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0.999...= 1

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Re: 0.999...= 1

#491

Post by schoen »

bcrosby95 wrote: post #457
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…

Yet the way we represent numbers is also a human construct. 1 = .999... is hard for people to understand because we think and write in base 10 rather than base 3. There's nothing that is fundamentally hard to reason about here.

Of course in base three, they'll have a hard time with .222...

Re: 0.999...= 1

#492

Post by gspr »

brlewis wrote: post #483
gspr wrote: post #477

Earlier quoted context omitted.

> They have to know that 0.999... means you never stop writing nines. How do they know that that's a real number?

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).

Re: 0.999...= 1

#493

Post by erodommoc »

petters wrote: post #381
LunaSea wrote: post #327

Earlier quoted context omitted.

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.

Personally, I think it makes perfect sense. Take two sets A and B. If we can assign every element in A to a different one in B, we say that |A|≤|B|. Makes perfect sense for normal, finite sets, right? As it happens, this definition extends to infinite sets as well.

Right, but many things make sense for finite sets that don't make sense for infinite sets. Just because you can extend that definition doesn't mean that it's "true" for infinite sets.

Re: 0.999...= 1

#494

Post by btilly »

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…

For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that an irrational number is one whose decimal representation doesn't repeat.

How is this a mistaken belief?

Every rational number winds up in a repeating decimal representation and every number with a repeating decimal representation is a rational number. We learn algorithms to go back and forth between the two in elementary school.

Therefore irrational numbers cannot have repeating decimal representations. Conversely numbers with decimal representations that don't wind up repeating cannot be rational and so must be irrational.

Re: 0.999...= 1

#495

Post by schoen »

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…

Apparently the New Math (https://en.wikipedia.org/wiki/New_Math) tried to address this kind of issue quite explicitly, by drawing a consistent distinction between numbers and numerals (where a numeral is a symbol that names a number). Reportedly most American math students found this kind of distinction extremely hard to grasp when they were presented with this kind of issue in elementary school. Maybe it would have worked better when they were a bit older.

I wonder if there's a way of teaching this kind of distinction and issue well in a way that would make sense for most students.

I think Feynman said somewhere that the New Math explicitly taught base representation and base conversions, probably as a way of trying to underscore the idea that "123" is a representation of a number rather than a number. Feynman found this to be of questionable value and thought that most students didn't manage to get the point.

Edit: there's a similar issue in linguistics because you have words, phonemes, phones, graphemes, and glyphs. You could say that "dog" isn't a word, but is rather the standard way of writing a particular word in the standard writing system for English (which would sometimes be indicated by in linguistic contexts). This idea lets you refer to and as ways of writing the same word, or and , or in the case of languages with multiple writing systems and , or and .

Re: 0.999...= 1

#496

Post by gspr »

crispyambulance wrote: post #354
gspr wrote: post #162

Earlier quoted context omitted.

> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thi…

> It is hard to have the patience to chase down the consequences of their ill-fated definitions, though. Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility. Whenever one sees a 0.999[... to however many digits] one can safely assume it's less than one or perhaps more realistically "almost 1". To say 0.999…

> Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility.

Oh absolutely. I'm not expecting STE(no M this time!) practitioners to necessarily be aware of why 0.999…=1 in their daily lives, but I do expect them to have encountered enough situations in their field of expertise where scraping the surface using shallow intuition and gut feeling lead them wildly astray. I'm therefore surprised that they're willing to deny this basic fact to the face of mathematicians. The ones I've interacted with also don't happen to be the types that'll start arguing Anatomy 101 facts with a heart surgeon at a bar, but somehow arguing over basic calculus with mathematicians is fine.

Re: 0.999...= 1

#497

Post by ubercow13 »

ubercow13 wrote: post #292
dahart wrote: post #260

Earlier quoted context omitted.

> For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. Yes you absolutely can, for exactly the same reason. 0.9bar7 is nonsensical precisely because you can move the decimal to the right an infinite number of times, and still have an infinite number of 9s befor…

In your proof you say >And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right This is not how induction works. The induction shows that you can shift the decimal point any finite number of steps to the right and there will still be infinite 9's after it. If you want to show something is still true after infin…

>because decimal representations only have countably many digits

This should rather say that it's because decimal representations have digits indexed by the natural numbers I guess, rather than by any larger countable ordinal

Re: 0.999...= 1

#498

Post by gigatexal »

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 infinitesimally small but it's still some non-zero number -- I dunno this is probably proving my ignorance it's just what it is.

Re: 0.999...= 1

#499

Post by erodommoc »

jl2718 wrote: post #402

The proof relies on the assertion that the supremum of an increasing sequence is equal to the limit. This is mathematical dogma, and should be introduced as such. Once that is accepted, it becomes obvious. This is illustrative of what I see as a fundamental problem in mathematics education: nobody ever teaches the rules. In this case, the rules of simple arithmetic hit a dead end for mathematicians, so they invented…

And this I think is the real issue. When someone says that 0.999... = 1.0, what they are saying is that this is true given a number of assumptions that we are taking for granted that would not be obvious to a non-mathematician. There's a lot of math hiding in those '...'.

Re: 0.999...= 1

#500

Post by andrewprock »

brlewis wrote: post #471
burmer wrote: post #450

Earlier quoted context omitted.

How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.

Apples and oranges. For any two different real numbers, there's a number between them. Integers work differently.

This is a bold assertion, and one that is not obviously true, especially in cases like 0.999... and 1.0
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