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0.999...= 1

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Re: 0.999...= 1

#111

Post by topaz0 »

agumonkey wrote: post #89
saagarjha wrote: post #77

Earlier quoted context omitted.

> but then they let an infinite sequence be treated as any finite number Technically it’s an infinite series (sum of an infinite sequence), which is a finite number in certain cases like this one.

certain cases, so define this special one but frankly it doesn't make my brain happy (but who am I)

It's not a special case. It's the standard general definition of the limit of a sequence. The limit of a sequence a_1,a_2,...,a_n,... exists if and only if there exists some L (the limit) such that, for any (arbitrarily small) epsilon > 0, there exists some N with the property |a_n - L| N. In this case a_n is n 9s after the decimal point (or a_n = 9*sum_{i=1}^n (.1)^i).

Re: 0.999...= 1

#112

Post by ristos »

I don't think that 0.999... = 1 is actually provable. I think this and all of calculus is actually axiomatic, which has the following axiom:

Given ε = 1/∞ then: ε = 0

Am I wrong in thinking this way? It seems as though there's no way to actually truly prove that an infinite series converging towards zero actually hits zero (from a constructivist pov)

Re: 0.999...= 1

#113

Post by smlckz »

Reminds me of the paradoxes of Zeno [1], especially the paradox of Achilles and the tortoise.

At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following:

    1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}}
    1 + 2 + 4 + 8 + 16 + ... [somehow] = -1
As well as the weird prime product (Product of 1/(1-(p^-2)) for p prime) and the sum of x^-2 from x=1 to [sigh] being equal to (pi^2)/6 are some example of infinite beauty of mathematics that I remember.

[1]: https://en.wikipedia.org/wiki/Zeno's_paradoxes

Re: 0.999...= 1

#114

Post by mFixman »

vfinn wrote: post #62

Sorry for my naivety, but why one couldn't prove by induction that adding 9s never close the gap, or let's say, that by definition the operation is such, that it never closes the gap. If you can always halve the pie, then you can continue eating forever. To me it would be much easier to accept that (1/3)*3 is not 1.

Let's try a naïve inductive proof of 0.999… ≠ 1.

  Base case: Given 𝑎⁰ = 0, 𝑎⁰ ≠ 1.
  Inductive case: Given 𝑎ⁿ⁺¹ = 1 - (1 - 𝑎ⁿ) / 10, 𝑎ⁿ ≠ 1 ⇒ 𝑎ⁿ⁺¹ ≠ 1
This proves that for every 𝑎ⁿ = 0.999…9, there's an 𝑎ⁿ⁺¹ that's a 9 larger and still different than 1, which is similar to your halving the pie example. However, you can see that it always happens that 𝑎ⁿ⁺¹ > 𝑎ⁿ, so the "last" infinite 0.999… is not part of the sequence of the inductive case.

My intuitive way to see this is that infinitely repeating decimals are an abomination that breaks the nice property of decimal notation where each number contains a single representation (without zeroes at the beginning or at the end of the decimal). Fractions are the one true way to represent rational numbers.

Re: 0.999...= 1

#115

Post by undecisive »

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it.

If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount.

People mistakenly believe that decimal numbers don't have multiple representations - which, in a way is correct. The bar or dot or ... are there to plug a gap, allowing more values to be represented accurately than plain-old decimal numbers allow for. It has the side effect of introducing multiple representations - and even with this limitation, it doesn't cover everything - Pi can't be represented with an accurate number, for example.

But it also exposes a limitation in humans: We cannot imagine infinity. Some of us can abstract it away in useful ways, but for the rest of the world everything has an end.

I wonder if there's anything I can do with my children to prevent them from being bound by this mental limitation?

Re: 0.999...= 1

#116

Post by recursive »

tomtomtom1 wrote: post #84
saagarjha wrote: post #73

Earlier quoted context omitted.

> but 1/3 != 0.33333... But the issue is that this is easy to verify experimentally via (in this case infinitely) long division that you can do by hand. So it’s hard to convince people of this.

but the long division algorithm never terminates. why would it terminate at countable infinity?

You can show that the long division algorithm is looping. Further, you can show that it will continue, for these inputs, to produce `3`s forever with no change in state. How could `...` be defined such that it wasn't 0.333...?

Re: 0.999...= 1

#117

Post by cochne »

smlckz wrote: post #113

Reminds me of the paradoxes of Zeno [1], especially the paradox of Achilles and the tortoise. At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following: 1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}} 1 + 2 + 4 + 8 + 16 + ... [somehow] = -1 As well as the weird prime product (Product of 1/(1-(p^-2)) for p prime) and the sum of x^-2 from x=1 to [ sigh…

It's less controversial than those because it's actually true. The latter statements are not true with the traditional definition of a sum.

Re: 0.999...= 1

#118

Post by gbrown »

ant6n wrote: post #86
notRobot wrote: post #29

Earlier quoted context omitted.

> if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3 I disagree. Any middle school student can calculate 1/3 to be 0.33333... using long division, but there's no immediately obvious way to go from 1 (or 1/1) to 0.9999...

> Any middle school student can calculate 1/3 to be 0.33333... using long division, ... ...the same way That Chuck Norris can count to infinity... twice!

Recursively?

Re: 0.999...= 1

#119

Post by traderjane »

undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

You must go up to something like limits to make ... meaningful.

Re: 0.999...= 1

#120

Post by upofadown »

eof wrote: post #104
upofadown wrote: post #102

> ...infinitely many 9s... How about we prove that an infinite number of 9s is impossible? Assume that we have a finite number of 9s. Add a 9. The result is not infinite. Add another 9. The result is still not infinite. We can repeat this process for an infinite amount of time and still not have an infinite number of nines. Any process that can not be completed in a finite amount of time can not complete and can not…

I tried this route as a counter to Cantor's diagonal argument, and got chastised by my then professor. I hope you have better luck, as I was never able to convince myself otherwise.

By that point you were already working in a universe where infinity was in some sense real. The universe of mathematics is different than the universe we live in.
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