[split] 0.999... equals 1

[leo-rcc]

Merged the threads.

[edk]

It is accepted because the definition of numbers requires a divisible space between each. Because there is no divisible space between 0.999... and 1, it is accepted as 1. The proofs, however, are all wrong. But we do not know what else to do with this number without shaking the foundations mathematics.

The proofs are correct. Any repeating decimal number is merely a representation of an exact rational number.

[IATIA]

It is accepted because the definition of numbers requires a divisible space between each. Because there is no divisible space between 0.999... and 1, it is accepted as 1. The proofs, however, are all wrong. But we do not know what else to do with this number without shaking the foundations mathematics.

The proofs are correct. Any repeating decimal number is merely a representation of an exact rational number.

You cannot use infinities in algebra and a repeating decimal falls into the infinity category. In calculus infinities are calculated as limits. As in "approaching 1", but does not equal one. Infinities are strange and do not conform to our mathematics. A piece of infinity is still infinity. Two infinities are not twice as big as one infinity. One cannot calculate an infinite progression because that would take an infinite amount of time, but through calculus, we can calculate the limit. Like I said though, it is accepted as 1 because the definition of numbers states that between any two real numbers there is another real number. So technically, 0.999... is not even a real number. (then again, maybe 1 is the unreal number )

[LastPoet]

You cannot use infinities in algebra and a repeating decimal falls into the infinity category.

False, you can use infinities in algebra under certain conditions, but it doesn't matter, but we're talking anout DECIMAL representation of a real number. 1/3 and 0.333... are the same Real number and you can perform operations on both.the number 0.2 is considered in the full decimal form to be 0.20000000... , but of course for practical reasond we don't do that!

A piece of infinity is still infinity.

Really? the Reals Set is infinite set and I can grab both finite and infinite pieces of it, eg. thr set {1,2,3,4,5} (finite) and the Rational Set number(infinite).

[Tiberius]

True, you cannot use infinite values in algebra, but we aren't. 0.999... isn't infinite, it is infinitely long. There is a big difference. Technically, all values can be represented as infinitely long:

1 = 1.000...
3.64 = 3.64000...

Etc.

There are various proofs that 0.999... = 1, not all of them are algebraic.

[edk]

You cannot use infinities in algebra and a repeating decimal falls into the infinity category. In calculus infinities are calculated as limits. As in "approaching 1", but does not equal one. Infinities are strange and do not conform to our mathematics. A piece of infinity is still infinity. Two infinities are not twice as big as one infinity. One cannot calculate an infinite progression because that would take an infinite amount of time, but through calculus, we can calculate the limit. Like I said though, it is accepted as 1 because the definition of numbers states that between any two real numbers there is another real number. So technically, 0.999... is not even a real number. (then again, maybe 1 is the unreal number )

1. It's not infinite, it just has an infinite number of digits.
2. Algebra does deal with infinite things sometimes. Finding the sum of infinite series, for example.
3. 0.999... is just a representation of a number.

I'll try to demonstrate it to you with a different rational number:

The decimal representation of 22/7 is infinitely long. It looks like this:
3.(142857)...

The number one million times larger than that is 3142857.(142857)...

The number 999999 times larger than 22/7 is 3142857.142857... - 3.142857... = 3142854 exactly. 3142854/999999 = 22/7.

[Tiberius]

A non-algebraic proof:



So, please explain how this proof is "wrong".

[fr0d0]

Here's my dick...

[IATIA]

A non-algebraic proof:



So, please explain how this proof is "wrong".

That is simply a limit. The problem still goes back to any infinite number or series. Infinity is basically a concept and not a real number.

1/0.999...=1.00...100...100...1
That answer is not 'allowed' unless 0.999... has a finite end as:

1/0.999=1.001001001...
1/0.9999=1.000100010001...

We are only 'allowed' to repeat a finite sequence.

The rules change whenever infinity is involved. There would be an infinite number of zeros before each 1 which means the first 1 would never be reached. Again, it is easier to accept 0.999... as 1 than to redefine our mathematics involving infinities. A WHOLE lot easier.

[Tiberius]

That is simply a limit.

No it isn't. The series converges to a real number, due to the completeness property of real numbers. For more information, see: http://en.wikipedia.org/wiki/Infinite_se...ent_series

The problem still goes back to any infinite number or series. Infinity is basically a concept and not a real number.

Yes, infinity is a concept; nobody is arguing against that. However, it is a concept that mathematics allows, and has studied at great length. Infinity doesn't need to be a real number in order to exist in mathematics; there are infinitely large sets of numbers (the natural numbers are countably infinite, and the real numbers are uncountably infinite) for example.

1/0.999...=1.00...100...100...1
That answer is not 'allowed' unless 0.999... has a finite end as:

1/0.999=1.001001001...
1/0.9999=1.000100010001...

Your answer is "not allowed" because it is demonstrably false. The calculation 1/0.999... ends up with a number with only one '1' in it; namely, 1.

We are only 'allowed' to repeat a finite sequence.

Agreed, but there is a finite repeating "sequence" in 0.999... it's the number '9' repeating over and over. With 1/0.999... the answer (1) has repeating 0's (1.000...), like most numbers do.

The rules change whenever infinity is involved.

Infinity isn't involved here though.

There would be an infinite number of zeros before each 1 which means the first 1 would never be reached.

NO! This is such an easy concept to grasp! You cannot have an infinite number of zeros "before" a 1...or ANY number. An infinitely long sequence of numbers by definition cannot have anything following it. The concept of 1.000...1 is completely invalid in mathematics; it is like trying to draw a square circle.

Again, it is easier to accept 0.999... as 1 than to redefine our mathematics involving infinities. A WHOLE lot easier.

There is no need to just "accept" 0.999... as 1, as long as you understand the mathematics behind infinities, which you clearly don't. There are various rigorous proofs of the concepts and calculations involving infinitely long numbers, and involving the concept of infinity itself. If you don't want to learn, that's up to you, but the fact is that these proofs do exist, and they are now well accepted and taught in mathematics.

This might help you out a bit: http://en.wikipedia.org/wiki/0.999...#Sk..._education