[split] 0.999... equals 1

[Retorth]

Good man! It is indeed a friggin awesome math phenomenon.

[Meatball]

You can also think of it as a side effect of the system we use to represent numbers.

1/10, though very simple in decimal(base10) notation, is infinitely long when represented in binary(base2).

.1(decimal) = 1/10 (decimal) = 0.00110011...(binary)
or
0.333...(base10) = 1/3 (base10) = 0.1 (base3)

Also,

In base2, 0.111... = 1.
In base3, 0.222... = 1.
In base16, 0.fff... = 1.

Sorta unrelated, but I suggest looking into other number systems. It's really fascinating and will change the way you think about math. For example, we all know that computers generally use binary(or hexadecimal as a means of representing binary). Did you know that the Mayans used a base20 counting system? Or that the Yuki tribe of Native Americans used a base8 number system, because they counted using the spaces between their fingers, rather than the fingers themselves, which one might conclude is the basis of our decimal system?

It's really fascinating stuff, even if you aren't a math geek.

[Edwardo Piet]

I am right in thinking it is a tautology yeah? The "proving" is just about explaining why it's true by definition. It's just another way to represent '1', so it's more of an explanation of how it is exactly the same thing than a "proof", because it's true by definition . Right?

EvF

[Tiberius]

The proofs are real math proofs, using logic and algebra usually. It just proves that two different numbers can be the same, which is an assumption most people don't make when considering the decimal system.

[Edwardo Piet]

Yes, they are different ways of representing the same amount right? That's what I meant, 0.9r is just another way to represent '1', the mathematical "proofs" are a way of explaining it?

Or am I missing something then?

EvF

[Retorth]

It's just hard to digest initially, but once you grasp the concept and understand the true nature of infinity, it becomes much clearer.

[Tiberius]

Well, 1 and 0.9r are different representations, but they have the same numerical value. Just as you can have a mathematical proof that 3/3 = 2/2, we have mathematical proofs that 1 = 0.9r.

[Retorth]

I rather enjoy explaining it with

1/3 = 0.3r


Very clear and concise imo. Tried showing a few friends today and they were all intrigued.

[Edwardo Piet]

Well, 1 and 0.9r are different representations, but they have the same numerical value. Just as you can have a mathematical proof that 3/3 = 2/2, we have mathematical proofs that 1 = 0.9r.

Well that's what I mean, yeah. I just find it funny thinking of it as proof, because 2/2 is 3/3 by definition for example - so it seems more like explaining a tautology to me, that it actually being something to be proven. 2/2 is another way of representing the exact same number as 3/3, 1, or 0.9r, right?

Anyway, I understand. I understand that that's what mathematical proofs are, I just don't see them as stuff needed to be proven (at least not the examples above anyway), I see them more as tautologies needed to be explained to those who are ignorant of it.

@Retorth, I definitely like the continuity proof best, and I think all the other proofs become clear once you grasp that one. I love that.

EvF

[Violet]

http://en.wikipedia.org/wiki/Limit_(mathematics) I'm posting this link because Adrian seems to think that the entire mathematical community is behind .9^ = 1... when Infinite Limits is an established mathematical concept. Infinite Limits is directly contradictory to .9^ = 1. An infinite limit is to state that infinite .9's will infinitely get closer to 1, but infinitely never reach it.

Ever heard of calculus? http://en.wikipedia.org/wiki/Calculus the first thing you will learn about in Calculus is limits. One of the earliest subjects you will be taught is infinite limits.

You can state the 'proofs' of http://en.wikipedia.org/wiki/0.999... but the fundamental fact of http://en.wikipedia.org/wiki/Infinity (that it is endless) means that an endless number of .9's will infinitely never reach one. I have stated this since the beginning of this thread... and this boils down to this question: Is calculus (specifically limits to a function) a fallacy? Or is the concept of infinite .9's=1 a fallacy?

If all of these 'mathematical proofs' of 1=.9^ are true, then you have disproved modern calculus