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0.999...= 1

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Re: 0.999...= 1

#431

Post by lmkg »

theincredulousk wrote: post #308
lmkg wrote: post #137

Earlier quoted context omitted.

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…

You can't fix irrationals by changing base. But, there are other approaches.

Continued fractions give very approachable representations for common irrational numbers like e and sqrt(2). While π doesn't have a good continued fraction, it has some very well-behaved generalized continued fractions.

https://en.wikipedia.org/wiki/Continued_fraction

Re: 0.999...= 1

#432

Post by brutt »

gspr wrote: post #428
brutt wrote: post #418

Earlier quoted context omitted.

I defined it: 0.000...1 = 1/10^∞ = 1/∞

That's not a definition. Neither 10^∞ nor 1/∞ is defined in any standard system, so you'll have to define those too if you want to use them to define 0.000…1.

See https://en.wikipedia.org/wiki/Surreal_number .

Re: 0.999...= 1

#433

Post by vfinn »

kosievdmerwe wrote: post #303
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 = 0.9…

"If 0.999...!= 1 then there must exist a number A, such that 0.999... Thanks for the reply, but I don't know how you can make the above claim, because to me the question of what we mean by 0.999... is intertwined with the claim itself. I mean to me it seems to be a matter of how you interpret the approach to infinity. I don't see why there has to be A in between, if you interpret 0.999... as the "biggest possible number below 1", as then there would be also a "difference of the smallest possible amount" between those numbers approaching 0, but not quite getting there. But then again, if it's by some fundamental definition (limit) that 0.999... = 1, then ok.

I'm slightly embarrassed I don't know more mathematics, but I'm trying to learn some more...

Re: 0.999...= 1

#434

Post by addicted »

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…

There isn’t a proof because it’s actually kind of arbitrary that 0.999... = 1. Fundamentally, this is true because we chose it to be true.

Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was always Fundamentally, 0.999... has no intrinsic meaning, and it’s value depends on the meaning we decide to give this representation.

Re: 0.999...= 1

#436

Post by brlewis »

gspr wrote: post #427
brlewis wrote: post #399

Earlier quoted context omitted.

If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...

> If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?

They have to know that 0.999... means you never stop writing nines.

Put 1.0 on top, 0.9 on the bottom. Start subtracting from left to right, and keep writing nines on the bottom as you go to the right. In no time you'll see that the answer is infinite zeros.

Re: 0.999...= 1

#437

Post by cthalupa »

novacole wrote: post #413

So 99.999..% of the speed of light is just the speed of light?

Yes.

How much faster than 99.999..% the speed of light would you need to go to get as fast as the speed of light?

If your answer is 0.00..1% the speed of light, this answer is nonsensical because infinity never ends, and you never ever have the ability to add that 1 at the end. So, the only answer can be 0.00.., and 0.00.. = 0, so 99.99.. has to = 1.

Re: 0.999...= 1

#438

Post by danharaj »

rrmm wrote: post #414
danharaj wrote: post #171

Earlier quoted context omitted.

If I give you two representatives of real numbers, say turing machines that write out on their tape the binary digits of those real numbers, in general you will not be able to order them.

Certainly not non-computable ones, but presumably they lie somewhere regardless of my inability to do it on a TM. Which presumably gives rise to all the weirdness uncountable infinities give you. I guess I shouldn't phrase it as "you can fully order it". :D Zermelo's theorem at that point right?

Even computable reals do not have computable ordering.

> but presumably they lie somewhere regardless of my inability to do it on a TM.

Why?

> Zermelo's theorem at that point right?

It is declared by fiat in standard set theory that infinite sets can be well ordered. This is no real mathematical justification. The real justification is social: that it is convenient for mathematicians to not care about the ontology of these nasty infinite objects so long as results are mostly reasonable for objects that mathematicians actually care about. You don't get into too much trouble pretending the reals are nice so long as you don't look too hard.

Re: 0.999...= 1

#439

Post by aliceryhl »

brabel wrote: post #422
brutt wrote: post #304

Earlier quoted context omitted.

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

If you accept that 1/∞ = 0 Then you accept that ∞ * 0 = 1 But the definition of 0 is exactly that anything multiplied by it must be 0. So this cannot be true. To take a more verbal route: you cannot take nothingness and repeat it. Repeating (or multiplying) nothingness (or 0) is fundamentally nonsense. Programmer explanation: one cannot loop through `null` even once, let alone a large number.

No, you don't have to accept that. In my Analysis 2 course we worked a whole bunch with [0, ∞], i.e. the positive real numbers together with infinity, and we defined 1/∞ = 0 and ∞ * 0 = 0. You lose some of your usual rules of arithmetic, but it gets a lot easier to talk about integrals.

Re: 0.999...= 1

#440

Post by twiceaday »

addicted wrote: post #434
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…

There isn’t a proof because it’s actually kind of arbitrary that 0.999... = 1. Fundamentally, this is true because we chose it to be true. Now, there are good reasons we chose it to be true, and that’s what people usually use as proofs. If it’s not true then a bunch of mathematical expressions become more inconvenient. But there is no reason as such why 0.999... could not have been defined as something that was alway…

.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.
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