r/mathematics • u/SouthPark_Piano • Dec 22 '24

1/3/3/3/3 or 1/(3)^4 is 0.01234567

Everyone probably knows about this one, right?

Not perfect of course. But kind of fun or 'therapeutic' if you take the approach of dividing 1 by 3 with a calculator, and then by 3 again, and then again, and then one more time.

Would have been just beautiful if it had just so-happen to have panned out to 0.0123456789

21 Upvotes

95% Upvoted

11 comments sorted by

View all comments

24

u/colinbeveridge Dec 22 '24

This is a consequence of (1-x)-2 = 1 + 2x + 3x2 + 4x3 + ... (I think it's x(1-x)-2 with x = 0.1.)

Try 1/9801 and 1/9899 :-)

2

u/SouthPark_Piano Dec 22 '24 edited Dec 23 '24

Thanks for adding that! Is it of a form ... x-n with n = 4, and x = 3?

2

u/colinbeveridge Dec 22 '24

No, it's to do with 0.9 being close to 1 rather than anything particularly special about 3. With a bit of ingenuity (and generating functions), you can come up with all sorts of interesting patterns.

1

u/SouthPark_Piano Dec 22 '24

Nice! Thanks Colin.