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0.999...= 1

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Re: 0.999...= 1

#381

Post by petters »

LunaSea wrote: post #327
laichzeit0 wrote: post #318

Earlier quoted context omitted.

> We cannot imagine infinity Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number

This is one of my pet peeves in maths. Although I do understand the concepts presented, the notion of "greater" makes no sense when applied to something without boundaries. Yet it's used all the time.

Personally, I think it makes perfect sense.

Take two sets A and B. If we can assign every element in A to a different one in B, we say that |A|≤|B|.

Makes perfect sense for normal, finite sets, right? As it happens, this definition extends to infinite sets as well.

Re: 0.999...= 1

#382

Post by [deleted] »

da_chicken wrote: post #298
thinkharderdev wrote: post #288

Earlier quoted context omitted.

That doesn't exist. An open interval doesn't have a smallest number.

Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly…

[deleted]

Re: 0.999...= 1

#383

Post by alanbernstein »

_Donny wrote: post #264
317070 wrote: post #192

Earlier quoted context omitted.

It depends on more than just ZFC, also on the definitions of the real/complex numbers. The crux of the proof is that 0.99999... is being constructed within the real/complex numbers, and in that system it is equal to 1. And at the point where students see this, the whole concept of real numbers and infinity is usually ill-defined. I actually understand the scepsis for this theorem and where it comes from. The proof re…

I think this is spot on, at least for me personally. I am not very good at mathematics, so I never questioned my professors when they said that "You cannot treat infinites as regular numbers". Perhaps due to that statement, I did not really pursue these kinds of equations. For instance, I do not really see how the algebraic argument on the Wiki is any different from: 2 * inf = inf inf + inf = inf (subtract inf from b…

There is this phrase, often used when describing the decimal expansion of pi - "keeps going infinitely". This phrase is not exactly incorrect, but I wonder if it misleads people into thinking that an "infinite decimal" is "a kind of infinity", which it really isn't in any meaningful way.

Re: 0.999...= 1

#384

Post by hinkley »

saagarjha wrote: post #66
champagneben wrote: post #51

Earlier quoted context omitted.

How do you jump from x = 0.999... to 2x = 1.999..

It’s less intuitive, but still relatively straightforward from performing the “long” multiplication. You do need to convince people that the “8 at the end“ is irrelevant, however.

You need to convince people there is no 8 at the end because there is no end.

If you see an 8 it’s because you didn’t carry the 1. Keep going.

Another way to think about it is that you have n digits, and I’m using the n and n + 1 digit at the same time. But since n goes to infinity, +1 hardly matters.

Re: 0.999...= 1

#385

Post by gspr »

brutt wrote: post #363
brlewis wrote: post #341

Earlier quoted context omitted.

You're repeating the same wrong thing you said earlier. It's 0.999... and not 0.999...0 In the same way, it's 0.000... and not 0.000...1.

0.999... = 0.999...9 0.999...9 + 0.000...1 = 1 0.999...0 + 0.000..1 = 0.999..1 0.000...1 = 1/∞ 0.999...9 = 1 - 1/∞ 0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞ If x/∞ = 0, then 0.999...x = 1. If x/∞ ≠ 0, then 0.999...x ≠ 1.

None of these, except 0.999… and 1 are well-known standard objects in this setting. You have to define what you mean.

Re: 0.999...= 1

#386

Post by 0xff00ffee »

tom-thistime wrote: post #224
0xff00ffee wrote: post #203

Earlier quoted context omitted.

Yep. Agree 100%. It is like the blue dress. I think the problem is the repeating function. Infinite things are non-intuitive and should be presented differently. Even here on HN you still see people confused about "convergence" and "identity". 0.999... doesn't CONVERGE, it literally is 1. I suspect this persists even with students that have had second year college calculus that discusses convergent series and sums.

Fine. Define 0.999... as the limit of the series sum(n=1 ... N)(10^-n), as N-> infinity. This is standard high school calculus. "Number" and "series" and "limit" and "convergence" don't all mean the same thing. However this number is defined as the limit of a convergent series. So the question really is meaningful. (One clue that this question is meaningful is the amount of space introductory calculus textbooks use t…

I'm not sure if you are asking for an answer or a rhetorical question? I'll assume the former.

Your terms are bit jumbled, so let's keep it simple: you're asking how to prove if an infinite sum converges and what its value is. Convergence proofs require analytic thought: meaning there may not be an immediate look-up. You need to convert the problem into the known corpus of convergent sums or use one of many tests (bounds test, integral test, etc) to show it converges analytically. Which you only learn through experience and memorization (unless you want to re-prove hundreds of series... maybe you do!) Fortunately this one is easily re-written as a known convergent sum.

First, you missed a term in your sum (9), re-written here:

sum(n=1..inf) 9 * 10^-n

Step 1: you pull out the 9 and it becomes 1/10+1/100+1/1000...

Step 2: Then we shift to n=0 by subtracting 1/10^0 from the series so that it is in the form n=0..k-1

1/10^0 + 1/10 + 1/100 + 1/1000 + ... + 1/10^-n - 1/10^0

Step 3: Now we've got ourselves a geometric series of just 1/10^n .. wikipedia does a great job explaining the sum convergence for GS from n=0...inf: https://en.wikipedia.org/wiki/Geometric_series

Step 4: compute geometric convergence

(1-r^n)/(1-r) = (1-(1/10)^n)/(1-1/10) = 1/(1-1/10) = 10/9

So we have 10/9 as the solution to Sum[n=0...inf](1/10^n)

Step 5: the remaining arithmetic

Now subtract our 1/10^0 ... and then * 9 = 1

Re: 0.999...= 1

#387

Post by enedil »

sebringj wrote: post #258
ping_pong wrote: post #160

My 5 year old stumped me with this, and I had to look it up. He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...? I didn't have a good enough answer for him, so I had to look it up and found this page. I tried to explain it to him but since I'm a terrible teacher and he's only 5, it was hard…

Is this problem simpler than we want it to be? Meaning 1/3 is a concept stating there is 1 part of 3 total. If you have 3 total parts, added then it is a whole. Trying to shoe-horn it into the decimal system, similarly to try to represent pie as a clean number into the decimal system etc. Isn't the issue representing the number in one for and another, not the actual logic of the issue? idk

Why this is an issue, when 0.9999... is exactly 1?

Re: 0.999...= 1

#388

Post by brutt »

gspr wrote: post #376
brutt wrote: post #362

Earlier quoted context omitted.

0.000...1 = 1/∞

And what does the right hand side of that mean? Division is commonly defined for a real numerator and a real, non-zero denominator. You are using the common symbol, but with ∞ in the place of the denominator. Since ∞ is not a real number, you must be using a non-standard definition of division, and have to define what you mean.

In some systems, division by ∞ is not defined at all (forbidden), in other it defined as 0, in another systems it defined as non zero.

Re: 0.999...= 1

#389

Post by gspr »

brlewis wrote: post #377
gspr wrote: post #353

Earlier quoted context omitted.

> There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.

You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.

> 1 - 0.999... = 0.000... proves that 0.999... = 1.

If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point

Re: 0.999...= 1

#390

Post by jonsen »

oconnor663 wrote: post #348
jonsen wrote: post #324

Earlier quoted context omitted.

The first is an infinite series: 9/10 + 9/100 + 9/1000 + ... + 9/10^n + ... The second is not?!

I mean you could if you want to: 1/10 + -9/100 + -9/1000 + -9/10000 + ... But I just meant them as series, not necessarily as sums.

Or

  1 - 9/10 - 9/100 - ...
or

  1 - ( 9/10 + 9/100 + ... )
So it becomes circular:

  0.00...1 = 1 - 0.999...
(In math a series is a sum:

https://en.m.wikipedia.org/wiki/Series_(mathematics)

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