/sci/ - Science & Math

>>11987247
If you want to assert numbers can operate 2 dimensionally and make areas of squares and you accept inverses then you have to allow this to happen universally to all numbers. You can't apply this universally in the system we made so sort of as a band aid we allow this to keep the axioms in place.

>>11987247
They “exist” just as much as any other number.

>>11987247
They complete the basic system of algebra you're used to. They don't come from nowhere, they come from the last gap left in that system.
Your intuition means nothing. Practical applications mean nothing.

>>11987247
>I understand they have practical uses but what's the reasoning or proof behind them?
The reasoning behind them is twofold:
- It is possible to extend the system of mathematics you're used to (real numbers) to a broader set of numbers (complex numbers) in which you can express certain ideas; and when you do this, importantly, almost all the usual rules and properties of mathematics still hold. That last past is what makes the complex numbers an extension of real numbers, rather than just an idle fantasy.
- The resulting system is useful for practical things.
That's it, really. Mathematicians are generally interested in richer versions of systems of math we are already used to, that still have all the traditional requirements of a system of math. If such a system is conceived of and also serves a practical purpose, people start using it.

>>11987247
>what's the reasoning
x^2 + 1 = 0 doesn't have a solution, let's see if we can come up with a reasonable structure where the solution exists.
integers from naturals is the same thought process, the equation is x + 1 = 0. same for rationals from integers, the equation is 2x - 1 = 0 for example.

>>11987307
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

>>11987423
this

>>11987247

Imaginary numbers make sense if they are defined. Imaginary numbers in Schrodinger's equation are bullshit.

>>11987247
[math] U(1) \cong SO(2) [/math]

>>11987555
and how so? Wave equation is a fucking wave so complex solutions are great to explain oscillations. Also if you're doing linear combinations only in reals your superpositions will just not work, period.

>>11987247
They're extremely important in analyzing stability of dynamical systems in engineering and also doing circuit analysis in electrical engineering, so yes they are very practical.

>>11987247
wow so contrived to extend the reals to satisfy algebraic completeness. Fucking normies.
[eqn] \mathbb{C} \cong \mathbb{R}[x]/(x^2+1) [\eqn]
Here you fucking go. Sooooo contrived and unrealistic. You know what's contrived? Extending the domain of the square root to include -1 for no reason. But apparently it's the only way to explain this stuff to normies without touching abstract algebra.

>>11987795
[eqn] \mathbb{C} \cong \mathbb{R}[x]/(x^2+1) [/eqn]
fucked that up

>>11987247
[eqn] \mathbb{C} \cong \mathbb{R}[x]/(x^2+1) [/eqn]
Sooooooo contrived, jeez.
>what's the reasoning or proof behind them
there's no proof for a fucking definition, retard. As for reasoning, it's defined as an algebraically complete extension of the reals.

>>11987247
Numbers are just an abstract thought that has uses. "Imaginary" numbers are similarly numbers that have uses in regards to the complex plane and more. Complex numbers allow for very easy rotation of vectors by simply multiplicating a vector with an imaginary number (which rotates a vector 90° anti-clockwise in the complex plane). Numbers are just as imaginary as "imaginary numbers" are.

Are natural numbers legit? There can be orange and orange, but claiming that there are "two oranges" is way too abstract.

>>11987762
>Wave equation is a fucking wave so complex solutions are great to explain oscillations

Made up bullshit. Exactly what does the imaginary unit mean? It's supposed to be a law of nature. Like F=ma F is force, m is mass, a is acceleration. It's a physical law it's supposed to be physical. What does I in the Schrodinger equation mean physically? Which physical quantity?

>>11988021
Why do you have to assert that imaginary numbers exist somewhere in physics? Most mathematicians are Platonists that don't think numbers are physical objects, but immaterial ones. It would mean you would have to defend the existence of immaterial objects, but it also means you don't have to ask questions like "do imaginary numbers exist in physics".

>>11988021
The i in schrodinger comes from operator of energy. The operator is defined in the way, the value of <psi|E|psi> is measurable, thus real

>>11987247
>what's the reasoning
Without imaginary numbers, the fundamental theorem of algebra fails sometimes. It is considered desirable for the theorem not to fail, ever.

Yes

>>11988021
you clearly never worked with second-order linear differential equations, which are essential in physics and in wave equation in particular. Using [math] e^{i\theta} [/math] saves you so much trouble when integrating/differentiation/doing algebra on it than sine and cosine.
>Made up bullshit
congrats, bro, so are all other numbers. One way to rigorously define natural numbers is through fucking empty sets.
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
Why aren't you flipping out on that, but complex numbers are a problem. Ah, let me guess, it's the word "imaginary". Are rational numbers actually rational? It's just stupid semantics.
>Exactly what does the imaginary unit mean?
It is an element of the set [math] \mathbb{C} [/math] with the property [math] i^2 = -1 [/math]. Now you can define this set using the algebraic definition in >>11987820.
>What does I in the Schrodinger equation mean physically?
what does force mean physically, you genius? Can you feel it? You feel acceleration, but that's it. The concept of force is completely useless in non-classical physics and even in some classical situations it is much more desirable to talk about energy and momentum, which are even more abstract than force.

Just a placeholder so we don't write square root(-1) with everything, so we have an equation to not overcomplicate anything.

>>11987247
The fact that imaginary numbers are used is proof that math is a human invention and not universal.

Complex numbers are a mathematical tool, just to aide in middle of physical calculations. A trick. Schrödinger's equation is a "half equation" meaning the result must be always multiplied with its complex conjugate to get a real probability density.

>>11988270
The square root of -1 is not i. i^2=-1.

>>11987265
>Your intuition means nothing. Practical applications mean nothing.
Most based math post in the history of /sci/

>>11987247
1-2=-1

Are negative numbers legit?

>>11987247
imaginary numbers are just the y-axis in cartesian coordinates. nothing imaginary about them. euler's formula is a relation between coordinates and angles. its hard to keep track of things if ur using the cartesian symbols

>>11987247
Imaginary numbers? They are an affront to God and nature. As such they must be cleansed from the Earth with HOLY FIRE. Along with those GOD CURSED SODOMITES who promote them. DEUS VULT!

>>11987247
They are kind of a stop-gap, but ultimately you could equally start questioning any kind of number.
Potentially except for the naturals, but even those took a Peano to "prove" them.

>>11988711
Yes, -1 is a real number does not have any real square roots. However, the ordered pair (-1,0) has two complex ordered pair roots, (0,1) and (0,-1), under some specific definition of the product of ordered pairs.

>>11988711
Yes, -1 is a real number and does not have any real square roots. However, the ordered pair (-1,0) has two ordered pair square roots, (0,1) and (0,-1), under some specific definition of the product of ordered pairs.

They are nonsense. Pic related shows how easily complex numbers fall apart.

>>11990507
>literally write -1 instead of i
>pic related shows how easily negative numbers fall apart

>>11990684
kek

>>11987247
You have missed your opportunity for the ultimate shitpost.

"Are imaginary numbers even real?"

>>11990684
(-1)^2 + 1^2 = 1 + 1 = 2
Retard.

I stopped caring about math when I was introduced to the concept of imaginary numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't exist, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

>>11991393
that's the point. no solutions for sqrt(negative) is a flaw. Imaginary numbers is one way of fixing it. They are not imaginary. It's just a name, and it stuck.

>>11987247
works on my machine

They're most useful when they're imagined away.

>>11992024

>>11987247
You can say the exact same thing about your bog standard integer. You'll never see an actual 4 in nature. They're abstractions invented without corresponding physical objects because they're useful for certain purposes.

>>11990934
[math]i\cdot\bar{i}+ 1\cdot\bar{1} = 1 + 1 = 2[/math]
retard

>>11987265
>>11988931
'i' does have practical applications though.

>>11992138
It definitely does but that has nothing to do with why it's valid

>>11991393
shit pasta
give up

>>11992134
i^2 + 1^2 = -1 + 1 = 0
Retard.

>>11987247
Complex numbers are the algebraic closure of real numbers.

>>11990507
[math]-i^2=i^2[/math]

[math]-i^2+1^2 = 2[/math]

>>11992757
you're not using the correct inner product anon

>>11994982
>muh inner product
It's just the pythagorean theorem, brainlet.

>>11995018
pythagorean theorem is a statement about inner products anon

>>11995018
>he doesn't know greeks invented the inner product

>>11995031
Look at the original pic, anon. There's nothing more than the plain pythagorean theorem, just simple geometry.

>>11995053
>plain pythagorean theorem
>the sum of the squares of the lengths of each of the triangle's legs is the same as the square of the length of the triangle's hypotenuse
>square of the length
[math]|x|^2 = \langle x,x \rangle[/math]

>>11995073
>>square of the length
x^2 = x*x
Don't try to justify the flaws in complex numbers with gibberish.

>>11995081
>>square of the length
[math]|x|^2 = \langle x,x \rangle[/math]

>>11995101
I accept your defeat.

>>11995118
>I have no argument

>>11995144
>You are right modulo 2Pi.

>>11995144
I know you don't, that's why I accept your defeat.

>>11987307
> implying ye know shite about splitting fields

>>11990507
Okay, but all joking aside: this was a whole thread a while back, and I saw a *lot* of different posters talking about how it destroyed the concept of imaginary numbers.
So, like... are there just that many trolls here? Or are there actually just tons of stupid people here on /sci, and I miss them because I assume they were being ironic?
Might make a thread about this later, want /sci's opinion on how stupid /sci is.

>>11987247
If a notion can be consistently and unambiguously defined, and can be shown to have no contradictions, why should it not be considered to have a bona fide mathematical "existence"?

>>11995947
lurk more buddy

>>11987884
came here to post this

>>11987247
Them being a useful tool is good enough for me. All that other bullshit is just philosophical nonsense.

>>11995902
?