/sci/ - Science & Math

wait wait wait you retard
if y = 1 - 0.999...
then 1 - y = 1 - 1 - 0.999... = -0.999... != 0.999...
fukken idiot
stopped reading there, rest of your post is probs full of bullshit as well

>>5391544

>>5391533
http://www.math.hawaii.edu/~kyle/files/Math%20100-3.4.pdf
http://en.wikipedia.org/wiki/Geometric_series
http://en.wikipedia.org/wiki/Repeating_decimal
See also:
http://en.wikipedia.org/wiki/Stupidity
http://en.wikipedia.org/wiki/Psychiatric_hospital

>>5391556
except you can't minus MINUS something.
that's not fukken possible.
i'm going to gib you the benefit of the doubt and assume that you meant "four minus NEGATIVE two equals six" in that statement.
unless you're trolling.
in which case, 3/10./

>>5391562
>implying I couldn't debunk all this circular reasoning bullshit if I want
Give up already and give me my Fields medal

1/3 = .333...

(.333..)3 = .999...

therefore .999...= 1

>>5391575
y = 1 - 0.999...
0.999... = 1 - y
0.333... = (1 - y)/3

Then your "demonstration":

1/3 = (1 - y)/3
((1 - y)/3)*3 = 1 - y
Therefore 1 - y = 1 - y

Doesn't look like 1 = 1 - y

>>5391592
...but you just substituted 1 with 1 - y.
please stop trolling, we get it, you're mentally challenged.

>>5391533

> 0.999... != 1

Take a course in real analysis, dickhead.

Or in fact any university-level course.

>>5391533

FUCK OFF YOU WORTHLESS SHITBAGGING FUCK FUCK FUCK

>>5391597
But I did.
And they've shown me these "demonstrations".
But I don't gobble up dogma without questioning.

This is the benefit of having a very high IQ.

Feel free to find the mistake in my equations.

Does anyone know of that extension of real numbers of the form
{<span class="math">a+b\epsilon | a,b \in \mathbb{R}, {\epsilon}^2 = 0[/spoiler]}?

I forget what they're called, but maybe that can help give a compromise on this issue. In the real numbers, .9999... = 1, but in this extension, we can think of .999... = 1 - epsilon.

>>5391619

construct e.

I dare you.

>>5391619
http://en.wikipedia.org/wiki/Dual_number

>>5391649
Thanks. Couldn't recall the name.

>>5391619
>http://en.wikipedia.org/wiki/Dual_number
I have learned something today

>>5391556
1 - y = 1 - 1 - 0.999...
1 - y = 0 - 0.999...
1 - y = -0.999

>>5391619
Even within the context of dual numbers, 0.99... is still equal to 1.

>>5391674
Shhhh we're trying to fool the trolls into shutting up

Reported for inane trolling.

>>5391680
You know if you go deeper into Wikipedia there are dozens of educated people arguing the same thing as me. (and they're right)

>This is a fallacy that has been propagated by ignorant academics. As long as there are fools and Wikipedia is the viewpoint of its editorial staff, this grossly incorrect article will serve to misled many students.
http://en.wikipedia.org/wiki/User:ConMan/Proof_that_0.999..._does_not_equal_1#Contesting_the_.22Digi
t_Manipulation_Proof.22_that_1_.3D_0.999...

Have some critical thinking instead of gobbling up everything you're told.

0.999... != 1

Can someone just post the geometric series solution and shut him up.

>>5391710
<span class="math">0.9999... = \sum_{k=1}^{\infty} 9*(\frac{1}{10})^k = \frac{9}{1-1/10} = 10.[/spoiler]

>>5391702
Don't get your math from the talk pages of wikipedia.

Math 101: Course Summary
Rich Schwartz
August 22, 2009
http://www.math.brown.edu/~res/DUS/Summary/M101Summary.pdf
>What is a Real Number? Most people have known about real numbers since grade school. A rough and ready way to describe a real number is that anything with a decimal expansion is a real number. Numbers like 17 and PI = 3.1415926... are examples of real numbers. With this definition, you have to be a bit careful. The two expressions .99999... and 1 both describe the same number. So, you would really have to say that a real number is a decimal expansion, but with the proviso that certain decimal expansions name the same number. To be formal about it, you could say that the decimal expansion 3.14159... is the limit of the series
> 3 + (1/10) + (4/100) + (1/1000) + (5/10000) + (9/100000) + ... .
>So, first of all, you would have to know about about series and limits. Then, you would have to say that a real number is really an equivalence class of such expansions. Making the decimal expansion definition work is actually a bit clumsy, and so a real analysis class usually takes different (but closely related) approaches.

http://www.math.utah.edu/~bertram/courses/4030/Reals.pdf
> Examples: (a) The natural number m expands as the terminating decimal:
> m.000000 · · ·
> The infinite decimal:
> (m - 1).99999 · · ·
> also represents m,

>>5391717
http://homepage.cs.uiowa.edu/~fleck/ratnote.pdf
>Assertion: Each rational number has a periodic decimal expansion, and every number with a periodic decimal expansion is a rational number.
0.9... (repeating) is a periodic decimal expansion, and thus it is a rational number, and thus (obviously) it is 1.

http://www.math.ubc.ca/~cass/courses/m446-05b/dedekind.pdf
> 1. Inadequacy of infinite decimal expansions
> In practical terms, one would be tempted to identify a real number with its decimal expansion. Of course this can’t be quite valid, since some numbers have two decimal expansions:
> 1.00000000... = 0.9999999999...

http://www.springer.com/cda/content/document/cda_downloaddocument/9780387980973-c1.pdf?SGWID=0-0-45-
813443-p173916008
[Slightly modified for formatting]
> One difficulty with using infinite decimal expansions to define the real num-bers is that some points have two names. For example consider the expansions
> 1.000000000... and 0.999999999...
> Call them 1 and z, respectively. Clearly these are different infinite decimal expansions. However, for each positive integer k,
> 1-10^-k = 0.9999999999999...9 [number of '9's = k] <= z <= 1.
> Thus the difference between z and 1 is arbitrarily small. It would create quite an un-intuitive line if we decided to make z and 1 different real numbers. To fit in with our intuition, we must agree that z=1. That means that some real numbers (precisely all those numbers with a finite decimal expansion) have two different expansions, one ending in an infinite string of zeros, and the other ending with an infinite string of nines. For example, 0.12500... and 0.12499999... are the same number.

>>5391702
>dat wiki page

Pure comedy gold. Thanks for posting.

>mfw I found lots of new ideas for future troll threads

>>5391617
You aren't questioning it, you're saying it's false after it's been demonstrated to be true. There is a difference.

>>5391617
You have not in any way proven that y isn't equal to 0

>>5391739

This, if y is equal to 0 the conclusion is still the same, 0.999...= 1. Since your "proof" doesn't draw any conclusion about y it does in fact not prove anything.

A number is rational if and only if its decimal expansion eventually repeats the same pattern forever.

0.999... obviously has such a repeating pattern. Therefore 0.999... is rational.

So there exist integers p, q such that 0.999... = p/q.

p can't be greater than q, since 0.999... <= 1.

Now assume q is greater than p, and consider (p+1)/(q+1). This new number is greater than p/q but still less than 1; therefore we have found a rational number greater than 0.999... and less than 1. Contradiction. So q isn't greater than p.

Therefore p = q, and 0.999... = p/p = 1.

>>5391760
>therefore we have found a rational number greater than 0.999... and less than 1. Contradiction.

Nice circular reasoning. Here you are assuming that there cannot be a number between 0.999... and 1. Thus you are assuming 0.999... = 1. You are assuming what you want to prove.

>>5391533
>Oh! What happened! It should be x = 1!
y=1-.999...
y=0
x=1-Cy=1

>>5391710
Nice try but it's circular reasoning, the demonstration of the theorem uses a limit to define a number.

To get rid of r^n
r = 0 when 0 <= r < 1 and n -> infinity

Which is the same as assuming 0.999... to be 1.
Since the construction of the series 0.999... also has infinitesimal terms at the end.
sum of 9/(10^n) from n=1 to infinity, ect...
I could prove it rigorously but I don't have time now

>>5391714
>>5391720
I'll read that tomorrow.

>>5391739
>>5391751
If it's not true for every y, then the equations are wrong. They don't follow logically.

>>5391775
If it's not true for every y, then the equations are wrong. They don't follow logically.
that is complete bullshit

>>5391781

>>5391763
Good point. Let me try again.

If 0.999... =/= 1, then there is a number x such that 0.999... < x < 1. Consider the decimal expansion of x. Since x < 1, its expansion is of the form 0.abc... . At least one digit to the right of the decimal point must be something other than 9, since 0.999... =/= x. This digit must be smaller than 9, implying that x < 0.999... . Contradiction.

>>5391803
what about
(.999....+1)/2

>>5391805
What about it?

>>5391803

think about it this way:
if you have 0.999...<x<1, 0.999...=0.(9) by definition, where (9) represents an infinite number of 9. your x could simply be 0.(9)9 where you have one more 9 after those infinite number of nines

>>5391803
Here you are assuming that every number can be expressed in decimals. What if (0.999... + 1)/2 cannot be expressed in decimals? You are committing the error of circular reasoning again. You are assuming that decimal representations are valid. But this is what you are supposed to prove, because it's equivalent to saying 0.999... and 1 are equal.

>>5391805
Reals are dense, but who said 0.999... is a real?

>>5391813
>What if (0.999... + 1)/2 cannot be expressed in decimals?

Read the article above on radix representation and then move on with your life.

>>5391816
what else do you suggest it is?

>>5391820
unreal

>>5391812
>One more 9 after those infinite number of 9s

Then that number of 9s really isn't infinite, is it?

Consider .999.... over GF(7). Clearly .999... = .2222...

Now we also know that 22 = 1 in GF(7) so .999.... = .0101010101....

Now take this number over GF(2). We note that .0101010101... is in binary so that every other number is a power of two. But a power of 2 is just 0 in GF(2). So altogether after these transformations, .9999... = 0 in GF(9) [this comes from GF(7)+GF(2) = GF(9) because 7+2 = 9].

This means that .999... leaves no remainder because it's congruent to 0 in GF(9), thus is must be a whole number. But the closest whole number is 1, so .999.... = 1.

>>5391829
What if we add it in the middle?

>>5391829
think about it this way:
if you have infinity(i'll use oo as a notation) then taking in consideration your assumption would mean that oo < oo+1 which is false because oo has that strange propriety that other numbers don't have that says that oo+1=oo (well they are different but none is bigger than the other)
that same propriety allows for that extra 9 to exist

>>5391830
Somebody save this for future .9999... threads

>>5391830

Assuming tha 0.99999... != 1 breaks all sorts of stuff, for example:

10*0.9999.... - 1*0.9999... = 9
9*0.9999.... = 8.9999.....

So really, there are perfectly valid reasons to define 0.9999.... as 1 and convince students of it.

>>5391533
x=.9999
10x=9.999
10x-x=9.999-.9999
9x=8.9991
x=.9999

if we are saying it goes to infinity then there is no point in not using approximations like .999...=1 but that does not mean it actually does equal one.

>tfw math implicitly assumes 0.999... = 1 in all it's proofs that 0.999...=1

somehow, it always makes this circular mistake.

0.9999... = 1
is actually false in reality

it implies that approaching the speed of light is the same as going at the speed of light.

approaching sex and having sex are very different.

>>5392406
Actually it doesn't, even OP's very simple picture doesn't assume anything. OP in his response is the one that makes an assumption, he makes the assumption that y isn't equal to zero and he failes to prove it.

>>5392414
Intresting hypothesis, prove it.

The picture she shows is wrong. The work OP shows is just making x=y-1 go all the way back to x=y-1. Just manipulation of the equation. The picture is should also hold true but I do not think it is fair to compare both of them becasue one uses two variables and the other uses one variable.

You will notice that Y=.0000000000001 or some shit super small.

These are two different equations solving for two different things I guess. I think the fact that the picture related equation does not implement a Y variable makes this so. If you plug in (1-y) into for x ( in the picture equation) you should get the same answer as as the written equation. Both equations are true as far as I know.

Spent like 15 minutes on this shit. This is my explanation.

>>5392430
>even OP's very simple picture doesn't assume anything

it does at step 3
subtracting .999.... from 0.999.... only works if you assume they are equal to 1.

You can't subtract them from each other, they aren't a number. They are an "approaching" quantity, or a "growth"

just like you can't subtract infinities from each other, they aren't numbers, they are growths

>>5392434
>Intresting hypothesis, prove it.


True by definition of "approaching"

>>5392458
It's a rational number, of course i can perform basic arithmetic on it

>>5392458
>nition of "a

To prove/disproof this, you will first have to define what you mean by "numbers" "quantities" or whatever.... you will then have to explain how you represent them. until then you are saying nothing....

I agree with this.

>>5392458
nope, its a rational number. try again.

>>5392458
>they aren't a number

0.999... is defined as the limit as n approaches infinity of the first n partial sums of the sequence 9/10, 9/100, 9/1000, etc.

A limit, if it exists, is by definition a number, not a "growth".

>>5392474
>It's a rational number

yes illegal operation on poorly named quantities.

a quantity that is growing and approaching a limit is not the same as a number

approaching a limit and being at a limit are two different states.

>>5392465
Do you know what the mathematical concept of a limit is, EK?

>>5392493
>Do you know what the mathematical concept of a limit is, EK?


It is a logically imprecise since it confuses approaching something with being at it.

Approaching death and being dead are very different matters.

Approaching X, and being X, are not the same.

>>5392492
Nope, its just a number. rational one at that.

0.999... is always approaching 1, but never reaching it.

1 is quite a different idea altogether.

>>5392498
That's not what the term "limit" means.

Educate yourself: http://planetmath.org/Limit.html

>>5392502

calling it a rational number doesn't solve the logical error.

>>5392507
Because there isn't a logical error to begin with.

>>5392504
>http://planetmath.org/Limit.html

It is a definition of a limit. Getting "arbitrarily close" or "approaching" a value is what a limit is about.

Every definition of a limit suffers from this same gap. Whether you use topological spaces or sequences.

>>5392503
>s always approaching 1, but never reaching i
Again, define the stuff you are talking about.....

There are definitions under which 0.999... is indeed 1 and others where it is not.

>>5392518
>Because there isn't a logical error to begin with.

Approaching X and being at X are not the same

0.999... is a growing towards 1, hence it is not 1.

>>5392529
No it is not growing towards 1, it is equal to it.

>>5392525
Have you even taken freshman analysis?

>>5392541
>No it is not growing towards 1, it is equal to it.

Right, that is the claim, and it is wrong.

>>5392536

Yes. Much of math is arbitrary and leads to problems. Continue your studies.

>>5392547
no its a proven fact.

>>5392550
>no its a proven fact.

The proofs have errors. Like the middle step in OP's pic.

All suffer from implicit illogical steps.

>>5392547
It's only "Growing towards it" on a calculator in base ten, it's a rational number not a limit.

>>5392553


Actually the first step is wrong also

you can't say x = 0.999...

you can say x = 0.99999999, sure

0.999... isn't a number, it's an infinite growth.

you can't subtract infinities.

>>5392553
No, it doesnt, you are wrong.

>>5392559
>0.999... isn't a number, it's an infinite growth.

Here's where you are objectivly wrong

>>5392547
>Right, that is the claim, and it is wrong.
Do you know what an infinitesimal is?

>>5392563
>Here's where you are objectivly wrong

It's an infinite geometric series that never ends.

Numbers end. I don't believe you can subtract or add numbers that have no termination, they aren't numbers.

>>5392565
No it's (1/3)*3. 1.

you can't subtract 0.999... from 0.999....

Do it by hand, step by step if you think it's possible. There isn't enough time in the universe to do this subtraction.

No matter what universe you are in, you will never complete this subtraction, by definition it can't be completed.

>>5392568
I convert it into base 3, then I do it. Very easy.

>>5392570
>I convert it into base 3

Can't convert it to base 3. This suffers from the same problem.

You can't manipulate infinite growths with arithmetic. You can't divide or add infinities.

>>5392568
These threads are an excellent demonstration as to why ultrafinitism is a retarded philosophy.

>>5392574
But it's not an infinite growth, it's just a number. (1/3)*3.

>>5392580

it's a sequence that approaches (1/3)*3. it never actually reaches it, ever.

/sci/ is really the worst board on 4chan

>hurrr deos 0.999...=1
>hurrrrrrrrrr

>>5392586
No, it is equal to it. By definition.

As soon as you specify the number system as <span class="math">\mathbb{R}[/spoiler], 0.999... is rigorously defined as 1. If you disagree, you do not understand the axioms of the real number system, or the axioms of analysis on the reals.

>>5392592
>No, it is equal to it. By definition.


0.999...is a concept much different than "1".

It is a decimal sequence of infinite 9's.
It has in it the concept of infinity, limit, approaching, etc.

>>5392574

Your'e making the mistake of mixing representation and what a number actually is... the fact that something requires infinite similar digits to represent it in one particular system does not mean that its not manipulable in another representation system.

>>5392587
If you ignore these threads and the Big Bang Theory spam as of late you might actually find semi-interesting discussion here.

>>5392593
Axioms are arbitrary and have no truth value.

The concept of 0.999... is much different than 1, they are not identical.

>>5392605

>big bang

unobservable speculation goes on /x/

>>5392608
Then imagine if humans used the binary system instead of the decimal one and tried to add 1/5 + 1/5.... are you saying this is not possible? because this is the point you are trying to make.

>>5392611
But I've already observed six seasons of it...

>>5392608
>are arbitrary and have no truth value
In real analysis, the axioms are taken as true. When we deal with calculus on the real numbers, the precise de finition of .999... as a real number is well known. If you wish to redefine the notion of an infinitesimal and appeal to some pathetic concept of intuitionism, you are free to do so, but I dare you to find a new system other than the hyperreals compatible with the transfer principle such that it conforms to all of the engineering applications of calculus.

>>5392615

How do you figure?

>>5392608
>are arbitrary and have no truth value
In real analysis, the axioms are taken as true. When we deal with calculus on the real numbers, the precise definition of .999... as a real number is well known. So when we use the notation "...", it is already defined like this. If you wish to redefine the notion of an infinitesimal, invent new notation, and appeal to some pathetic concept of intuitionism, you are free to do so, but I dare you to find a new system other than the hyperreals compatible with the transfer principle such that it conforms to all of the shitty engineering applications of calculus.

>>5392611
Please read http://physics.stackexchange.com/questions/11136/what-has-been-proved-about-the-big-bang-and-what-ha
s-not

>>5392635
>In real analysis, the axioms are taken as true.

Yes and the bible is taken as true in Christianity.
Doesn't mean it is.

>>5392647
>Doesn't mean it is.
You have no idea what objectivity means, pussybaby. Thanks for showing off your lack of education.

>>5392647
>es and the bible is taken as true in Christianit
Do you even know how math beyond 1+1=2 works?

>>5392656
>>5392655

>saying X is an axiom because you can't prove it

hehe
take a logic class.

>>5392659
Good one, you unenlightened lamebrain. Give me the axioms of the bible. Mathematics deals with proof - i.e. inference or deduction from axioms, using only the rules defined in the logical calculus you are dealing with. By rules of which logic do you think one has to accept existing laws?

>>5392664
Or, to add, not accept existing laws in an objectively-defined framework. You can redefine the axioms of ZFC, but you cannot call it ZFC.

>>5392659
>>saying X is an axiom because you can't prove it

One terrible aspect of math. It just assumes what it can't call into question or prove.

>>5392685
I'm sorry it's (n)deep(n+1)u. Maybe you'd be better as my waiter.

>>5392685
I'm sorry it's (n)deep(n+2)u. Maybe you'd be better as my waiter.

>>5392685
How are you supposed to prove anything if you do not start with axioms? Take a course on formal logic, you fool.

1/9=.1111111.......
2/9=.22222222.......
and so on.
9/9=.999999999999.........
9 divided by 9 is 1

therefore .9999999.......=1
1 - .999999....= 0.000000000....1 <<infinite zeroes followed by a one

.99999999...... +0.000000000...1 = 1

therefore ,99999999.... is not equal to one.

but is 0.00000000000...1 equal to zero?

>>5392685
>One terrible aspect of math. It just assumes what it can't call into question or prove.

Ya but if you call it an axiom then it's okay.

10 - y - x = 10 - y - (1 - y) = 10 - y - 1 + y = 9

>>5392702
Do you question the axioms of first-order logic? Do you think arbitrary mathematical systems have to have a real-world application? Do you think that you can on the fly change the rules (axioms) of how to play chess while you play it with someone else? When you've both agreed upon the initial rules of the game? You're fucking retarded.

>>5392699
>0.000000000....1 <<infinite zeroes followed by a one

>infinite zeroes
>followed by a one

wut

>>5392721
the smallest number greater than zero

just like how .999... is the largest number smaller than one.

>>5392740
n = 0.000...1
0<n/2<n

Explain that, trollfag.

There should be a new Internet law about how autism and trolling on a math board tend to become indistinguishable

>>5392758
that's it, the new trollface should be an autistic child's face.

>>5392752
congrads you just made a proof explaining why 0.00000000...1 is equivalent to zero.

perhaps you can use the same idea to prove that .999999... is equal to one.

if 0.99... = 1, then there would be no number between it and 1

but obviously there are infinite numbers between the two, since each 9 added to the sequence is a new number

and by definition we need to keep adding 9's infinitely, forever to get to 1

The real numbers are, by definition, the set of all Cauchy sequences of rational numbers, under the equivalence relation that sequences are equivalent if and only if their difference converges to zero.

The moment you understand the formal definition of the real numbers, it becomes obvious that 0.999... = 1. Here's why: 0.999... is the real number represented by the Cauchy sequence whose n-th term is <span class="math">1 - 10^{-n}[/spoiler]; 1.000... is the real number represented by the constant Cauchy sequence 1. (This is just the definition of decimal expansion.) The difference of these sequences is <span class="math">10^{-n}[/spoiler], which converges to zero.

Therefore, the Cauchy sequences 0.999... and 1.000... represent the same number.

Now, can this misconception die already?

.9999... = 1 in canonical mathematics while .999... equals whatever the fuck you want in your own little world.

Deal with it autists.

>>5392781
Give me a reason to use these fallacious "real numbers".

>>5392780
That's not how infinity works. You're not "adding more and more 9's forever" or something. There are already infinitely many there. The number represented by 0.999... is a single, unchanging number, a number also known as 1.

can someone explain three things to me? I just finished calc 3 but our professor didn't have time to go over them.

1. greens theorem

2. stokes theorem

3. divergence theorem

I'm told these are very important to understanding later math. Please help.

>>5392786
>There are already infinitely many there

Infinity isn't a number. It isn't static.

This is where your conceptual problems begin.

>>5392789
The other two are special cases of Stokes's theorem:

The integral over the boundary of an orientable manifold of a differential form is equal to the integral of the exterior derivative of the differential form over the entire manifold.

Hope this helped!

>>5392785
That's what the real numbers are. When a mathematician talks about the real numbers, it's understood that they're talking about that set of equivalence classes of Cauchy sequences of rational numbers.

If you're talking about some other number system, don't refer to it as "the real numbers", because it's something else. It'd be like saying that the rational numbers include pi; sure, you can call it a rational number if you want, but you'd be going against the established meaning of the term.

>>5392796
Why should I use them? Why were they defined in this way?

>>5392797
Because they are the only complete ordered field up to isomorphism.

>>5392802
Archimedean*

>>5392794
All mathematical concepts are static unless the notation of time is explicitly introduced. In particular, even though infinity isn't a natural number, it's still an actual, complete mathematical concept.

Your difficulties with this are understandable; the ancient Greeks were also uncomfortable with this idea: https://en.wikipedia.org/wiki/Actual_infinity

It took until relatively modern times for it to be formalized, but it is certainly fully formalized at this point.

Also, look up cardinal and ordinal numbers. There are ways to treat the different infinities (and yes, there are multiple sizes of infinity) as things that work like numbers in some ways.

Hundreds of pages later and I don't actually see anybody pointing out the error in the OP.

/sci/ is fucking pathetic.

>>5392795
the first sentence is the only part that helped me.

I would only be able to understand the next sentence if i already knew their definitions.

im gonna study stokes theorem first

>>5392806

can't add or subtract infinities
you're conceptual problems are showing.

>>5392802
And why should I use a fields satisfying the Archimedean property?

>>5392797
Why should you use any mathematical object? Because a lot of interesting things come from those definitions. That's really the fundamental reason.

The real numbers form a very interesting and useful object, so out of convenience, it's given a name. Once a name is established, it's best not to suddenly go against convention, because it'll just cause unnecessary confusion.

There are plenty of alternate systems where 0.999... isn't equal to 1; these systems all have infinitesimals. It turns out that they're less interesting and useful than the real numbers, so they're not studied as much.

>>5392812
I wouldn't. Stay away from that non-algebraically closed pleb shit.

>>5392815
>they're less interesting and useful
Why?

>>5392795
>>5392789
In english/math, those three theorems relate surface integrals to line integrals/volume integrals.

1. Greens Theorem states that the line integral of a simply-connected and closed 2-D surface is equivalent to the double integral of the perpendicular curl over the surface.

2. Stokes Theorem generalizes that to multi-dimensional surfaces (where the line integral of the boundary of the surface is equal to the double integral of the curl in all dimensions)

3. The Divergence Theorem relates the surface flux through a closed and self-contained volume to its surface integral over the boundary of the volume.

>>5392810
https://en.wikipedia.org/wiki/Cardinal_arithmetic#Cardinal_arithmetic

And I didn't say they're like numbers in every way. They just have some of the properties of numbers, and they do some of the same things: cardinal numbers measure size (include infinite sizes), and ordinal numbers extend the notion of counting in order.

>>5392808
If you just want the applications of calc 3 without any theory, then study Stokes's theorem in the form of a surface integral, which you can find on google. The one posted was the generalized Stokes's theorem, which also generalizes the other two you asked about.

However, the typical Stokes's theorem given in calc 3 isn't a generalized of Green's/Divergence theorem.

>>5392818
You do realize the complex numbers are Archimedean, right?

>>5392827
I was saying to stay away from the Real numbers.

>>5392819
That's a really hard question to answer. Take a course in real analysis and you'll get some idea of why. Basically, a ton of really nice theory comes from the real numbers; it has to do with the fact that the real numbers are the only complete, Archimedean, totally ordered field.

To paraphrase Fermat, there isn't enough room in this text box to contain a proper explanation of what those are and why they're important.

>>5392818
You can have non-archimedean fields with algebraic closure. Read Robinson.

>>5392835
>implying he actually cares about the properties of real numbers and isn't just goading you into embarrassing yourself by accidentally giving incorrect information

Ignore the trolls.

>>5392835
>he's never heard of non-standard analysis
>he doesn't know about the transfer principle

>>5391575

3*(1/3)=(3/3)=(1/1)=1

>>5392840
Interesting

Is there such thing as a non-Archimedean manifold? Can you have vector spaces over non-archimedean fields?

>>5392844
>he thinks anyone cares about his high school courses

>>5392854
>non-standard analysis
>high school
10/10, comeday gold.

>>5392840
or for a less ridiculously complicated example, take the algebraic closure of the completion of a p-adic field.

>>5392820
>>5392825
much better thank you :)

youtube here i come for some examples.

>>5392856
>they didn't teach non-standard analysis in his high school

Wow, inner-city kids are so underprivileged!

>>5392844
I've heard of non-standard analysis. And yeah, now that I think about it, I was understating their importance; I retract my previous statement. The real numbers are certainly important, but systems with infinitesimals also are.

Thanks for reminding me to stop forgetting a whole branch of analysis.

>>5392852
You can have vector spaces over any field.

>>5391533
also
0.333333.....
">.>

>>5392880
> infinitesimals
> important

pick one.

>>5392921
>completely arbitrary and baseless archimedean property
>important
pick one.