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# Classic puzzle: .9999..... AND 1

### #1

Posted 20 October 2013 - 12:32 AM

Here is a classic exercise: Which of the following are true and why?

0.99999..... < 1

0.99999..... = 1

0.99999..... > 1

### #2

Posted 20 October 2013 - 03:24 AM Best Answer

### #3

Posted 20 October 2013 - 11:00 AM

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #4

Posted 20 October 2013 - 02:16 PM

So are we saying only one case is true?

### #5

Posted 20 October 2013 - 04:28 PM

So are we saying only one case is true?

Yes. Both sides of the relations specify real numbers, which are well-ordered. So, exactly one of the relations must be true.

### #6

Posted 04 November 2013 - 05:31 PM

hmm. there's a mathematician on youtube, norm wildberger, whom i kinda like who would disagree.

the problem being that we cant actually "know" infinity. try writing infinite 9's. you can't.

so at best we can say 0.9999999 or wherever you care to stop approximately equals 1.

to prove this point, accurately calculate pi +e +sqrt(2). if you're being honest, you would calculate it as pi + e +sqrt(2).

### #7

Posted 04 November 2013 - 06:06 PM

Past prime, actually.

### #8

Posted 05 November 2013 - 03:12 PM

hmm. there's a mathematician on youtube, norm wildberger, whom i kinda like who would disagree.

the problem being that we cant actually "know" infinity. try writing infinite 9's. you can't.

so at best we can say 0.9999999 or wherever you care to stop approximately equals 1.

to prove this point, accurately calculate pi +e +sqrt(2). if you're being honest, you would calculate it as pi + e +sqrt(2).

The notation convention 0.999 ...

**denotes**an endless string of 9's.

You can't

**write**the decimal representation of 1/3, but you can write 0.333 ...,

Thus, given the meaning of that notation, one can meaningfully write 1/3 = 0.333 ...

Similarly one can meaningfully write 1 = 0.999 ...

*The greatest challenge to any thinker is stating the problem in a way that will allow a solution.*

- Bertrand Russell

### #9

Posted 05 November 2013 - 06:38 PM

well, not entirely sure i agree. for example, how would you denote 1/13?

you need a consistent way of denoting any rational number, not just ones that repeat the last digit.

personally, i like the bar method, where the repeated values have a bar over the top over the length of the repeating part.

so

_ 0.9 = 1

would be more accurate (although more cumbersome.)

### #10

Posted 05 November 2013 - 07:00 PM

here's my challenge for you.

accurately add 1/13 to 45/89 in decimal representation.

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