadratic problem before that cubic one

[Anonymous No.12598259]

ax^3+bx^2+cx+d=0

[math] \displaystyle
ax^3+bx^2+cx+d=0 \\
S=2b^3-9abc+27a^2d \\
T=b^2-3ac \\
Q= \sqrt{(2b^3-9abc+27a^2d)^2-4(b^2-3ac)^3}= \sqrt{S^2-4T^3} \\
C= \sqrt[3]{ \frac{1}{2}(Q+2b^3-9abc+27a^2d)}
= \sqrt[3]{ \frac{1}{2}(Q+S)}
= \sqrt[3]{ \frac{1}{2} \left ( \sqrt{S^2-4T^3}+S \right )} \\
x_1= - \dfrac{b}{3a}- \dfrac{C}{3a}- \dfrac{b^2-3ac}{3aC} = - \dfrac{b}{3a}- \dfrac{C}{3a}- \dfrac{T}{3aC} \\
x_2= - \dfrac{b}{3a}+ \dfrac{C(1+i \sqrt{3})}{6a}+ \dfrac{(1-i \sqrt{3})(b^2-3ac)}{6aC} \\
~~~~ = - \dfrac{b}{3a}+ \dfrac{C(1+i \sqrt{3})}{6a}+ \dfrac{(1-i \sqrt{3})T}{6aC} \\
x_3= - \dfrac{b}{3a}+ \dfrac{C(1-i \sqrt{3})}{6a}+ \dfrac{(1+i \sqrt{3})(b^2-3ac)}{6aC} \\
~~~~ = - \dfrac{b}{3a}+ \dfrac{C(1-i \sqrt{3})}{6a}+ \dfrac{(1+i \sqrt{3})T}{6aC} \\
[/math]

[Anonymous No.12598260]

>>12598223
How can you “solve” something with a broken tool?

No wonder there are so many paradoxes in the fields you’ve just mentioned.
Even if we don’t get metaphoric, imaginary numbers are basically adding “what if..” into every place they are used, and it seems that mathematicians just take it at face value and use these numbers without regard of the side effects of the statement. The onus is on (You) to prove imaginary numbers CAN be used in mathematics, as the way I see it implicitly taints the whole chain of thought/proof that uses the concept.

[Anonymous No.12598265]

fuck off brainlet

[Anonymous No.12598270]

>>12598255
>the quadratic problem
nah they just shrugged it off because the ANSWER is imaginary
x^3-5x+1=0 has only real solutions, so it couldn't be ignored

[Anonymous No.12598285]

>>12598265
Dilate

Imaginary numbers are literally for trannies

You can’t rely on a good goy number even as a crutch as I said, it might make sense but it can’t be used, simple as

[Anonymous No.12598288]

>>12598260
Actually, nobody here is responsible for convincing you of anything. Everything about i and C is searchable. You haven't said anything specifically critical about C; instead, you have proven how sad you must be:
>today is the day guys
>I am going to change the world and reveal the lies for what they are
>/SCI/ IMAGINARIES AREN"T REAL!
>haha, I just showed the status quo
Why do you flounder about like a moron?

[Anonymous No.12598289]

>>12598260
>'broken tool'
You can define the field of complex numbers as R[X]/(X^2+1). nothing broken here retard, if in hs we say let's take i such that i^2=-1, it s for slow brains.

[Anonymous No.12598316]

>>12598289
Kys, that’s a strawman and not what I’m talking about
>>12598288
Imaginary numbers are used in proofs as a starting point to build theories in complex plane, all of this is straight bullshit, you are literally building upon something that not only doesn’t exist, but taints any and each ensuing statement
I gave you an example, Riemann zeta function, if you wish we can get more detailed

As I would expect the thread is getting filled up quickly with soi NPCs that wouldn’t even question their HS algebra textbook lmao

[Anonymous No.12598326]

>>12598316
>that's a strawman
>building upon something that not only doesn't exist

Okay you are bait, retard or high. Choose wisely faggot

[Anonymous No.12598333]

>>12598316
Do negative numbers exist?

[Anonymous No.12598335]

>>12598326
Low IQ NPC.

I’m going to sleep now and will respond with actual math tomorrow that will literally obliterate imaginaries and all cuckholds that waste their lives playing with useless theories touching them

[Anonymous No.12598338]

>>12598335
Here, read this tonight
http://libgen.rs/book/index.php?md5=3622F970CE96288F525DF3F56D8D0F38

[Anonymous No.12598342]

>>12598335
>useless
without them, you couldn't even send this fucking message eurofag

[Anonymous No.12598343]

>>12598333
Yes, they exist because they don’t have a paradox engrained into their definition

“Exist” is a dumb wording, it’s just better to say can be used in math and can’t be used

Negative numbers can be used. Imaginaries CANT

[Anonymous No.12598351]

>>12598338
I will. Be there in 12 hours because I’m asleep right now. I’m all set up right now finance wise, and have my whole life (60 years approx) to obliterate imaginary number cucks.

[Anonymous No.12598464]

>he gets thrown off by the name imaginary
Negative numbers are 'imaginary' too

Imaginary numbers and Real Numbers are like 2 independent direction vectors to each other. When you do basic physics of motion you use complex numbers

[Anonymous No.12598541]

>>12598343
>because they don’t have a paradox engrained into their definition
sure they do, you can hold two apples but you can't hold negative two apples

[Anonymous No.12598564]

Whenever I encounter them, I think: not these fuckers again.

They have no information solely. With their companion, c*mplex conjugate, they might have. The good of them is to battle quickly against sine and cosine identities.

But: they are useless. We can do math without them. Also: if you ever try to proof some physical thing with them, it is not going to happen. Quantum mechanics is a shitshow and vague non-physical theory of nothing.

Long live reality.

[Anonymous No.12598590]

>>12598259
The sheer willpower that allowed you to type this in detail is astounding

[Anonymous No.12598662]

Anyone who genuinely has a problem with complex numbers and isn't just trolling, deserved to be shot.

[Anonymous No.12598687]

>>12598662
Haven't been shot yet.

Quick Rundown:

Girolamo Cardano introduced c*mplexes in "Ars Magna" of year 1545. Cardano didn't invent them, he stole them from Niccolo Tartaglia which he promised not to :(


René Descartes did not accept complex numbers and ridiculously called them imaginary in "La Géométrie" of year 1637.


I trust Descartes.

[Anonymous No.12598717]

>>12598199

Its not absurd, you just don’t understand abstraction.

[math] \sqrt{-1} [/math] only has meaning when you are doing algebra with numbers that can’t be organized in a one dimensional line.

You will be doing algebra with [math] 2x2 [/math] matrices when you search for the roots of the characteristic polynomial of your linear transformation in order to find the two dimensional subspaces that said linear transformation maps back onto themselves.

Obviously, you can’t meaningfully organize the [math] 2x2 [/math] matrices into a line, but that doesn’t make polynomial expressions of the [math] 2x2 [/math] matrices meaningless. [math] \sqrt{-1} = \sqrt{-1 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} } = \sqrt{\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}} [/math] is not only meaningful for finding the roots of characteristic polynomials, the ability to solve it is the sufficient condition for finding [math] n [/math] roots of your [math] n[/math]th degree characteristic polynomial.
When your [math] n [/math] roots belong to distinct equivalence classes (and when the subset of matrices belong to an integral domain field), it proves that the whole linear transformation can be completely understood as linear transformations, each within [math]0[/math], [math]1[/math], and [math]2[/math] dimensional, independent subspaces.

The abstraction surrounding [math] i [/math] matters because of how broad the sufficient condition for finding [math] n [/math] roots of the characteristic polynomial is.

[Anonymous No.12598748]

>>12598564

You can divide and conquer a linear transformation without using the rotation-scaling model of Complex Numbers.

You can build an eigenbasis of a linear transformation without the notion of conjugate eigenvectors, or conjugate eigenvalues.

However, it is IMPOSSIBLE to guarantee (in 99.999% of cases) that you will find n eigenvalues in n distinct equivalence classes UNLESS you can find the square root of negative 1.

[Anonymous No.12598893]

>>12598748
Eigenvector means vector that points same directin when matrix multiplied. Talking about non real eigenvalues and vectors are nonsense.

We just need to live with n or less than n possibly eigenvectors.

[Anonymous No.12598908]

>>12598893
Non-real eigenvalues are completely legitimate because eigenvalues don’t have to be real numbers.

[math] \sqrt{-1} [/math] is just an abstraction of the sufficient condition that the field of eigenvalues must satisfy to (almost) always get as many distinct equivalence classes of eigenvalues as there are dimensions of the linear transformation.

[Anonymous No.12598915]

>>12598893
No, “eigenvector” doesn’t mean that the vector goes in the same direction when multiplied.

[Anonymous No.12598917]

>>12598199
Quantum particles have imaginary spin and they are real concrete things that exist.

[Anonymous No.12598925]

>tainted, absurd object
Quite clearly b8, but I'll still bite.
How about Fourier transforms? The most elegant / complete formulations rely on complex exponentials, and frequency domain is not only the truth, but an extremely useful (and in some cases necessary) tool to analyze LTI systems, solve certain classes of PDEs, etc.

[Anonymous No.12599452]

>>12598717
You summed it up really well my fren.
OP be like
>Oh no [math] \begin{pmatrix} a & b \\ -b & b\end{pmatrix} [/math] doesn't exist. It s all fake by Low IQ NPCs, muh imaginary

[Anonymous No.12599458]

>>12599452
*a b -b a . sorry

[Anonymous No.12599476]

>>12598199
>Complaining about imaginary numbers
You need to be 18 to post here.

[Anonymous No.12599480]

>>12598199
what do you mean "how is that allowed"? math is all made up, you retard, you MAKE the rules. if you don't like complex numbers, then do your math without them

[Anonymous No.12599513]

>>12598199
>thinking 2+2=4 is any different from "imaginary numbers"

[Anonymous No.12599520]

>>12598199
Imaginary numbers exist as a way to discuss phenomena that need to operate in a repetitive manor. For instance if you try to graph a circle in Cartesian coordinates, there is no function that exists that will complete a full circle in a non piece wise manor.

This is the biggest reason for imaginary numbers, to be able to graph circular functions and other repetitive functions on a graph.

For your zero and one question, I would propose that math is a language and takes its fundamental basis in language. Therefore we cannot have quantities of any consequence without a distinction between nothing (0) and something (1,2,3 etc) the foundation of "real math" is the language used to discuss it. This is why math is so universal, it is purely tangible and based off quantity and count.

[Anonymous No.12599732]

Alright guys, riddle me this:

What is the decimal number of i?

Pi = 3
e = 2.27

What is i? Is it larger or smaller than pi?

[Anonymous No.12599745]

>>12599732
i = 1i

[Anonymous No.12599748]

>>12598199
It's useful for sone real life problems.

[El Arcón No.12599752]

>>12599732

[Anonymous No.12600149]

>>12599732
If you want to talk about something being larger or smaller, you need an ordering. There are many orderings you can put on the complex numbers but none of them retains arithmetic the same way the real numbers does, if that is what you mean?

[Anonymous No.12600174]

>>12598564
>if you ever try to proof some physical thing with them, it is not going to happen.
what about EE

[Anonymous No.12600182]

>>12598199
Just dont refer to them as numbers. They are important structures that have propeties oridinary numbers dont have, are an useful compact way of integrating notion of rotation extending already existing rules. Some caveats related to regarding i as sqrt(-1) exist but again- think of them as useful structures that behave almost like ordinary numbers, which makes them convenient.

[Anonymous No.12600246]

>>12598335
You seem to be an expert. What are some some other math fields that I should distance myself from, and what are some based math fields?

[Anonymous No.12600286]

>>12598223
>You like number theory? Without complex numbers many problems couldn't even hope to be solved.
Such as?

[Anonymous No.12600317]

>>12598687
>I trust Descartes.
He thought the pineal gland was the resting place of the soul.
He thought eye sight was caused by tiny balls of light pushing against hoses and tubes in our bodies.
He thought animals didn't have feelings. You know who else thinks that's true? People who eat dogs in China.

[Anonymous No.12600341]

>>12598199
complex numbers = conformal geometry. there are more geometries than just Euclidean, there's affine, projective, hyperbolic, spherical and perhaps more. conformal geometry is the geometry where angles are defined, but not lengths. an example of a conformal transformation which is not Euclidean is the circle inversion. complex numbers are an algebraic structure which is the best for handling this stuff. everything you can do with complex numbers represents something in conformal geometry. if I showed you this geometric/algebraic structure without calling it "complex numbers", you would be amazed how convenient and useful it is.
it's a totally different story that complex numbers were discovered as an answer to an algebraic problem about real numbers. that's why they're called ""numbers"" in the first place and that's where all the confusion comes from.

[Anonymous No.12600679]

>>12598260
"broken" does not mean what you really mean, which is "counter-intuitive". Mathematical ideas, processes, and results are not always intuitive, but they should be logically correct nonetheless if done properly. If the math conflicts with your "common sense", then your common sense is wrong, not the math.

[Anonymous No.12600687]

>>12598343
Again, "paradox" =/= "counter-intuitive". Pick up a book and learn about the field rather than bitch about how mathematical results don't fit neatly into your narrow view of the world.

[Anonymous No.12600700]

>>12598343
what makes "1 + x = 0" different from "x^2 + 1 = 0", so that one deserves a solution and the other does not?

[Anonymous No.12600758]

>>12600317
every second of human experience is equivalent to everything a dog experiences in his life

[Anonymous No.12600965]

>>12598199
natural numbers = discrete magnitudes
positive real numbers = continuous magnitudes
integers = discrete magnitudes with a discrete sign
real numbers = continuous magnitudes with a discrete sign
complex numbers = continuous magnitudes with a continuous sign

why is the final line so controversial? what makes it different?

[Anonymous No.12601116]

>>12599732
i.0

[El Arcón No.12601126]

>act like the Lord's presence is a common public space
>don't understand why he gets angry

[Anonymous No.12601133]

>>12598259
What this guy said.

Basically, people were trying to solve cubic equations back in the 16th century or so, and they stumbled upon complex numbers. Unless you accept that complex numbers are just as good as real numbers, you can't find the roots even if they are all three real.

[Anonymous No.12602027]

>>12598915
Av=pv

Sounds same direction to me.

[Anonymous No.12602044]

>>12601133
Of course one can. This formula just fucking accepts cucklex solutions too

[Anonymous No.12602079]

>>12602044
>Of course one can
tell us how

[Anonymous No.12602084]

>>12598316
You should checkout quaternions it would blow your mind.

[Anonymous No.12602125]

>>12600679
common sense has no actual meaning, simple as that

people who try to rationalize abstraction are retarded

[Anonymous No.12602130]

>>12598285
t reddit formatting

[Anonymous No.12602134]

>>12600246
this is your brain on literature

[Anonymous No.12602139]

>>12600700
if this entire thread is not bait i am dreaming

[Anonymous No.12602141]

>>12598233
> casus irreducibilis
you can't state some real numbers without the use of imaginary numbers, so imaginary numbers aren't a technique that was invented to make thing easier, they exist.

[Anonymous No.12602146]

>>12598564
>a+bi
>matrix

literally the same thing, are you dense?

[Anonymous No.12602162]

>>12600182
it is a field, i will call them numbers lol

[Anonymous No.12602172]

>>12600965
the young brain reaches a point when it just stops believing shit and starts to question when complex numbers are taught.

Imagine 17yo's picking into the teacher when she's teaching long division

[Anonymous No.12602399]

>>12598199
>Why imaginary numbers have a right to exist
They're numbers with angles: 1=1∠0°, -1=1∠180°, i=1∠90°
They add geometrically: 1+i = √2∠45°
They multiple rotationally (phase shiftingly) M∠a * N∠b = MN∠a+b

Of course they don't make sense in scalar contexts

[Anonymous No.12602413]

>>12602027
That is the one dimensional subspace case
In the two dimensional subspace case, you will replace the vector with a two column matrix, and the scalar with a 2x2 matrix

[Anonymous No.12602418]

>>12598199
There's extensive proofs for imaginary numbers having direct real-world applications and existence. They were built from observing physical phenomena and structures. You're literally just triggered by the word "imaginary", and nothing else.

That's the entire solution to your problem. Get over crying about the word "imaginary". If they had called them something like "co-planar roots", you would have never even thought twice about their """""right""""" to exist.

[Anonymous No.12602447]

>>12602413
>In the two dimensional subspace case, you will replace the vector with a two column matrix, and the scalar with a 2x2 matrix
that's what the complex notation stands for, yes

[Anonymous No.12602450]

>>12598199
All numbers are imaginary.

[Anonymous No.12602474]

>>12598199
>Any anon can change my mind? Why imaginary numbers have a right to exist?
learn to speak, please

[Anonymous No.12602627]

>>12602447
...which means the vector doesn’t go in the same direction when multiplied by a complex eigenvalue

[Anonymous No.12602763]

>>12599732
[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]

[Anonymous No.12602772]

>>12602450
BASED

[Anonymous No.12604854]

>>12602627
so?

[Anonymous No.12605166]

>>12604854
I guess you are a retard for not reading the reply chain

[Anonymous No.12605528]

>>12598199
Please answer seriously: how does it feel to be a midwit? You're struggling with the basic concept of mathematics.

>why people think the initial distinction between 0 and 1 makes the solid foundation for all the modern math we have right now
One of these is the additive identity and the other as the multiplicative identity in the field of integers. What you're arguing is as retarded as
>why does the initial distinction between true and false make a solid foundation for all formal logic

[Anonymous No.12605547]

>>12598199
>Why not also create 2+2=4 mathematical objects
Quaternions?

[Anonymous No.12605577]

>>12598199
Well the point of Math is to uncover and catalog all possible rules mathematical objects can interact.
Unfortunately for you, it turns out complex numbers have too many interesting properties to ignore.

[Anonymous No.12605590]

>>12602450
This
>he thinks numbers are real

[Anonymous No.12605599]

>>12598199
Learn2logic faggit

[Anonymous No.12605611]

>>12598199
Let's be honest ALL numbers are pozzed and should never have been discovered by humans.
Even natural numbers produce crazy shit like diophantine equations and Fermat's last theorem.
This is not what evolution intended for us at all.

Counting was a mistake.

[Anonymous No.12605944]

>>12598199
Anon, if you're not shitposting, you *do* know that there's a very simple formal algebraic construction of the complex numbers right? If you accept the construction of the reals, you can accept the ring R[x] and its ideals. The set of complex numbers C is simple R[x] / (x^2 + 1), where we can identify the class of x under the quotient as being *exactly* i, the imaginary unit.

[Anonymous No.12605951]

>>12598316
>Kys, that’s a strawman and not what I’m talking about
Anon, you complain about the complex numbers not being rigorously defined...and then someone rigorously defines it for you, and you call it a shitpost.

[Anonymous No.12606555]

>>12598199
Because anything is allowed in math if it has utility (i.e. doesn't break rules you consider more important). Stop thinking about math as working with quantities or whatever.

[Anonymous No.12606658]

>>12605611
>Let's be honest ALL numbers are pozzed and >should never have been discovered by >humans.
>Counting was a mistake.
Well spoken!
Although, do you think it would be right to deny other species the ability to count as well? I once went to the Li river in China's Guangxi province, where the fishermen use cormoran birds to catch fish for them. A cormoran is allowed to eat every 8'th fish that it catches. So they have learned to count to 8. It is debatable whether it would be cruelty to animals to disallow them to count.

[Anonymous No.12606667]

>taking an object's name too seriously
This is the peak sign of brainletism

[Anonymous No.12608806]

>>12598199
If you are mad at basic reality you need to take your meds.