/sci/ - Science & Math

>>12799348
Recall that the matrix form of the SoE looks like [math] Ax = b [/math], where in this case the x vector corresponds to your [x, y, z], and b the RHS of the equation. So taking the inverse of A allows you to manipulate the equation like so: [eqn] Ax = b \implies A^{-1} A x = A^{-1} b \implies x = A^{-1} b [/eqn]. You'll use this method when you're solving larger linear systems not by hand, and understanding how it works is essential for more advanced topics in linear algebra.

>>12798252
Yeah, I'm not gonna pick up an onahole, that's a bit much for me. And the daki is going to completely SFW, so there's even less to be embarrassed about. Just have to get over that initial hump of embarrassment.

>>12615929
>Can u prove that pi is infinite?
As some anons have already noticed ITT, the number pi is actually not a well-defined object. The reason is that there is currently no logically sound theory of real numbers of which pi is a part of. For all practical purposes, pi has something to do with several different algorithms which generate rational numbers within narrower and narrower range, which approximate the area of a circle.
But in the modern illogical framework, pi is said to be a real number that doesn't have a definition as a finite object, in the sense that all definitions require you to do infinite amount of work, such as integrate or sum an infinite amount of terms. In contrast to numbers like sqrt(2), which has a purely finite definition as the positive root of x^2=2.
If you look at it set theoretically, then all real numbers are infinite, as they contain infinitely many elements as sets, but this is deceiving, because clearly the number 1/3 or 2 are perfectly finite and understandable objects. What happened here is that set theorists wanted to accommodate infinite objects like pi into the system of numbers that they had to ruin it for all the other numbers by transforming them into mysterious infinite beasts that totally disguise what they are.
Another, trivial sense in which pi is infinite is that no finite amount of digits will specify it correctly, but that is also shared by most rational numbers, which are very finite objects, so sometimes looking at the decimal expansion is the wrong thing to do to understand things.
cont.

>>12513185
Start with solving the following puzzle to see if you're worth the salt:
On the opposite sides of a long stick there are 2 armies of ants, each army containing 100 ant, making up a total of 200 ants. Every ant in one army moves towards an ant in another army. Once two ants touch, they both immediately turn around and begin walking in opposite directions. Every ant walks with the same constant speed.
When an ant reaches the end of the stick, it falls off and dies.
The question: how many total ant collisions happen until all the ants fall off?

>>12394540
I already said that I didn't mess up the problem. There's no point in lying about that. I posted the version that I wanted to post. It's there in one of the links anon posted. You still insisting that I messed it up makes you the troll, not me.
I can assure you that I'm not trolling, there is an actual solution to this and that everyone ITT so far has gotten it wrong. If you weren't so quick to pass it off and instead spent a little bit of time thinking about it you would realize I'm right.
Also
>>>can dish it out but can't take it himself
Lol what? At this point it's just random accusations with no basis in reality.

>>12230175
>Okay /sci/, does free will exist or not?
It exists.
>Are the movements and interactions of the smallest units that make up the universe predetermined, random or neither?
Neither.
>If it is random or predetermined then, since my body is inherently made up of these units, I am the one making the decisions am I not?
No. If they're predetermined or random you have no control over them.
>I am made of those units and however they choose to move, through whatever force or lack thereof, is however my body and mind chooses to act.
You seem to be ascribing choice to a place where there is none. The units completely determine what you do, and all "choice" we have is an illusion (if you accept that these units are random or predetermined).
>Therefore "free will" is real, even if the nature of the universe is random or predetermined
Doesn't follow.

>reading Serre's "Faisceaux Algébriques Cohérents"
Comfy as fuck

>>11825539
Why would I lie about being a tranny? Trannies are always proud of being trannies.

Whenever someone brings up this question I am reminded of Shafervich's remarks in his essay "On Some Tendencies of Development of Mathematics" (http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mo&paperid=562&option_lang=eng))
>Viewed superficially, mathematics is the result of centuries of effort by many thousands of largely unconnected individuals scattered across continents, centuries and millennia. However, the internal logic of its development muchmore resembles the work of a single intellect developing its thought in a continuous and systematic way, and only
using as a means a multiplicity of human individualities, much as in an orchestra playing a symphony written by some composer the theme moves from one instrument to another so that as soon as one performer is forced to cut short his part, it is taken up by another player, who continues it with due attention to the score.
>Truly, this is not a figure of speech! The history of mathematics is full of examples in which the discovery of one scholar remained unknown only to be reproduced later with amazing accuracy by another. In a letter
written the night before the duel which led to his death, Galois formulated some fundamental assertions about the integrals of algebraic functions. More than twenty years later, Riemann, who certainly did not know of Galois' letter, posed a new and proved those very same assertions. To give another example: After Lobachevsky and Bolyai independently founded noneuclidean geometry, it came to light that Gauss and Schweikart had independently arrived at the same results more than ten years earlier. One experience a strange feeling when one sees the same diagrams, drawn as if by the same hand, in the writings of four mathematicians working independently of one another.

If you accept that
1. All numbers can be represented by infinite decimal expansions in the form of a function f: N->{0,1,2,3,4,5,6,7,8,9, ".", "-"}, (for example, 0.99999... is the function f(0)=0, f(1)=. , f(2)=9, f(3)=9, f(n)=9 for all n>3 . 1 can be represented by g(0)=1, g(1)=., g(2)=g(3)=... = g(n) = 0). A sequence is valid if it has at most one "-" in front and exactly one "." somewhere in the sequence. We assume that every valid representation determines a unique number but not necessarily that every number has a unique decimal representation (for this is what we will want to disprove).
2. You can do arithmetic on numbers in expected ways.
Then you are forced to have
0.9999....=1
Proof:
By comparing each digit, we see that 1 is not less than 0.999....
Now subtract the two numbers (which you can do by assumption 2)
1-0.999... = x >= 0
As every number can be represented by an infinite decimal expansion (assumption 1), there is a sequence of digits for x.
If there are any nonzero digits of x after the dot ".", there is a first nonzero digit and it has a well-defined natural number position N. But clearly that cannot be, since then it would mean that x is bigger than some fraction 1/10^N. Thus every digit of x must be zero and so x itself is zero.
So 1-0.999... = 0
now again, by assumption 2., you can add 0.999... to both sides to get
0.999... = 1
QED

>>11601548
Can we just lay off the ad-hominems, it's really not productive. Do you have an elementary proof using complex numbers that you can break down in terms understandable to a person who's just learning trig? If so, I would genuinely, unironically would love to see it. That's the whole reason why I made this thread, after all!
I know of such a proof but it requires proving a lot of unintuitive facts like sin(x)=x-x^3/3! +... and properties of the exponential.
I'm entirely open to the idea that you have a simpler proof. Please provide it!