>In foundations class >We've been constructing numbers

>>8374258
>whiteboard

How to spot a shit uni.

dear diary

>>8374281
This is the 21st century, not the 1800s

>>8374288
But it is /sci/ related. I never thought the 0.99... = 1 thing was an actual real life debate. I thought it was a /sci/ grown meme but now I can see with better eyes and I see opportunity to look way smarter than I am.

what, there's no way your professor believed that
also, what's up with all of these silly 'proofs' that play with infinitely repeating decimals? just say "given x,y, if there exists no z (that is not x,y) such that x<z<y WLOG then x=y"

>>8374321
>what, there's no way your professor believed that

I am just as surprised as you are. This professor specializes in pure mathematics and you'd think that he would know better. I think this is something he never actually debated with anyone before and because number theory is not his specialization he saw that 0.5 and the 0.499... and thought that obviously the second one was smaller. Then he's been teaching this every year and because no one knew the argument they just accepted it.

When he realized the proof was correct he was smiling really weirdly. Like he was fucking happy. You know that numberphile meme where the guy is talking about the number 5 and he 'smiles the smile of a 1000 suns' or some bullshit like that? That's how he looked.

>given x,y, if there exists no z (that is not x,y)

And how are you going to proof this.

How can you really show there is a number between 0.49... and 0.5?

The obvious one is doing (0.49... + 0.5)/2 but how does this decimal look like?

0.2499999... + 0.25

Then you'd have to prove that 0.249... = 0.25 and then you would just keep repeating yourself infinitely.

There's a much simpler way to prove that 1 = 0.9999....

It's been rigorously proven that between any two distinct rational numbers, there must exist another rational number. However, there exists no rational number between 1 and 0.999..., so the two cannot be distinct from eachother

>>8374345
>It's been rigorously proven that between any two distinct rational numbers

Coincidentally, we just proved this theorem.

>>8374327
>>8374258
>professor
>specializes in pure mathematics
you can't make me believe this. this isn't even a university I bet

>>8374359
>this isn't even a university I bet

This is a university...

Where else would they be teaching foundations?

File: zero-point-nine-repeating.jpg (32KB, 423x895px) Image search: [Google]

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>>8374319
1 = 0.999... only if you don't know how to divide.

>>8374258
I don't get this proof. So you let x = 0.44... in one and x = 0.4999.. in the other and then x = 4/9 but x also equals .5?

>>8374573
I think the lower rightmost column stands alone and is read top-down.

>>8374573
>>8374606

this.

>>8374327
>And how are you going to proof this.
By contradiction. Assume 0.999.. < z+0.5 < 1 Consider the first decimal were z+0.5 differs from 0.9999.. that decimal must be different from 9 and therefore lower than 9 and it follows that z+0.5 is smaller than 0.9999..

>>8374281
From my experience the best teachers write on boards instead of just going through slides

>>8374697
On chalkboards, not whiteboards

>>8374757
>On chalkboards, not whiteboards
>Chalkboards
>16th century technology
>Better than Whiteboards invented in the 1950s

Hipster go on /killyourself/

>>8374258
But 0.4999... Is not the same as 0.5 and same applies for 1 and 0.999...

Have you even looked at the proofs? Theyre nonsense

>>8374803

yeah lets have this fucking thread again for the 0.999... millionth time

>>8374803
Nice proof m8 I r8 8/8.

> it's come up with a thousand first way to prove 0.(9) = 1

HOW MANY TIMES FAGGOTS: .(9) = 1 BY DEFINITION OF RATIONALS YOU FUCKING RETARDS

>>8374327
The fuck? The way you construct the reals is literally by identifying sequences like 0.4999... and 0.5. Your teacher should be fired.

>>8374327
This is obvious you buffoon.
Suppose there was number between 0.9... and 1. Call it x, so 0.9... < x < 1.
Clearly, x = 0.x1x2x3... where xi is the number in the ith position following the decimal in the decinal representation of x. If x != 0.9... then there exists an n such that xn != 9. But then x < 0.9, a contradiction. Hence 0.9... = 1.

>>8375063
How can there be number between 0.9.. and 1?

>>8374892
>.(9) = 1 BY DEFINITION OF RATIONALS

By what definition of rationals? Rational numbers are equivalence classes of integers, in the way we defined them.

How does that immediately imply that 0.99... = 1?

In what prehistoric intuitionistic way are you defining rationals?

>>8375063

You obtained a contradiction by assuming that x_n != 0.999, not x_n != 1.

0.999... is equal to 1 in the sense that it would be if we could naturally obtain the infinite sequence of 9s. Consider a string treatment of 0.999...: 0.999... is the infinite-th element in the length-ordered set {0.9, 0.99, 0.999, ...}; we see that, as the index in the order increases to infinity, the value at that index converges to 1. More accurately, rather than saying 0.999...=1, it's more accurate to say that the limit of the sequence s_n= s_{n-1} + 9 * 10^(-n) (s_0 = 0) is 1 as n tends to infinity. From this point of abstraction, you can form your own convenient subjective opinion based on how you feel about infinity.

>>8374757
What the hell is the difference, except that whiteboards are easier to clean?

>>8375172

I thought that rationals were equivalence classes on Z^2, not Z.

>>8375354
Equivalence classes on pair of integers...

>>8375272
>is the infinite-th element in the length-ordered set {0.9, 0.99, 0.999, ...}
please stop talking

File: IMG_20160927_173748.jpg (3MB, 3120x4160px) Image search: [Google]

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Did everyone in /sci/ forgot how to math? In math being close to something does not means equals just like limits don't as well.

>>8375547
>4/9 = 0.49...
Maybe you should learn basic arithmetic before trying to do big boy math.

>>8375300
well, they are also cleaner in general, allow better and more understandable drawings, the chalk does not makes your hands dirty, you can rub your eyes after writing without infecting your eyes for two weeks and you can make better multicolor drawings because everyone knows every not-white chalk on black or green boards looks like shit.

>>8374258
Nah, you definitely are fucked if your teacher isnt readily aware of how to convert repeating decimals to fractions. Good luck learning.

>>8375573
Coloured chalk on blackboards is fine. The problem is that the majority of blackboards are actually green and coloured chalk on greenboards is ass cancer

>>8375147
>Suppose... a contradiction

>>8375272
reread my post moron, you completely misunderstood it

>>8374757
>chalkboard are more advanced than whiteboards
Dare I say more?

>>8374258
You must go the absolute shittest of shit-tier universities.

>>8374258
Fess up, OP. What university is this so I can avoid it like the plague?

File: Georg_Cantor2.jpg (223KB, 587x770px) Image search: [Google]

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when we run through cantor's diagonalization argument and produce a new decimal expansion that differs from every element in a supposed list of all real numbers, how do we know that it's genuinely not on the list, i.e. not just an alternative representation of a number already on the list like in this 0.9999 and 1 case?

>>8376539
this is something that has to be explicitly dealt with and it usually isn't.

one thing you can do is just say "if the digit is 1-7, then increase by 1, else make it 2," or similar. Then there are no 9s in the new number.

>>8376564
Different anon here that's barely following.
Won't the problem remain with any repeating numbers? Even if they're not 9.

File: 1473808104380.jpg (17KB, 326x272px) Image search: [Google]

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>>8376539
This is a very good question and one not adequately addressed by most presentations of Cantor's argument, imo.

Probably the cleanest way to do is to note that Cantor's proof implies without any problem that the set T of all binary sequences (never mind real numbers for now) is uncountable. Let [math] f: T\to [0,1] [/math] be the function that takes a binary sequence to the corresponding real number (more precisely, f(a1, a2, ...)=0.a1a2...).

This function is at most two-to-one (because a real number can have at most two different decimal expansions).

Now we can show that [0,1] is not countable. Suppose for the sake of contradiction we could enumerate the elements x1, x2, ... We can write [math] T=f^{-1}(x_1)\cup f^{-1}(x_2)\cup\dots [/math]. But each [math] f^{-1}(x_i)[/math] has at most two values, and in particular is finite. Thus we have expressed T as a countable union of finite sets, which implies T is countable, a contradiction.

>>8376539
Diagonal arguments are unfortunately quite subtle.

Try this paper anon:
http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Gray819-832.pdf

>>8376539
use binary. it can be shown that any number which differs from another by exactly one digit is not that other number. since this is the exact number constructed in the diagonal argument, we are done.

>>8377301
This does not address that anon's question at all.

>>8374258
¿A qué universidad vas anon?

>>8377208
>http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Gray819-832.pdf

why can't we do the same thing with natural numbers and use the diagonal method to construct a transcendental?

>>8377329
Consider that our usual rules are
1) you have a finite number of symbols, read left to right
2) you have a radix point
3) you have a possibly-infinite number of symbols, read left to right

This is just a convention. You could have
1) a finite number of symbols, read right to left
2) a radix point
3) a possibly-infinite number of symbols, read right to left

This would be a p-adic type construction. Then indeed the diagonal argument works on p-adic integers.

It abuses much; but, if you carry out such a construction on the naturals in base 10 you can construct the "number" [math]\overline{9}.0[/math]. If you consider 10-adics (please don't), then this would be -1. -1 is not a natural number, so of course it doesn't appear on your list.

>>8374319

What thr fuck is your math background to think thst is a meme?

>>8375547
>4/9=0.499..
wow

>>8376127
>>8374281
>>8374321
>>8376112
Also since you guys wanted to know

I go to UC Irvine

>>8377534
topkek that is an american university and you can clearly see spanish written in the board, fag.

You are not OP. I know because I am OP and I am NOT fucking telling because I am not retarded. You think I want to turn my university into a meme? Fuck that. I work hard for my shit, y'all boys are just fucking whack

>>8375172
I think he meant by definition of decimal system.

>>8377534
Nice try, I've taken classes there

>>8377301
Actually it works fine in any base OTHER than binary, lol.

>>8375459
Agreed, he should be precise: 0.(9) is the [math]\omega[/math]th element in the ordered set {0.9, 0.99, 0.999, ...}.

>>8377613
>mexican intellectuals

>>8376564
Now prove that the only ambiguous decimal representations are those involving infinite strings of 9s.

>>8377301
>list is 0.1000..., 0.1010..., 0.1100..., 0.0110..., ...
>"new" element generated is 0.0111...
Binary isn't immune to this either, anon.

>>8374781
MIT

>>8377773
>He doesn't know in 2016 MIT stands for MEMES AND IT

>>8374757
retard

>>8375354
You're korrect.

>>8374258
>Some proofs that 0.999… = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999… must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.
>However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean.
You're literally like ancient greeks realizing there is no natural number that corresponds to the square root of 2 and then insisting that this means the square root of 2 doesn't exist.

>>8378919
Did you even read the post lol

Why do autists get so excited when they correct the professor?

>>8377613
>that is an american university and you can clearly see spanish written in the board, fag.
Sounds about right.

>>8379535
t. Donald 'Builddewall' Trump

>>8374697
While this seems to be generally true, I've always thought that presentation slides have so much more potential for laying out examples and graphs in a clearer or more detailed fashion, and allow more material to be covered in a shorter time because you won't be spending as much time waiting for the guy to finish writing things down or spend half a minute wiping the board.

>>8374866
OP started it

>>8376599
Nice, thank you.

Is this some epic meme that people pretend to be stupid by refusing to believe that 0,999... = 1? I don't get it. Like, haha, you made me believe that you're an idiot. Joke's on me I guess.

>>8375272
This is correct, however, in terms of notation
[eqn]0.999... := \lim_{n\to\infty}(1-\frac{1}{10^n}):=1[/eqn]