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0.999...= 1

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Re: 0.999...= 1

#511

Post by btilly »

obastani wrote: post #358
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.

This is a good point, but an even more basic issue is that the question "what is a number" is a matter of definition. There isn't a "correct" definition of numbers; only one that we've accepted as standard. The accepted definition of a "real number" is actually quite complicated [1], and it's certainly not easy to convey why this complexity is necessary. Other definitions are also possible [2], but nonstandard. The s…

Sorry, no.

There are TWO standard definitions of the real numbers. Namely Dedekind cuts and Cauchy sequences. (They are completely equivalent.) The usual decimal representation of a number turns out to be a Cauchy sequence.

The "simplest definition" that you provide turns out to be rather non-simple in practice. Try proving that multiplication is commutative to see the difficulty.

There are plenty of other number systems out there. Try https://en.wikipedia.org/wiki/Surreal_number or https://en.wikipedia.org/wiki/P-adic_number or the complex numbers.

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

Re: 0.999...= 1

#512

Post by SkyBelow »

gspr wrote: post #345
brlewis wrote: post #316

Earlier quoted context omitted.

Why is 0.000... bigger than 0?

It isn't.

Or is it? Say I'm a layman and I decide that in the system of math as I understand it, 0.000... is larger than 0. Yes, if I was going to be completely form with my own system of math I would eventually have to face the problems this introduces and resolve it, but until then I can generally adopt a self contradictory system and continue to live my life unaffected. Much like many people live their whole lives using naive set theory for their understanding of sets.

Re: 0.999...= 1

#513

Post by cthalupa »

ubercow13 wrote: post #488
cthalupa wrote: post #452

Earlier quoted context omitted.

>What if you have 0.9̅4? Well, you fundamentally can't. If the 9s go on for forever then you never reach a point where you can add the 4. The definition of infinity precludes anything after infinity, because it never ends so you can never get there.

That is, until Cantor showed otherwise

Are you saying that Cantor showed you can have 0.9..4?

I'm not even remotely an expert here, so I might certainly be wrong, but I don't understand how Cantor's theorems show an ability to stick a finite number and stop on the end of an infinite number.

Re: 0.999...= 1

#514

Post by [deleted] »

samatman wrote: post #503
barrkel wrote: post #344

Earlier quoted context omitted.

I expect mjd is thinking of irrational bases. The number might still be written as 10, in digits that look decimal.

Thank you! This thread is full of people insisting on something wrong because they were taught incorrectly; an irrational number is defined in terms of integer ratios for a reason. It's not like those people haven't worked with an irrational base before, either! Radians have an irrational base. When we talk about 2π radians, or 1/4π radians, that's exactly what we're doing.

[deleted]

Re: 0.999...= 1

#515

Post by edanm »

I think if you're trying to "prove" this using axioms, you've already lost.

The problem isn't that you can't come up with axioms to convince people you have a proof - the problem is with people not understanding that 0.99999.... is not a number - it's one representation of an abstract entity called a number.

The problem is, the maths required to actually define the concept of a number is fairly complicated, so it's hard to explain to someone why all of these axioms make sense in the first place.

Re: 0.999...= 1

#516

Post by esmi »

tom-thistime wrote: post #272
esmi wrote: post #261

Earlier quoted context omitted.

> I go on to claim, though I'm more unsure of this, that the real numbers are not even the only way to make calculus work. That claim is certainly true. The proof is that calculus (Analysis) exists for the complex number system too. Although I don’t think that’s what you meant and I doubt the complex number system is “more intuitive.” Just out of curiosity, have you heard of real analysis? How do you define calculus?

A Riemann integral in complex analysis uses the same definitions as one in real analysis. Derivatives ditto. In the thread we're comparing real analysis to nonstandard analysis. I'm eager to be corrected if you can tell me something I said that's wrong. I'm not interested in gradually upping the ante with you until it's clear who really has more math background.

I didn’t really see anything wrong in the thread but you use words like “calculus” and I honestly don’t know exactly what you mean. Do you mean the plug and chug methods often taught in high school? Or do you mean the application of a set of theorems derived from the axioms of a given system of numbers? If you mean the former then perhaps we can clear up some misconceptions which are clarified by the latter.

Re: 0.999...= 1

#517

Post by brlewis »

gspr wrote: post #492
brlewis wrote: post #483

Earlier quoted context omitted.

They don't have to know that it's a real number. Knowing that you never stop writing nines is sufficient to perform the calculation.

You're asking them to perform subtraction. They probably know how to do that with real numbers, but problably not with much else. So they'll have to know that they're real numbers (or whatever numbers you are demanding that they be – you're still unclear on this point if it's not actually the reals).

Internally I'm saying they're real numbers. In what I say to the person trying to intuit that 0.999... = 1, I'm deliberately avoiding talking about number systems. I'm assuming this person thinks of numbers as sequences of digits, possibly with a decimal point.

Re: 0.999...= 1

#518

Post by ben509 »

gigatexal wrote: post #498

I hate to say it but I still don't believe this, it just goes against all intuition that I have, but people much smarter than I have proven it so I take it on faith for doing things like calculus etc just my lizard brain won't let me accept something that looks like less than 1 being 1 the same way that the limit of 1/x as x goes to infinity is zero but it doesn't seem like ti should be. The number gets infinitesimal…

Maybe try this: do longhand division of 7 by 7 or 9 by 9 or whatever, but instead of putting 1 in the first place, start with 0.

        0.99
    7 | 7.00000
        0
        7 0
        6 3
          70

Re: 0.999...= 1

#519

Post by austincheney »

mjd wrote: post #280
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.

I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that a…

Proof: bring up removing time zones, such that there is only one time zone.

The responses are comical, even here on HN, such as not being able to wake up or not knowing when morning is or when meals are. Let’s not forget that time zones are a human invention less than 200 years old.

Re: 0.999...= 1

#520

Post by __s »

calibas wrote: post #295
__s wrote: post #287

Earlier quoted context omitted.

I think what's important here is that if you're making that claim, 0.333... = 1/3 - 1/∞ Which implies 3/∞ = 1/∞

Apologies, I should have explained it differently. 0.999... implies a number infinitesimally smaller than 1. You wouldn't use 0.999... in a hyperreal system because you can represent it directly. I shouldn't have mixed different systems and claimed they're mathematically equivalent, you've proven that doesn't work.

Agreed, I think the main lesson from this topic is that decimal numbers are a pain. Stick to integers & symbolic operations. You can spit out a decimal approximation at the end

Computers agree: never trust precision to floats

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