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Some people just can't be convinced about that!

Try convincing them that 0.000...001 = 0. (Actually, it's a contradiction; the [...] implies numbers without end, while the 1 is at the end. What?)

I was a grown-up before I realised that (independently of any maths teacher). I knew the decimal for 1/9, 2/9, 3/9 etc. and saw the pattern.

To be fair, until you've seen some analysis and are aware and fine with limits, it's not entirely obvious. (about 99.999999...% obvious)

Reminds me of a funny story from high school. On my first test in one math class I did some erroneous rounding of answers and so got a few points off here or there. To make sure that didn't happen again, on the next exam I wrote out all my answers to like 12 decimal places. Having gotten everything correct, my math teacher actually gave me a 99.9999 (repeating) on the exam.

It's really fabulous each time, the students add some decimals after a period.

I'm going to start doing this for my students.

Actually there's a simple algebraic proof, you don't need any limits:

Let x = 99.99...

10x = 999.99...

9x = 900

x = 100

Done!

That's not a proof because it assumes that 99.999.... is a meaningful number that can be manipulated that way.

Are you not using continuity of scalar multiplication? Infinite decimal notation has no sense without limits. Actually, real numbers don't exist at all until we push around rationals to their limits :D

Actualy, one can define the set real numbers as the set of Q-Dedeking cuts (a Dedeking cut is a partition of Q in two subset A and B such that B is greater than A, and such that A has no greatest element.) That way, the number 99.99… is define as the couple of the union A of the sets (A_n) of the rationals number lesser than 99.99…9 (with n 9 after the point) and the intersection B of (B_n), define a similar way. Again, it's not limits, it's infinite unions. And finaly, it is easy to prove that 99.99… is exacly the Dedekind cut of the rational 100. So, 99.99…=100, without limits.

Yes, of course, you are absolutely right that this does not involve topology, but it is induced by the order on cuts in such a natural fashion that I was just being overly picturesque.

I try to make this concept a little simpler by using the following example.

We know that 1 = 1/3 + 2/3. We also know that 1/3 = .333333... and 2/3 = .666666.... Then 1 = 1/3 + 2/3 = .333333... + .666666... = .999999....

People still don't believe me.

What is there to believe? No, really, what IS THERE to believe? It's a $*%!"# notation, it was never meant to be believable, just convenient...

People will cling to their petty opinions and beliefs even if it doesn't reflect in their lives in any possible way. Like sum of series. Why would you even wish to argue about that if you are not mathematician?

He's arbitrarily close to perfect, no matter how you cut it.

Yes, of course, you are absolutely right that this does not involve topology, but it is induced by the order on cuts in such a natural fashion that I was just being overly picturesque.

The condugate of 9 is 9.

Thus, 99.9bar = 99.9

No! The bar meant that the nines repeat forever!