Print

Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

[Graknorke]

It's not helped by the fact we aren't good at imagining an infinity.

I can though...

Would you guys rather I use the notation 0.000...1   
(for an endless number of zeroes before the one)

How can you have a one at the end if the zeroes are endless?

[Vorthon]

Everything that came out of my stepdad's mouth today. I'm surprised I'm not permanently walleyed from over-exposure to stupid.

[Valid_Dark]

How can you have a one at the end if the zeroes are endless?

why couldn't you.  It's to denote a number smaller than imaginable.
How can somebody say there is an infinite number of numbers between 1 and 2 without there being a number infinantly small.
It has other uses, but they're few and far between.

[Darvi]

How can you have a one at the end if the zeroes are endless?

why couldn't you.

Because there's no end on which to append the 1?

[Slayerhero90]

How can you have a one at the end if the zeroes are endless?

why couldn't you.

Because there's no end on which to append the 1?

Without the one, it would be exactly zero. There may not be a real end, but the one dictates that it is one of the infinitely smallest unit away from 0.

[Darvi]

See, this is why we invented calculus. Unlike reality, maths doesn't have an infinitely smallest unit.

[miauw62]

And wouldn't -1 be smaller than 0.00...1?

[PanH]

It's not really that infinitesimals are ininteresting, but well ...

[USEC_OFFICER]

And wouldn't -1 be smaller than 0.00...1?

0.00...1 is supposed to be the theoretical smallest unit possible in math. Which I find a bit silly, but I don't know enough to determine it's existence or not. So yes, -1 < 0.00...1, but that's not really what it's supposed to be about.

[MagmaMcFry]

And wouldn't -1 be smaller than 0.00...1?

0.00...1 is supposed to be the theoretical smallest unit possible in math. Which I find a bit silly, but I don't know enough to determine it's existence or not. So yes, -1 < 0.00...1, but that's not really what it's supposed to be about.

Actually, from the very definition of an ordered field it follows that it has no smallest positive number. That's because one of the prerequisites of an ordered field F is that for every element a, b and c of F with a < b and positive c, the inequality ac < bc holds. So let's assume there is a smallest positive number p. If we let a := 1, b := 2 and c := p * 1/2, we get p * 1/2 < p, and since another prerequisite states that the product of two positive numbers is again positive, we get 0 < p*1/2. This shows that p/2 is a smaller positive number than p, which is a contradiction to the definition of p, therefore the assumption we made (namely that a smallest positive number exists) is wrong.

[USEC_OFFICER]

So my suspicious were correct then. Good to know.

[andrea]

my favorite way to think about 0.999... = 1 is perhaps the one that involves least numbers ( although it relies on definitions and/or other proof).

Real numbers are dense, which means that for any A and B (A<B) there is a C such that A<C<B.
there is no such C between 0.999... and 1 , therefore they are the same number.
even the "smallest number"0.0000...01 imagined by previous posters, if added to 0.999... would make 1. which means that there is no C between A and B, therefore they are the same number, and 0.00...1 is equal to 0.

or else(same thing, said in other way): the real numbers can be seen as the points of a line. between any 2 points A and B of a line there always is a point C. and it continues as above.

of course, you then have to prove that real numbers are dense. but at least for me, this thinking helps accepting the equation.

It should be noted that if you are willing to move away from reals,this doesn't apply anymore.
according to wikipedia, for example, in hyperreal numbers, there are non-zero infinitesimals and 0.999... is not equal to 1

[Dutchling]

But couldn't you just continue doing that until you 'prove' 1= 0?

I mean, between (0.99...... - 0.00...01) and 1 is nothing either, since 0.99.... equals 1 and can thus can not be in-between them etc.etc.etc.

[andrea]

that would be because 0.9999-0.00...1 IS 0.999... , by the above reasoning. therefore it is neither strange nor worrying that 0.999...- 0.00..1 is equal to 1.

the point is that in real number, 0.00... 1 is 0 ( can't really find anything between 0.00...1 and 0 either). so any crazy sheningans involving it are void.

But there are better proofs in this thread and on the internet. This is just a way to try to make sense of it.

[penguinofhonor]

I just assume 0.999... is identical to 1 about 99.999...% of the time.