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0.999...= 1

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Re: 0.999...= 1

#301

Post by black_knight »

WORLD_ENDS_SOON wrote: post #278
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…

I wonder if this is related to "intuitionist" math. This is an alternative formulation of math which doesn't have the law of excluded middle, recently discussed on Hacker News relating to this physics research: https://www.quantamagazine.org/does-time-really-flow-new-clu...

If you want to work with the real numbers intuitionistically (or constructively), you quickly find out that infinite decimal expansions are not what you want.

In classical mathematics all the usual definitions of real numbers (decimal, Cauchy sequences and Dedekin cuts) are equivalent. If you overthrow the Law of Excluded middle, these are all different.

Infinite decimal expansions are bad intuitionistilcally for several reasons. The first one which come to mind is that you cannot add numbers together. Imagine your numbers started 0.33333 and 0.66666. OK, so far it would seem that the sum would start 0.99999, but somewhere down the line one the firs number could contain two 4s, making a 1 carry all the way up and leaving one behind, so that it should in reality be 1.00000000001…

On the other hand there could also show up a 2 later, making it 0.999999998. Thus, you cannot decide weather the first decimals should be 1.00 or 0.99 without looking at infinitely many decimals. And the fact that 0.999… = 1.000 will not help you out, since 1.00000000001 ≠ 0.999999998.

Being able to define addition on decimal expansions is equivalent for constructivists to solving the halting problem. It cannot be done.

It turns out Cauchy sequences are better behaved, and (with a bit computational improvement) you can make a lot of things work out. See Bihshop's book, Foundations of Constructive Analysis, for details.

Re: 0.999...= 1

#302

Post by heinrichhartman »

0.9999 = 1 is a consequence of the way we define rational and real numbers and limits. There are alternative definitions of numbers where this equality does not hold: Non Standard Analysis https://en.wikipedia.org/wiki/Nonstandard_analysis being the most famous one.

But for the sake of argument, let's just define numbers as sequences of digits with a mixed in period somewhere:

    MyNumber := {
      a = (a_1, a_2, ...) -- list of digits a_i = 0 .. 9; a_1 != 0.
      e -- exponent (integer)
      s -- sign (+/- 1)
    }
Each such sequence corresponds to the (classical) real number: s * \sum_i a_i * 10^{i + e}.

We can go on and define addition, subtraction, multiplication and division in the familiar way.

Problems arise only when we try to establish desireable properties, e.g.

(1/3) * 3 = 1

Does NOT hold here, since 0.9999... is a difference sequence than 1.000....

So yes, you can define these number systems, and you will have 0.999... != 1. But working with them will be pretty awkward, since a lot of familiar arithmetic breaks down.

Re: 0.999...= 1

#303

Post by kosievdmerwe »

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.

Other people have pointed out that induction never makes the jump from a finite number of 9s to an infinite number of 9s.

I feel the easiest "proof" is a proof by contradiction.

First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x If 0.999...!= 1 then there must exist a number A, such that 0.999... Now since 0 For instance, we might have A = 0.99998999... .

However, this would mean A Now this isn't a rigorous proof, the thing that makes me most uncomfortable is the bit that I state A [1] If this is not clear think about how you would compare two decimal expansions to see if one is smaller than the other. You go through every digit until you find one that is different between the two numbers and then you compare those.

Re: 0.999...= 1

#304

Post by brutt »

tom-thistime wrote: post #121
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…

(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.) In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof. But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. T…

    0.999... + 0.000...1 = 1
    0.000...1 = 1/∞
    0.999... = 1 - 1/∞
1/∞ is zero or not?

Re: 0.999...= 1

#305

Post by iamgopal »

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…

5 year old is curious or asking such question is mind blowing.

Re: 0.999...= 1

#306

Post by pwdisswordfish2 »

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…

There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely.

Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?

Re: 0.999...= 1

#307

Post by GolDDranks »

OskarS wrote: post #184
amelius wrote: post #163

Earlier quoted context omitted.

I think you should look more closely at what I did with the order of operations, and the fact that "<" now is part of the expression acted over by the limit.

Limits are defined for functions and "1 - eps lim[eps->0]( 1 - eps is equal to 1 - 0 which is obviously false. Again though, limits are defined for functions, not inequalities. If you have the limit lim[eps->0] (x - eps) Then it is equal to x - 0 which is equal to x. You're simply wrong here. You can read that wikipedia article if you don't trust me, or you can watch any of the thousands of youtube videos of mathemat…

You know, inequalities can be interpreted as boolean functions. The function under the limit produces consistently true as you approach the limit. (Of course one-sided one; the case where epsilon is getting smaller).

It isn't a valid argument for 0.999... != 1 but an interesting one nevertheless.

Re: 0.999...= 1

#308

Post by theincredulousk »

lmkg wrote: post #137
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…

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.

great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3.

I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in base-10 decimal.

Re: 0.999...= 1

#309

Post by jonfw »

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…

Is this a mental limitation, or is it a simple defense mechanism against diving into rabbit-holes of thought with no end and no real productivity? It seems much easier to come to the conclusion that .99999... and 1 are different numbers, is it really worth the effort to consider otherwise?

We create these abstractions to simplify our thought- and analyzing or over-analyzing these simplifications can have the opposite effect.

Re: 0.999...= 1

#310

Post by da_chicken »

tom-thistime wrote: post #121
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…

(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.) In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof. But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. T…

The easiest way I know to explain it is fractions.

  1 / 3 = 0.33333....

  2 / 3 = 0.66666....
So what's 3 / 3?

Some people don't like that one. They might like this one better:

  1 / 11 = 0.0909090909...
What's 10 times that?

  10 * 0.0909090909... = 0.90909090...
So, let's do some addition and let the values zipper together because a nine will always line up with a zero:

  10 * 0.0909090909... + 0.0909090909... = 0.90909090... + 0.0909090909... = 0.9999999999...
However, 10 * 1 / 11 = 10 / 11. And 10 / 11 + 1 / 11 = 11 / 11. So 11 / 11 must be the same as 0.99999....

This works for any repeating fraction. You can do it with 1/7 and 6/7. You add the decimal representations of the numbers up and the value will be 0.99999...

Technically, it works for any repeating fraction in any base. This is great because a lot of fractions are only repeating fractions in certain bases. So if 0.1 in base 10 is a repeating decimal in base 2 (it is) then you can show that (in decimal) 0.1 + 9 * 0.1 will represent (in binary) 0.11111...., which is equal to 1.

The issue is that 1 / 11 + 10 / 11 (in decimal) must still equal 1 in ALL bases. Well, guess what? In Base 11 the decimal looks like:

  0.1 + 0.A = 1.0
And 1.0 in base 11 is 1.0 in any base.
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