0.999...= 1 • Hacker News Board

Re: 0.999...= 1

#311

Post by blauditore » Tue, Apr 28, 2020, 4:33 PM UTC

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…

>the mistaken belief that an irrational number is one whose decimal representation doesn't repeat

...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?

Re: 0.999...= 1

#312

Post by greenshackle2 » Tue, Apr 28, 2020, 4:33 PM UTC

_bxg1 wrote: post #204

Okx wrote: post #182

Earlier quoted context omitted.

I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

Why not 0.00...1?

What is 0.00...1 times 34?

Re: 0.999...= 1

#313

Post by alpple » Tue, Apr 28, 2020, 4:34 PM UTC

Why do we accept .999... as a valid notation. Why not only allow 1 to denote this concept?

Re: 0.999...= 1

#314

Post by tom-thistime » Tue, Apr 28, 2020, 4:35 PM UTC

CogitoCogito wrote: post #293

tom-thistime wrote: post #282

Earlier quoted context omitted.

Strictly speaking I brought up the surreal numbers.

Then I guess you're the smart one. My apologies.

Sorry, sorry. I've been trying to bring up nonstandard analysis, and repeatedly getting poked by people saying "what do you think 'calculus' is?" "have you ever heard of a limit?" and so forth. In the process I have apparently become even more ornery than usual.

Re: 0.999...= 1

#315

Post by shrimpx » Tue, Apr 28, 2020, 4:36 PM UTC

A dumb consequence of the axiom of choice? The reals are like a membrane with no atomic pieces. You can move in either direction infinitely and you can zoom in infinitely without reaching any “Planck unit” so to speak. So what does it even mean to pick out a “real number”? To me anything built on this concept is nonsense.

Re: 0.999...= 1

#316

Post by brlewis » Tue, Apr 28, 2020, 4:36 PM UTC

traderjane wrote: post #259

2OEH8eoCRo0 wrote: post #221

Earlier quoted context omitted.

Ask for a number between .9 repeated and 1

It's the smallest number bigger than 0.

Why is 0.000... bigger than 0?

Re: 0.999...= 1

#317

Post by brlewis » Tue, Apr 28, 2020, 4:37 PM UTC

brutt wrote: post #300

2OEH8eoCRo0 wrote: post #221

Earlier quoted context omitted.

Ask for a number between .9 repeated and 1

0.00...1

No, there's no 1. 2OEH8eoCRo0 is exactly right. Subtract 0.999... from 1 and you get 0.000...

Re: 0.999...= 1

#318

Post by laichzeit0 » Tue, Apr 28, 2020, 4:37 PM UTC

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…

> We cannot imagine infinity

Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number

Re: 0.999...= 1

#319

Post by smitop » Tue, Apr 28, 2020, 4:38 PM UTC

missedthecue wrote: post #299

parski wrote: post #214

Earlier quoted context omitted.

I asked my math teacher this when I was a kid. He told me to accept that's the way it is so I did.

Kinda like why 5 rounds up instead of down

When 5 rounds up, you can easily implement rounding as floor(x + 0.5).

Re: 0.999...= 1

#320

Post by astrobe_ » Tue, Apr 28, 2020, 4:39 PM UTC

heinrichhartman wrote: post #302

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, ...) -- lis…

1 = 0.9... is the consequence of purposely ambiguous and questionable notation. That's an old teachers' trick to make students talk and listen about mathematics.