iheartchaos:

There’s all sorts of magic to be had with numbers, and many mathematicians have made entire careers in finding these little tricks that are mostly useless, but fun anyway. Unfortunately, a lot of calculators are going to truncate the results of this trick, but if you manage to get a hold of…

Now, I find it funny that they never actually bothered explaining this, and I’m not okay with that.

As the OP has implied, many things in Mathematics don’t just happen because of coincidence. So, we’re going to look at what’s so special about 998001. One thing that I would immediately check for is whether 998001 is prime, because if not, then that will yield us possibly more info. And of course, it isn’t:

998001 = 3^6 * 37^2

Bingo. Why, you say? 37 is quite a special number, considering 3*37 = 111, a three-digit number with the same digit. And, any three-multiples of 37 will consequently give a three-digit repeating number, up to 999.

Now made easy to tell with prime factorization, we know 998001 is a square, i.e. square of 3^3 * 37 = 999…!

So 1/998001 = (1/999)^2. Well what is 1/999? …

1/999 = 0.001001001…

A-ha! This looks familiar; now we’re just one step away from this pattern to that of 1/998001.

We know that dividing 1 by 999 creates this three-away 1 pattern, so dividing it by 999 again would duplicate 0.001001001… three-away as well. More formally:

1/999 = 0.001001001…

= 0.001 + 0.000001 + 0.000000001 + …

= 1 * (10^-3 + 10^-6 + 10^-9 + …) (Dividing 1 by 999 repeats 1 every 3 digits)

1/998001 = 1/999/999 = 0.001001001…/999

= 1/999 * (10^-3 + 10^-6 + 10^-9 + …)

= 0.001001001… * (10^-3 + 10^-6 + 10^-9 + …)

+0.000001001001001…

0.000000001001001…

0.000000000001001…

0.000000000000001…

=0.000001002003004…

And that’s the pattern we’re looking for. While this makes it not so magical anymore, it allows you to realize, that this **MUST** happen… there are no coincidences many times in Math.

(via dataanxiety)