Doesn't 1=0.9999 forever?

Question

Because 1 divided by 3 equals 0.3333 forever, and 0.999 forever divided by 3 equals 0.333333 forever also. If you disagree go to collage and get a math major, and give a good explanation why.

Answer 1

yes it does. why? some people aren't smart.

Answer 2

This is not quite so obvious as you think.  When you were taught long division, you will have used terminating decimals.  To write 0.99999.../3=0.333333... is actually very different to a mathematician, because we define 0.99999... (and other recurring decimals) in a very different way: when we write 0.5, we are using the point as a shorthand to mean 5/10.  When we write 0.9999..., what the shorthand actually means is lim_{N->inf} sum_{n=1}^{n=N}9*10^-n, which reads "the limit as N tends to infinity of the sum from n equals 1 to N of 9*10-n".  This is actually a rather more complicated idea, but we can show using elementary methods that this limit does evaluate to 1 (monotonic bounded sum implies existence of limit, then use contradiction).

Your explanation using 'division' is nice and intuitive, but is actually no more mathematically grounded than any argument showing that 1=/=0.999... (it just has the luck property of giving the right answer).  Actually, there is a bit more that luck to it, because you could probably develop your argument a bit in terms of redefining division to be a property of decimal limits and so on, but that is surprisingly hard to do well too.

Conclusion: you are right, but you need university level maths to really argue the point properly.

Answer 3

1 = 1

.99999 = .99999

that's your answer

no math major required

Answer 4

Yes, this is correct.

The simplest way of showing this is simply noting that there are no real numbers between 0.999~ and 1, and consequently, the two cannot be different real numbers.

0.999~ I sometimes used due to rounding errors, especially with computer software.