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0.999...= 1

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Re: 0.999...= 1

#621

Post by andrewprock »

naikrovek wrote: post #613
gspr wrote: post #594

Earlier quoted context omitted.

> > What is an "infinitely small" number? > What is an infinitely large number? Neither is a well-defined concept within the standard reals, and completely unnecessary for understanding that 0.999…=1. > > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer? > By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, beca…

0.9 is not equal to 1, 0.99 is not equal to 1, 0.999 is not equal to 1, 0.9999 is not equal to 1, 0.99999 is not equal to 1, 0.999999 is not equal to 1, and so on, ad infinitum. Saying that if you add enough "9"s it suddenly equals 1.0 makes absolutely no sense to me, and I seriously doubt that anyone will be able to convince me that it does make sense. I've read every single post in this thread and none of you have…

This is correct.

No finite representation of repeating 0.9s can equal 1.0

The ask that people accept infinite representations as valid is a big one.

Re: 0.999...= 1

#622

Post by tomtomtom1 »

mFixman wrote: post #591
tomtomtom1 wrote: post #175

Earlier quoted context omitted.

wouldn't 0.999.. be equal to 1 - 10^w since it's only a countably finite series of nines.

0.999...9 with a countable finite amount of nines is clearly less than 1. 0.999... with infinite nines is equal to 1.

sorry I typoed I meant, countably infinite. by w I meant the ordinal.

Re: 0.999...= 1

#623

Post by CogitoCogito »

dahart wrote: post #617
CogitoCogito wrote: post #600

Earlier quoted context omitted.

I'm only going to respond to a few chosen points here because I find your post kind of meandering and hard to follow. I'm not cherry-picking (I don't care about "winning" any argument), I'm just trying to focus my response a bit so it increases the likelihood that you'll understand what I'm trying to say. > I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or…

Sorry, I misread the blog post’s statement about infinitesimals, I thought it said surreals. How did Euler actually write his proof, do you know, or have a link? I’ve poked around online but can’t find it.

I usually hear people say that he wrote the algebraic one (i.e. the one that I'm saying is invalid). But I also have no idea where he supposedly wrote it so I can't verify it. For me it's more hearsay.

By the way I wouldn't hold it against him if he did write the proof that way. I'm pretty certain all the logic/model theory that comes into play came long after his death. The surreal numbers certainly did.

Re: 0.999...= 1

#624

Post by gspr »

TheRealPomax wrote: post #611
gspr wrote: post #598

Earlier quoted context omitted.

> 1. we assert that 0.666...7 is a sequence of digits. A sequence in the mathematical sense is a function whose domain is the natural numbers. Please define that function for the creature you're working with here. Otherwise you're trying to prove things about an object with no definition. You will end up in trouble.

Why? https://news.ycombinator.com/item?id=23009160 posited that we were in a situation where (1) holds, so that's where we're starting. If we assume (1) holds, then (2) cannot hold. We can abandon (1) but then we're no longer replying to that specific comment, now we're trying to prove something else.

My point is that it cannot be clear what assuming (1) entails when you aren't properly defining the quantities involved. You have to answer in clear and mathematical language what the quantity in (1) is defined as. What is the definition of "0.666…7"?

As it currently stands, assumption (1) is similar in nature to me saying "gnarfgnarf is an imaginary number". It's completely meaningles unless I define what I mean by gnarfgnarf.

So: what do you mean by 0.666…7?

Re: 0.999...= 1

#625

Post by Pudini »

perl4ever wrote: post #583
Pudini wrote: post #557

Earlier quoted context omitted.

Obviously saying it is not obviously true is false if 0.999... == 1.0

Can something be true and not obvious?

Obviously not.

Which is the point in using the word obvious, obviously. Namely, using it to feel superior or to not provide a better argument.

Re: 0.999...= 1

#626

Post by ant6n »

mywittyname wrote: post #143
ant6n wrote: post #86

Earlier quoted context omitted.

> 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!

I smart middle-schooler is absolutely capable of understanding that dividing 1/3 results in an infinitely repeating sequence of 0.33333... Even without understanding the concept of infinity, they will quickly realize that there's no reason to believe the problem will stop adding a 3 to the end of the result with each iteration.

> I smart middle-schooler is absolutely capable of understanding that dividing 1/3 results in an infinitely repeating sequence of 0.33333..

And how will I smart middle-schooler know that the result of running the long division algorithm is exactly 1/3, rather than some approximation.

Re: 0.999...= 1

#627

Post by [deleted] »

andrewprock wrote: post #619
tom-thistime wrote: post #606

Earlier quoted context omitted.

Could you please explain a little more what "not a countable representation" means?

A countable set is one where you can reach any element in a finite amount of steps. See: https://en.wikipedia.org/wiki/Countable_set

[deleted]

Re: 0.999...= 1

#628

Post by [deleted] »

andrewprock wrote: post #619
tom-thistime wrote: post #606

Earlier quoted context omitted.

Could you please explain a little more what "not a countable representation" means?

A countable set is one where you can reach any element in a finite amount of steps. See: https://en.wikipedia.org/wiki/Countable_set

[deleted]

Re: 0.999...= 1

#629

Post by tom-thistime »

andrewprock wrote: post #619
tom-thistime wrote: post #606

Earlier quoted context omitted.

Could you please explain a little more what "not a countable representation" means?

A countable set is one where you can reach any element in a finite amount of steps. See: https://en.wikipedia.org/wiki/Countable_set

If you're saying we never get from the initial 0 to the trailing 1 in a finite number of steps, that's true and that's what DavidVoid is saying. But I haven't made the list of digits uncountably large by adding one element. Instead I put that element in a transfinite position in the ordering and I have to watch out for weird consequences.

I'm no expert, but countability doesn't depend on how the set is ordered. It depends on whether the elements can be placed in 1:1 correspondence with the integers. 1,0,0,... has a countable number of elements, and so does 0,0,0,...,1. They can be put in 1:1 correspondence with each other. This definition of countability is described in your link.

I meant to ask, does "countable representation" some kind of detailed definition that I can look at?

Re: 0.999...= 1

#630

Post by effie »

hota_mazi wrote: post #608
effie wrote: post #579

Earlier quoted context omitted.

Irrationals have a unique decimal representation in the mathematical sense: given a definition such as $x^2 = 2 $, any digit of the decimal expansion of $x$ can be determined.

> Irrationals have a unique decimal representation in the mathematical sense Not all of them do. Actually, so many don't that mathematically, the number of them that do is zero. Sure, there are exceptions like sqrt(2) and sqrt(3), but there is an uncountable infinity of irrationals between these two numbers that just don't have a representation.

For irrationals, the problem with infinite sequence of digit 9 does not occur. So for any given irrational (given its definition), any digit is determined by this definition, because any irrational number can be approximated with arbitrary precision by a rational number (which has unique decimal expansion). If you think otherwise, where is the problem?

This is not affected by the fact that irrationals cannot be counted. Given the irrational, a rational close enough exists which has the same decimal expansion for the first n digits, for any n.

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