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0.999...= 1

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Re: 0.999...= 1

#561

Post by karatestomp »

pasquinelli wrote: post #356
tom-thistime wrote: post #190

Earlier quoted context omitted.

Possibly people are looking at two different symbols and asking "can you show me logically why those are equal." If they're given a definition of "equal" and they still object, that's a different problem. I have this problem every time I play with group theory again. You get the axioms for a group, which say there is some identity but don't explicity require the identity to be unique. You can easily prove that the id…

I think a lot of people don't think of math in terms of definitions and proof. Math was just something they were taught as kids. And even if they've gotten into more advanced math, i think the 1 = .9... question hits their kindergarten brain and they just say "no" to it the same way they'd say "no" to someone singing the alphabet song in the wrong order.

I think a lot of people see “find me a number between 0.9999... and 1” as no more or less valid (plus, perhaps, no more or less pedantic and dickish) than “ok smarty pants, why don’t you just keep adding 9s until you reach 1, then we’ll talk.”

Re: 0.999...= 1

#562

Post by jdashg »

gowld wrote: post #548
naikrovek wrote: post #541

Earlier quoted context omitted.

I don't understand how they're the same number. I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.

What is an "infinitely small" number? Is 9999..... the same as infinity? What is 1.0 - 0.99999.... = ? What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?

Those are great questions that not every system is required to address in the same way. (In a similar vein, +0 != -0 in Java) This is breakdown in notation and/or convention. There is no ground truth, just what's true within the system.

Re: 0.999...= 1

#563

Post by mjd »

blauditore wrote: post #311
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…

>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?

It's _true_, but it's not good as a definition, because it's hard to reason about. It drags in all sorts of contingent facts about base-10 representations that are not usually of interest.

The equivalent property, that a number is irrational if it's not equal to m÷n for any integers m and n, is much simpler. So we use that as the definition, and from that simple and intrinsic definition, we prove the _theorem_ that the decimal representation of an irrational number never repeats.

Re: 0.999...= 1

#564

Post by mjd »

barrkel wrote: post #344
tom-thistime wrote: post #326

Earlier quoted context omitted.

Yeah, can we have an example of an irrational number whose decimal representation repeats or terminates? Or a rational number whose decimal representation doesn't repeat?

I expect mjd is thinking of irrational bases. The number might still be written as 10, in digits that look decimal.

That is not what I meant at all. (My phrasing was unclear. Sorry for the confusion.) Jordi understood what I meant though.

Re: 0.999...= 1

#565

Post by mjd »

tom-thistime wrote: post #326
blauditore wrote: post #311

Earlier quoted context omitted.

>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?

Yeah, can we have an example of an irrational number whose decimal representation repeats or terminates? Or a rational number whose decimal representation doesn't repeat?

My phrasing was bad. I should have said "the mistaken belief that an irrational number is * defined to be * one whose decimal representation doesn't repeat”.

Usually we define it like this: an irrational number is one that isn't a quotient of two integers. Starting from that definition, we then prove the _theorem_ that the decimal representation a number repeats if and only if the number is rational.

It's much easier to start from the intrinsic properties, and use those to prove things about the representation, than the other way around. But if you don't distinguish the representation from the thing itself, you can't tell which way you are going.

Re: 0.999...= 1

#567

Post by jdashg »

I think this is a notation and definition problem. To me, it behaves differently in `Y = 1 / X`, which distinguishes quite strongly between `X = 1 - 0.9999` and `X = 0.9999 - 1`! If 0.9999 ought be exactly equivalent to 1.0, there ought to be no difference between `1 / (1 - 0.9999)` and `1 / (0.9999 - 1)`.

To me, 0.9999 indicates a directional limit, which can't necessarily be evaluated and substituted separately from its context.

Re: 0.999...= 1

#568

Post by mjd »

btilly wrote: post #494
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…

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…

My phrasing was poor. See https://news.ycombinator.com/item?id=23013090

Re: 0.999...= 1

#570

Post by grawprog »

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'm willing to believe, but every proof on that page I read came down to basically, it might as well be 1, so it is 1. The way i see it, it comes down to accuracy like any of our measurements and falls under rounding error. There's no way we can ever actually measure the infinite amount of space between .999... and 1 so effectively they're the same. As far as math and anything practical and even theoretical is concerned, they're the same, but...conceptually in my brain it just feels that little bit smaller. I know I'm wrong for all intents and purposes, but I dunno, that.
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