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0.999...= 1

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Re: 0.999...= 1

#391

Post by gspr »

brutt wrote: post #388
gspr wrote: post #376

Earlier quoted context omitted.

And what does the right hand side of that mean? Division is commonly defined for a real numerator and a real, non-zero denominator. You are using the common symbol, but with ∞ in the place of the denominator. Since ∞ is not a real number, you must be using a non-standard definition of division, and have to define what you mean.

In some systems, division by ∞ is not defined at all (forbidden), in other it defined as 0, in another systems it defined as non zero.

> In some systems, division by ∞ is not defined at all (forbidden), in other it defined as 0, in another systems it defined as non zero.

Fine by me. Define whatever notion you're using. You can't just throw out non-standard things and expect people to know what you mean.

Re: 0.999...= 1

#392

Post by JJMcJ »

Usually the concept of a limit, which assigns a meaning to 0.999..., isn't studied until calculus.

There are approaches to mathematics that avoid infinite constructions, and a "strict finitist" would not assign 0.999... a meaning.

The stunning success of limit based mathematics makes finitism a fringe philosophy.

Remember, class, for every epsilon there is a delta.

Re: 0.999...= 1

#393

Post by enedil »

hilbertseries wrote: post #372
TheRealPomax wrote: post #238

Earlier quoted context omitted.

Can you explain what you mean with "real" in that sentence? Because in the context of maths, a real number is "a number in ℝ", which this absolutely qualifies for. Whereas in plain English the term doesn't really have a clear definition. You might consider "real" numbers (in plain English) to mean physically measurable quantities, but there are plenty of numbers that we can write out because they're infinitely long,…

Infinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does not end. The representation has infinitely many sixes and does not end. If there is a 7 in the representation then it must be at a specific decimal. If it was at the end of the representation then it would be a finite decimal representation. The notation is m…

Yes, exactly, .666...7 means a number that has 6 at each decimal position and 7 for index k, where k is greater than any natural number. This exactly is .666... (just as parent commenter explained).

Re: 0.999...= 1

#394

Post by xyzzyz »

da_chicken wrote: post #310
tom-thistime wrote: post #121

Earlier quoted context omitted.

(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.) In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof. But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. T…

The easiest way I know to explain it is fractions. 1 / 3 = 0.33333.... 2 / 3 = 0.66666.... So what's 3 / 3? Some people don't like that one. They might like this one better: 1 / 11 = 0.0909090909... What's 10 times that? 10 * 0.0909090909... = 0.90909090... So, let's do some addition and let the values zipper together because a nine will always line up with a zero: 10 * 0.0909090909... + 0.0909090909... = 0.90909090.…

This is basically how we were taught to convert decimal fractions with periodic decimal expansions to regular fractions in school. You can even do that without that lining up of zeroes and nines, just multiply by the period: if x = 0.090909..., then 100x = 09.090909... (we shift the decimal point by two positions), and since the stuff after the point is the same, after subtracting it cancels out: 100x - x = 9.0, and so 99x = 9, from which we obtain x = 1/11.

Re: 0.999...= 1

#395

Post by throwaway894345 »

barrkel wrote: post #339
throwaway894345 wrote: post #325

Earlier quoted context omitted.

Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since…

No, dates are harder than that. Humans use time to coordinate; the representation of time is fundamentally about communication. Only timestamps of events in the physical world are easy (ish). But that's not always, or perhaps mostly, what people are interested in. When people receive a time, they may (usually) want it in their own time zone, but they might instead want it in the time zone of the entity they're gettin…

You say "no, dates are harder than that", but it sounds like you agree with me exactly based on the rest of your post. Time is hard because human representations of it are varied, complicated, and often arbitrary--not because there's anything fundamentally hard about the math of time (again, relativity notwithstanding). Contrast that with numbers which are inherently difficult to intuit about apart from issues of representation.

Re: 0.999...= 1

#396

Post by laichzeit0 »

LunaSea wrote: post #327
laichzeit0 wrote: post #318

Earlier quoted context omitted.

> We cannot imagine infinity Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number

This is one of my pet peeves in maths. Although I do understand the concepts presented, the notion of "greater" makes no sense when applied to something without boundaries. Yet it's used all the time.

Things quickly fall apart if you rely on your intuition.

Let's imagine all the odd numbers: 1, 3, 5, etc. Now imagine all the even numbers: 2, 4, 6, etc.

Can we agree that there is an "infinite" amount of numbers in each of those groups?

Now imagine all the odd numbers and even numbers together. That's also infinite right?

Would you say there are more "all numbers" than just "all odd numbers", or would you say that there are an equal number of them? (hint: the answer is equal).

Re: 0.999...= 1

#397

Post by jordigh »

lmkg wrote: post #137
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…

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.

Decimals are easier to compare than fractions. I can't easily work out in my head which one is bigger, 457/790 or 580/924, but I can easily see that (approximately) 0.57848 is smaller than (approximately) 0.62771.

Since the fundamental thing most people want to do with numbers is see which one is bigger, they favour decimal expansions. And since decimals worked so well for fractions, why not use them for everything else?

Re: 0.999...= 1

#398

Post by da_chicken »

pwdisswordfish2 wrote: post #306
da_chicken wrote: post #298

Earlier quoted context omitted.

Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly…

There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely. Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?

That's a funny way to say, "No, I think you misunderstand. I mean to say no single infinitesimal number exists. Like infinity, the concept exists, but as a literal single number, no."

Re: 0.999...= 1

#399

Post by brlewis »

gspr wrote: post #389
brlewis wrote: post #377

Earlier quoted context omitted.

You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.

> 1 - 0.999... = 0.000... proves that 0.999... = 1. If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point

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...

Re: 0.999...= 1

#400

Post by klodolph »

hilbertseries wrote: post #374
klodolph wrote: post #219

Earlier quoted context omitted.

You would define it as a digit sequence, using ω + 1 as the indexing set. I would consider it to be most naturally an element of the hyperreal numbers, although it is also contained in smaller extensions of the real numbers.

Sure, you could define some sort of number system that has an infinite point digit. I doubt that’s what the original poster had in mind though.

These number systems already exist, I’m not just making them up.

If someone says 0.666…7 then you have a couple different ways you can take the discussion. You can say, “No! Real numbers don’t work like that!” or you can talk about what number systems would look like if you can do that.

It turns out that there’s a lot to learn from the alternative number systems, including formulations of calculus without limits that are easier to understand from an intuitive perspective, yet equally rigorous. The field is called “nonstandard analysis”. It’s not taught in college.

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