does .999 repeating equal 1 excatly?

- Chessmaster
**Posts:**191**Joined:**21 December 2005

Chessmaster wrote:does .999 repeating equal 1 excatly?

See http://en.wikipedia.org/wiki/0.999...

For adic-tional fun, don't miss the part about what integer might be represented as "...999" (repeating to the left).

- r.e.s.
**Posts:**337**Joined:**31 August 2005

Here's a wiki link to Zeno:

http://en.wikipedia.org/wiki/Zeno_of_Elea

And Zeno's paradoxes:

http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

MCC

http://en.wikipedia.org/wiki/Zeno_of_Elea

And Zeno's paradoxes:

http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

MCC

- MCC
**Posts:**1275**Joined:**08 June 2005

Chessmaster wrote:does .999 repeating equal 1 excatly?

Yes. One third is .333 repeating.

One third plus one third plus one third = 1, exactly.

.333 repeating plus .333 repeating plus .333 repeating = .999 repeating,

which is one.

- wapati
- 2010 Supporter
**Posts:**527**Joined:**13 September 2006**Location:**Brampton, Ontario, Canada

Or, to look at it another way:

I challenge anybody to come up with a number larger than .9999.... but less than 1.

Bill Smythe

I challenge anybody to come up with a number larger than .9999.... but less than 1.

Bill Smythe

- Smythe Dakota
**Posts:**536**Joined:**11 February 2006

Thanks i suppose that proves they are equal at least to me.Smythe Dakota wrote:Or, to look at it another way:

I challenge anybody to come up with a number larger than .9999.... but less than 1.

Bill Smythe

- Chessmaster
**Posts:**191**Joined:**21 December 2005

Smythe Dakota wrote:Or, to look at it another way:

I challenge anybody to come up with a number larger than .9999.... but less than 1.

Doesn't that mean that 0.9999.... and 1 are seperate numbers

Suppose I said, can anybody come up with an integer (whole number) greater than 1 but less than 2, with the implication that if they can't then

1 = 2

However large 0.9999.... gets, it never equals 1.

Pedantically.

MCC

- MCC
**Posts:**1275**Joined:**08 June 2005

MCC wrote:However large 0.9999.... gets, it never equals 1.

I'm inclined to agree with this. However small the difference is, there will always remain a difference, therefore they are not equal.

Just because you don't get enough change from £2 having paid £1.99 for your item to make a difference to your finances doesn't mean that the amounts are mathematically equal - I think !

Luna

- lunababy_moonchild
**Posts:**659**Joined:**23 March 2005

MCC wrote:Smythe Dakota wrote:Or, to look at it another way:

I challenge anybody to come up with a number larger than .9999.... but less than 1.

Doesn't that mean that 0.9999.... and 1 are seperate numbers

Suppose I said, can anybody come up with an integer (whole number) greater than 1 but less than 2, with the implication that if they can't then

1 = 2

You're right that Bill's argument is not a proof, however, you should not neglect the perfectly valid proof udosuk gave. If you prefer a variation on Bill's argument, consider the following: I define two numbers to be equal if and only if their difference is 0, i.e., a = b if and only if a-b = 0. An immediate consequence of this is that if a<b, then b-a = ε for some positive real number ε.

Assume that .999... ≠ 1. Then 1-(.999...)=ε for some positive real number ε. For any real number x, there exists a unique integer k such that 10^k ≤ x < 10^{k+1}. Fix such a k for our ε. Since 10^k is no bigger than ε, adding it to .999... cannot get you bigger than 1. However, it is easy to see that .999...+10^k > 1, a contradiction. For example, if k = -3, then .99999999...+10^{-3} = .9999999...+.001=1.000999999....

MCC wrote:However large 0.9999.... gets, it never equals 1.

.999... is a fixed number, it can't get larger, it just is what it is, a real number equal to 1. Your statement reminds me of the joke 2+2=5 for large values of 2.

- re'born
**Posts:**551**Joined:**31 May 2007

This is certainly an interesting scenario but I have to conclude 0.999repeating will never equal 1. Here's my logic.....

If 0.9999repeating equals 1 then the difference between these two numbers would be zero but this can never be because...

1.000000000000000 etc. minus

0.999999999999999 etc.

=0.000000000000001 and so on but never becomes zero.

Cec

If 0.9999repeating equals 1 then the difference between these two numbers would be zero but this can never be because...

1.000000000000000 etc. minus

0.999999999999999 etc.

=0.000000000000001 and so on but never becomes zero.

Cec

- Cec
**Posts:**1039**Joined:**16 June 2005