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0.999...= 1

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Re: 0.999...= 1

#521

Post by ubercow13 »

cthalupa wrote: post #513
ubercow13 wrote: post #488

Earlier quoted context omitted.

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.

Yes, one thing Cantor showed is that it is somehow meaningful to define numbers like infinity+1, where there are an ordered infinity of elements followed by one more element. Sets like this are called ordinals

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

So you could, if you like, define 0.9...4 to be a bunch of digits indexed by the ordinal ω+1. However the thing you have now defined isn't really a representation of a real number any more, unless you just ignore all the bits after the ... I guess.

Re: 0.999...= 1

#522

Post by tom-thistime »

DavidVoid wrote: post #505
tom-thistime wrote: post #485

Earlier quoted context omitted.

Ok. (a) I think you're saying that 0.00 ... 1 is not a real number. (b) Do we agree that lim n->inf 10^-n is a real number? (I.e. zero?) (c) Then you're saying that limit in "(b)" is not a reasonable definition of the string "0.00 ... 1". Is that correct?

That is correct. The limit of lim n->inf 10^-n is exactly 0, it is not 0.00...1.

Since we have no common definition of what 0.00...1 might possibly mean, let's say we agree.

Re: 0.999...= 1

#523

Post by bmn__ »

austincheney wrote: post #519
mjd wrote: post #280

Earlier quoted context omitted.

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.

> Let’s not forget that time zones are a human invention less than 200 years old.

What's the point of that statement? Before the introduction of formal time zones, we had thousands of informal ones, one for each settlement, calibrating noon to the zenith of the sun.

Re: 0.999...= 1

#524

Post by drran »

gspr wrote: post #470
brutt wrote: post #432

Earlier quoted context omitted.

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

You're working with surreal numbers? This is not what people would expect unless it's explicitly stated. In addition, you're likely going to have a hard time explaining surreal numbers to someone who struggles to grasp that 0.999... = 1 in the ordinary reals.

Surreal numbers and infinistemal are simpler to work with when you need to work with infinite series.

Here John Conway explains them: https://www.youtube.com/watch?v=1eAmxgINXrE

Re: 0.999...= 1

#525

Post by 0xff00ffee »

tom-thistime wrote: post #468
0xff00ffee wrote: post #444

Earlier quoted context omitted.

You asked for a "law" of equality (whatever that is) and provided an answer that proves it converges to 1. What more could you possibly want?

We seem to agree on this: you don't think there's any need for a way to determine whether two real numbers are equal. For ordinary math, though, using some criterion for equality (for example x>=y and y>=x) is basic and not controversial. So it seems unconvincing (to me) when you seem to imply the opposite.

> you don't think there's any need for a way to determine whether two real numbers are equal.

You are putting words in my mouth.

And you clearly do not understand the answer.

I guess I'm not very good at ELI5 because I very clearly answered your question with your own proposal.

Maybe when you get to college a professor can do a better job explaining it to you (if you actually make it to college, because you're going to struggle very hard if that's how you think when an answer is spoon-fed to you).

Re: 0.999...= 1

#526

Post by drran »

brutt wrote: post #363
brlewis wrote: post #341

Earlier quoted context omitted.

You're repeating the same wrong thing you said earlier. It's 0.999... and not 0.999...0 In the same way, it's 0.000... and not 0.000...1.

0.999... = 0.999...9 0.999...9 + 0.000...1 = 1 0.999...0 + 0.000..1 = 0.999..1 0.000...1 = 1/∞ 0.999...9 = 1 - 1/∞ 0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞ If x/∞ = 0, then 0.999...x = 1. If x/∞ ≠ 0, then 0.999...x ≠ 1.

Nice notation. I will steal it.

Re: 0.999...= 1

#527

Post by ISO-morphism »

austincheney wrote: post #519
mjd wrote: post #280

Earlier quoted context omitted.

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.

200 years ago we didn't need timezones because we couldn't communicate or travel fast enough or often enough for them to matter. With railroads and telegraphs the level of granularity required for "some time 300 miles away" went from days to hours and minutes. No one in the U.S. west wanted to be told "The sun is at its zenith around 8 a.m. for you." For most people talking about time in their day to day lives it's far more useful to communicate a relative time of day with people near them than to communicate an absolute moment in time and do the math to figure out how bright it is outside.

Re: 0.999...= 1

#528

Post by jturpin »

dvt wrote: post #429
supercasio wrote: post #419

Earlier quoted context omitted.

What? 0.999... = 1 is not dogma. Please don't spread misinformation. And at least read the link before commenting on something.

Did you read what @jl2718 posted? Namely: > the supremum of an increasing sequence is equal to the limit -- this is not misinformation (and to anyone familiar with some introductory analysis, correct[1]). Of course, calling it "dogma" is a bit inflammatory, but not technically wrong. It's kind of of a made-up rule to help us work with infinities (particularly in ℝ -- but it happens all the time in set theory, as well…

I do think it is technically wrong to call it dogma - the decimal is a geometric series with a limit, right? And limits have an unambiguous definition, it's the smallest value that the series approaches but never exceeds as it tends to infinity. I think the part that is admittedly weird is that the notation "0.999..." refers to the limit as the series tends to infinity, and it kind of hides that fact from you. Even just writing the geometric series down and plopping "infinity" as the value for x would be wrong, as it's the limit that is equal to 1 as x tends to infinity. So there's arguably more hidden notation than the ellipses implies, but nothing is pulled out of a hat here or defined for definitions sake.

Re: 0.999...= 1

#529

Post by drran »

brutt wrote: post #443
aliceryhl wrote: post #439

Earlier quoted context omitted.

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.

https://en.wikipedia.org/wiki/Surreal_number Quote: There are also representations like { 0, 1, 2, 3, … | } = ω { 0 | 1, 1/2, 1/4, 1/8, … } = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numb…

I recommend watching this lecture about surreal numbers by John Conway: https://www.youtube.com/watch?v=1eAmxgINXrE .

Re: 0.999...= 1

#530

Post by hansvm »

brlewis wrote: post #408
theincredulousk wrote: post #308

Earlier quoted context omitted.

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…

Whatever base you use, it will only be useful for rationals. You can't use any base to significantly improve the representation of pi. In base 7, 3.1 would be a nice approximation, but you can't go beyond approximations.

It doesn't have a terrible representation in base pi ;)
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