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0.999...= 1

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Re: 0.999...= 1

#221

Post by 2OEH8eoCRo0 »

gspr wrote: post #162
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…

> 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…

Ask for a number between .9 repeated and 1

Re: 0.999...= 1

#222

Post by ramshorns »

tom-thistime wrote: post #140
Scarblac wrote: post #131

Earlier quoted context omitted.

A number is just a number, it doesn't approach anything. A series can approach something, but a number can't. In any system where 0.99... is valid notation for a number, it doesn't approach anything.

Sure. Instead of "0.99..." please substitute lim n->inf sum(1..n)(9 times 10^-n). The point I'm making is that the "obvious truth" 0.99... = 1 that we're all talking about depends on the assumption that we're working in the real numbers. I claim that the real numbers are not something intuitively obvious to every sufficiently intelligent person; instead they are kind of weird and technical. I go on to claim, though I…

Well, a limit of a series is just a number and doesn't approach anything either. If the series approaches something, we say the limit exists and is equal to that thing.

Anyway, in the surreal numbers you could probably make up a notation where 0.999... actually denotes 1 - ε or something. But I daresay it might not be very useful because then how do you denote 1 - ε/2 or anything else.

Re: 0.999...= 1

#223

Post by 2OEH8eoCRo0 »

solotronics wrote: post #191
TomGullen wrote: post #167

Earlier quoted context omitted.

I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1

black magic! I wonder if there are any programming languages that are able to handle this properly?

https://www.wolframalpha.com/input/?i=.9...+%3D+1

Re: 0.999...= 1

#224

Post by tom-thistime »

0xff00ffee wrote: post #203
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…

Yep. Agree 100%. It is like the blue dress. I think the problem is the repeating function. Infinite things are non-intuitive and should be presented differently. Even here on HN you still see people confused about "convergence" and "identity". 0.999... doesn't CONVERGE, it literally is 1. I suspect this persists even with students that have had second year college calculus that discusses convergent series and sums.

Fine. Define 0.999... as the limit of the series sum(n=1 ... N)(10^-n), as N-> infinity. This is standard high school calculus. "Number" and "series" and "limit" and "convergence" don't all mean the same thing. However this number is defined as the limit of a convergent series. So the question really is meaningful. (One clue that this question is meaningful is the amount of space introductory calculus textbooks use to address it.)

Because I can still ask, in black and white, what law of "equality" do I use to establish that my limit equals 1? (It does, if I import the definition of "equality" from the real numbers. That's what they do in calculus class. )

Re: 0.999...= 1

#225

Post by cesarb »

ginko wrote: post #9

I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious: 1/3 = 0.333.. 3 * 1/3 = 3 * 0.333.. 3/3 = 0.999.. 1 = 0.999..

I personally like using fractions of 9.

  1/9 = 0.111...
  2/9 = 0.222...
  3/9 = 0.333...
  ...
  8/9 = 0.888...
  9/9 = 0.999...
What's neat is that this trick works for any repeating decimal, with any number of digits in the repeating part. For instance:

  123/999 = 0.123123123...
  999/999 = 0.999999999...
Multiply or divide by powers of 10 as necessary to shift the decimal point, and add the non-repeating part.

Once you accept this mapping, it's trivial to treat 0.999... as 9/9 (or 99/99, or 999/999, etc). Which can be simplified to 1.

Re: 0.999...= 1

#226

Post by ubercow13 »

dahart wrote: post #210
CogitoCogito wrote: post #193

Earlier quoted context omitted.

> You can. The proof still works if you do that. > What you’re refusing to accept here is the definition of infinity. I'm refusing to accept the definition of infinity? I have no idea what you mean by that. Would you make the same statement were you aware that I do in fact have a PhD in mathematics in the field of analysis? That I have in fact studied logic? Just as a hypothetical scenario.

The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatically…

>if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right

>0.9bar7 is a completely nonsensical number

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. It's like you're trying to perform transfinite induction on the set of numbers generated by moving the decimal point. You can only move the point a countable number of times, so there is no sense in which the property can still be true after infinity many times.

Re: 0.999...= 1

#227

Post by L_226 »

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.

Reminds me of Terry Pratchett's description of the mathematical reasoning of camels in Pyramids:

Lack of fingers was another big spur to the development of camel intellect. Human mathematical development had always been held back by everyone’s instinctive tendency, when faced with something really complex in the way of triform polynomials or parametric differentials, to count fingers. Camels started from the word go by counting numbers.

Re: 0.999...= 1

#228

Post by lukifer »

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…

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear.

This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly paradoxical, and fascinatingly debatable in a "is a hot dog a sandwich" sort of way. But reframe it as a process (or as code), and all confusion disappears: for how many loops would you like to proceed? If you never stop, 0.99999... clearly approaches 1, without ever reaching it, and asking if they're the "same" is as academic as asking if the Ship of Theseus is the same ship, or if an electron is the same entity from one picosecond to the next.

Re: 0.999...= 1

#229

Post by taylodl »

TomGullen wrote: post #167
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…

I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1

That's how I was taught in my Algebra class

Re: 0.999...= 1

#230

Post by 2OEH8eoCRo0 »

tomtomtom1 wrote: post #68
knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

if we say that infinitesimals exist. that 1/3 != 0.33.. and 1 != 0.9999... and the probability of possible events is never 0. what are the properties that we would lose?

1/3 does equal 0.3.. though.
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