Maths query

[d000hg]

Two things can not be exactly the same thing in all respects; otherwise they are not two things, it is one thing mentioned twice.

Some theories suggest there is only one electron, one proton, etc, and we just see it many times.

Are you saying every 1 is a different entity, or are they all the same 1 used a lot of times?

[NoddY]

...
Therefore 0.99999... and 1 are simply different representations of the same number.
...

That's correct. They are two decimal representations of the same real number.

The proof comes from infinite geometric series. Mathwords: Infinite Geometric Series
which is: a/(1-r)

a + ar + ar^2 + ar^3 + ... = a/(1-r)

0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...
= 9/10 + 9/10(1/10)^1 + 9/10(1/10)^2 + 9/10(1/10)^3 + ...

Which in a/(1-r) we can say a = (9/10) and r = (1/10) (where r in the range: -1 < r < 1)
Substituting values in:

[(9/10)]/[1-(1/10)]
(9/10)/(9/10) = 1

[MyUserName]

There is no number (in the real number system, with standard analysis) between 0.99999.... and 1. Therefore 0.99999... and 1 are simply different representations of the same number.

Anyone who says different either doesn't know what they're talking about or is using non-standard analysis. I suspect strongly the former.

So would 0.1 recurring also be the same as 1?

[NotAllThere]

So would 0.1 recurring also be the same as 1?

Don't be silly. There's many numbers between 0.1 and 1. For example, 0.2.

[OwlHoot]

So would 0.1 recurring also be the same as 1?

I know that was a joke (at least I hope to God it was!) But for any reader not too comfortable with recurring decimals

All you have to do is note that if x = 0.111 .. then 10 * x = 1.111 .., so subtracting the first from the second gives 9 * x = 1, or equivalently x = 1/9

This is similar to the geometric series method explained in a previous post, and the same approach works for any recurring decimal.

[MyUserName]

Don't be silly. There's many numbers between 0.1 and 1. For example, 0.2.

Ah yes, well done. You ahem spotted the deliberate mistake.
Anyway I was thinking or rather not thinking about whether the rule you mentioned would work from the opposing side with 1.1r and 1 but now I have been at my desk for a few hours and got a little more sleep it occurs to me that obviously there are numbers between 1.1r and 1 - eg 1.01.

I am not sure if the phase '1.0 recurring apart from the very very very very last one which is 1' has a mathematical shorthand though.

[Bacchus]

There is no number (in the real number system, with standard analysis) between 0.99999.... and 1. Therefore 0.99999... and 1 are simply different representations of the same number.

Anyone who says different either doesn't know what they're talking about or is using non-standard analysis. I suspect strongly the former.

That's feeble

There are no integer values between 1 and 2; doesn't make them the same thing.

By its very definition 0.999... gets increasingly close to 1 but never reaches it, mathematically they simply are not the same thing in any shape, size, or form. Arithmetically they may as well be.

[d000hg]

That's feeble

There are no integer values between 1 and 2; doesn't make them the same thing.

By its very definition 0.999... gets increasingly close to 1 but never reaches it, mathematically they simply are not the same thing in any shape, size, or form. Arithmetically they may as well be.

That rather negates the principle of limits, summation of converging infinite series, etc.

[OwlHoot]

By its very definition 0.999... gets increasingly close to 1 but never reaches it, mathematically they simply are not the same thing in any shape, size, or form. Arithmetically they may as well be.

How many more fecking times?

The infinite sequence {0.9, 0.99, 0.999, ...} "gets increasingly close to 1 but never reaches it" (to use your words).

But "by its very definition" 0.999 .. is not that sequence but defined as the _limit_ of that sequence, which is 1.

The persistently muddled thinking (and it isn't just you) around these sequences and limits is astounding!

[Bacchus]

How many more fecking times?

The infinite sequence {0.9, 0.99, 0.999, ...} "gets increasingly close to 1 but never reaches it" (to use your words).

But "by its very definition" 0.999 .. is not that sequence but defined as the _limit_ of that sequence, which is 1.

The persistently muddled thinking (and it isn't just you) around these sequences and limits is astounding!

The key word here being "limit"

This is the value beyond which the sequence can not progress, it is NOT the value that the sequence reaches. They are not the same thing.