1 = .9999.....

[NeoCracker]

FOr some reason a guy I went to school with couldn't bring himself to believe this statement, though I shall show you two ways in which it is true, and you can try to disprove it.

1/3=.333.....
2/3=.666.....

1/3 + 2/3 = 3/3 = 1

.333... + .666... = .999...

There fore, 1=.999....


Next up, imagine a number line. No two different numbers can occupy the same space. So, mathametecly, there will always be space between numbers, therefore there if there is no space between two numbers, they must occupy the same point.

SO, can anyone here tell me what number exists between 1 and .999....?

If not, logically, they occupy the same space, and are therefore the same number.

For those of you who dont' get the .... proceeding the nines, it means the nine is infinitely repeating.

Discuss.

[demondude]

It was about time this topic came up again. D:

[NeoCracker]

When did it come up before?

[blackmage_nuke]

I believe 0.99... doesnt equal 1 but approaches, similar to how the function y=e^x approaches the line y=0 but never touches it.

[Levian]

It's a very long time ago, but I still hate the topic.

In my humble opinion, 0,999 does not equal 1. :mog:

[demondude]

Lots of times, lots of different websites. =p

[Timekeeper]

Yea, they are the same.
When you reach infinity, anything becomes possible

EDIT: The number itself is different to 1, as you can never reach infinity.

If you want to use the number and don't want to reach infinity then you round and it becomes 1 anyway.
If you decide you can reach infinity then my earlier statement goes.

[Darkswordofchaos]

.999.... is a smaller number than 1 althought the diffrence is infinitly small so small in fact you couldent measure it. no matter what the space between them can be always get smaller but never disapear.

[Aerith's Knight]

And that's why you learn in the first year that you have to round up to the least accurate number in your calculation.

3 * 0.3333... = 1 * 10^1

3.0 * 0.3333... = 1.0 * 10^1

[Captain Maxx Power]

The major problem here is that an infinite series isn't and can never been a true representation of a fraction as given here. Take for example the fraction 1/4. Decimalised it is 0.25. The number 0.25 is a complete decimal representation of the fraction 1/4. However, the decimalised number for the fraction of 1/3 is of course 0.333... Because of this it becomes impossible for us to use it as the true fraction of 1/3. You would have to use a finite decimal in calculations such as 0.3333, or 0.3333333, but neither of these is the correct value. It would be like using 0.24 to represent 1/4; close but no cigar. It's from this misunderstanding that the equation 1/3 + 2/3 = 3/3 / 0.333... + 0.666... = 0.999.... can arise, because you're essentially adding together decimals that aren't a correct representation of the fraction being shown. You can't add infinite series of decimals together even if logically they make sense because they are infinitely long. Better to try and add together the distance of two infinitely long pieces of string.

Next up, imagine a number line. No two different numbers can occupy the same space. So, mathametecly, there will always be space between numbers, therefore there if there is no space between two numbers, they must occupy the same point.

SO, can anyone here tell me what number exists between 1 and .999....?

If not, logically, they occupy the same space, and are therefore the same number.

This is a falsity because it only applies to finite decimal values. If a given value is an infinite series then it becomes impossible to locate it within a number line because of this as the inherent value of that number is impossible to place. For example, 0.999 would not be in the same place as 0.9999. Because of this attempting to place any reoccuring decimal, such as 0.999..., into a theoretical number line is a pointless task because no matter where you choose to place that number, chances are someone could come along, add another 9 on the end and cause it to have to move.

For those with attention deficiency disorders, here is a summarized snippet of the general ramblings for you to ponder over - version: Infinite series and decimal values do not a good combination make.

[Rantz]

Doesn't make a practical difference to me whether they're the same or not. The fact that the decimals go on infinitely is just a limitation of the decimal system anyway, so the discussion is vain to me. A third is a third and the fact decimals can't account for that doesn't make it some big mysterious number that goes on for infinity. It's still just a third. Three thirds together make a whole, so the number .999... is a human construct that doesn't exist in the practical world. .333... symbolises a third and .666 symbolises two thirds, so .333... + .666... equals 1, not .999...

Edit: Yeah, CMP said it first

[Dignified Pauper]

Actualy, 1/3 stands in for the exact decimal number .33333... because you can multiply and divide by it and get an answer. If you take 9 * 1/3 you get a definite 3. If you take 9 * .333 you get a slightly different answer.

[Captain Maxx Power]

Actualy, 1/3 stands in for the exact decimal number .33333... because you can multiply and divide by it and get an answer. If you take 9 * 1/3 you get a definite 3. If you take 9 * .333 you get a slightly different answer.

Exactly. It's the unnecessary conversion to decimal numbers that creates the issue.

[oddler]

1 != 0.9... as 0.9... approaches 1 but never equals the limit. That's my 2 cents.

[Timekeeper]

.333... symbolises a third and .666 symbolises two thirds, so .333... + .666... equals 1, not .999...

Raps it up nicely methinks.