/sci/ - Science & Math

As Dedekind noted, at least part of the idea relies on the Axiom of Choice, which in it's time has thrown up quite a few problems.
In this case, I actually would change my answer depending on whether or not the Axiom of Choice is brought into the original question.
If it is, then they are not equal, because 1 - 0.9999... is a valid arbitrary infinitely long number, and therefore there is a 'distance' between these numbers.
But in the general case I prefer to answer no, as when I do mathematics I assume that infinites and therefore infinitesimals aren't real numbers.

>>7571505
>1 - 0.9999... is a valid arbitrary infinitely long number,

0.000... is a "valid arbitrary infinitely long number,"
...but it's still equal to zero

Yes but so is 0.0000...1,
so as 0.99... + 0.0000...1 = 1, 0.99... is not equal to 1.

>>7571505
>>7571392
W. likes the p-adic notation

>>7571529
>Yes but so is 0.0000...1,
That's just gibberish.
"an infinite series of zeros, followed by a one"

>>7571392
dedekind cuts for pi

http://math.stackexchange.com/questions/488800/dedekind-cuts-for-pi-and-e

the question is : how to know in a finitist manner that we can define pi as this series ? how do discover this series with only finitist means ?

>>7571550
1 - 0.999...

>>7571529
What's the definition of 0.0000...1?

>>7571620
It's pretty obvious, isn't it?

<div class="math">10^{-\omega}</div>

>>7571624
Actually, it's not quite so obvious - they might also mean <div class="math">10^{-(\omega+1)}</div> instead

>>7571626
Yes but you'll never get there to prove it's wrong. So it must be true.

>>7571626
>>7571624
Note, incidentally, that even in regular ordinal arithmetic these are completely meaningless as far as I can tell. So you have to go to some fairly exotic branches of math before these *don't* just equal zero.

These are different numbers in the hyperreals and surreal numbers, which do actually contain infinitesimals.

But in the math used the vast majority of the time, the space of numbers as defined simply doesn't contain those as special and distinct numbers, and they simply equal to zero. (Or possibly are undefined).

This doesn't mean that real math is "wrong" for not having those numbers in it - since the whole concept of discrete, symbolic numbers is essentially something we made up, we get to say what the rules are. And most of the time, leaving out infinitesimals makes the structure of math much simpler and easier to work with. (Sometimes we need the extra "power" available from infinitesimals - e.g. for analysis, of which "pretending dx/dy is actually a fraction" is one notable case)

>welcome to /sci/, the comment section to numberphile and vsauce

>>7571708
what are you talking about? youtube's comment sections are full of mongoloids who don't know anything about math or science.
oh wait...

For all practical purposes, .999 repeating = 1

But theortically speaking, it does not.

Since we're not being very formal I'll speak liberally.
It makes sense to imagine 0.999... = sum from 1 to infinity of 1/(2^n), because the series converges to 1 but it intuitively doesn't quite equal 1.
Now consider 1+ sum from 1 to infinity of ((5^n)-9)/(10^n). This series is crafted so that its partial sum equals the first series' one -1 for every n.
By this perspective, can you say that the two numbers that those series represent are equal or do you consider them to be different?
I'm not stating anything (even if I think the series described above represent the same number), just giving another perspective

0=1

>>7571708
>>7571814
>>7571823
saved

>>7571392
This again.

Think of it like this. 0.99999 + dx = 1

Dx is infinitesimaly small, so it doesn't really matter whether you add it to your repeating number or not, except for aesthetics.

It all goes back to the fact that limits in mathematics are poorly defined.

It's purely a matter of definition. You can say .99... is the hyperreal number 1-h, where h is the infinitessimal element, and not be wrong. If you require .99... to be real, then it's reasonable to define it such that .99...≤1 is an upper bound to every interval of the form [0,a) with 0<a<1 under the total ordering. Then clearly .99... is in every interval [a,1]. Compactness implies that the intersection of all such intervals contains exactly one element, .99... by hypothesis. But the intersection clearly contains 1, hence .99...=1.

>>7571505
>therefore there is a 'distance' between these numbers
Not quite. A metric function must have real codomain and be well-defined. Any small positive real you assign as d(1, .99...) is greater than the rational 1/n for a sufficiently large n, and density clearly furnishes a rational p with terminating decimal expansion such that d(1,p)<1/n<d(1, .99...), a contradiction. Letting d(1, .99...)=0 implies equality or requires you reject the reals as a metric space.

yes it is, a few ways to show this, one simple way is noting that 1/3=0.333333333333......
and multiplying both sides by 3 gives
1/3*3=3*0.333333333333
1=0.999999999......
Another is a little less satisfying depending on your math world view. So for two numbers to be different numbers there should be a number inbetween the e.g. between 1 and 2 there is 1.5. My favourite way though is this. Imagine the infinite sum 9/10+ 9/100+9/1000.....=0.999999...=x
now multiply both sides by 10
10x=9+9/10+9/100.......=9+x
so 10x=9+x
9x=9
x=1

>>7571505
>>7571392


1/3 = 0.33 repeating

0.33 repeating x 3 = .99 repeating

1/3 x 3 = 1

.99 repeating = 1

>>7571529
>Yes but so is 0.0000...1,
>so as 0.99... + 0.0000...1 = 1, 0.99... is not equal to 1.

True if 0.0000...1 > 0, else
0.99... + 0.0000...1 = 0.99 + 0

So if 0.000...1 > 0, they must be different real numbers, and even if one argues that 0.00..1 is not a real number, they still must be different, so it must be possible to find a number x>0 so that 0 + x = 0.000...1, which means you could take x as the second real number.
Two different real numbers have infinitely many other real numbers inbetween.
Since you won't find any, the assumption that the two are different must be wrong, x=0.

>>7571392


>does .999999 the "one" song by metallica

>>7571611
0

>>7572020
No they're not. They approach being well defined :^)

>>7572010

>>7572572
>>7572526
>>7572505
american kids are out of school

Just watch the video ffs
you can't add 0.99999999... and 0.00000...

>>7572608

no, it's .0000000000000000000000000000000000000000000000000000000000...

>>7572630
Yup, 0

>>7572629
The video is idiotic and clearly wrong. Addition and multiplication of irrationals is well defined and Wildburger has done nothing to disprove it.

>>7572629
I'm Scottish you bitch

>>7572634

Just like .999999999999999999999999999999999999999999999999999...

is actually 9

>>7572656
Yup :^)

>>7571888
Underrated.

0.999... = 1, over the reals

>Even Euler knew that 9.999... = 10
>4chan users think he was wrong based on youtube videos

Okay.

>>7573544
>knows nothing about nothing
>comes to throw their shit on a good thread
Piss off euler fanboy

>>7573547
Sure thing mememaster, you sure proved me wrong with those spitting hot arguments. Don't you have youtube videos to be commenting on right now though?

I see most of the people refusing that 0.999... = 1 just seem to have an inherent difficulty in thinking about the concept of "infinity".

>>7571392
No one has posted this yet.
For all you newfags
> pic related

why did we ever allow infinity

>>7573622
As I said yo some other anon spitting his bs all over us, watch the video, this is different

>>7571529
by that value

.000...1 it approaches ZERO

if limits have taught you anything

as one number approaches zero infinitely and the other approaches one.

At all points .9999... and .000...1 will always add to 1. so long as the .001 ends with respect like .9n + .n1 = 1 and on.

So if .9999... is always approaching 1, and .000...1 is always approaching 0 as the "..." increases they can be represented as 1 and 0.

.0001 will never reach a definite value even though it is put like "but after all the zeros goes a one" it will never have a numerical value because the zeros are infinite making the last digit insignificant and actually negligible.

The .9999... will always keep increasing in value as you infinitely add more nines, but it is written in a form that doesn't just say "round up to one"

In base ten there is no way to represent perfectly a base 3 number for example:

2/3 = .6666..

1/3 = .3333...

1/3 + 2/3 = 1

.3333... + .6666... = 1

/thread

>>7571392
>0.9999...
It all depends on your definition of those three dots at the end of the last 9.

>>7573855
>Aproaches ZERO
What the hell kind of number are you talking about? This number is a moving target? It's ACTIVELY approaching zero? Can you say "well 6.75 is clearly approaching seven". Numbers are fixed. This is why 0.000...1 is meaningless nonsense.

>>7574605
What's nonsense is that you think there is actually a "1" at the "end" of that string. You can only approach the end of that string.

2/10 I actually replied to your 4th grade concept of reality

>>7571392
sum(10^-n,n,1,N-1)=(1-10^N)/9
As N grows large the sum grows closer and closer to 1/9. The difference between the sum and 1/9 can be made arbitrarily small. The statement "lim N->∞{(1-10^N)/9} = 1/9" does not imply equivalence between the series and 1/9. It only implies that the limit is 1/9.

>>7571513
>I cant read

>>7571529
>0.0000...1
How many times do I have to see this bullshit on here? Goddamn it.

>objects to the real numbers on ultrafinitist grounds
>immediately assumes a positive number 0.00...01 with infinitely many zeroes exists
Don't ever change you beautiful summerfag high school kids

>>7571392
>1:10:06
tl;dw

>>7576766
cool story bro. Want to tell us what hand you wipe with as well ?