Decimal curiosities

There's not much of a story to this post, except for a few curiosities the decimal system throws up (largely as a result of the binomial expansion).

Some time ago, I looked at some Fibonacci witchcraft: $\frac{1}{999,998,999,999} =
000\,008\,...$, neatly enumerating the Fibonacci sequence in six-digit blocks (which, like all of the following, can be lengthened by judiciously adding 9s to the denominator).

I came across some other neat ones. There's the power series:

$\frac{1}{999} = 0.001\,001\,001\,...$
$\frac{1}{998} = 0.001\,002\,004\,...$
$\frac{1}{997} = 0.001\,003\,009\,...$

These are based on $(1-x)^{-1} = 1 + x + x^2 + x^3 + ...$, for small $x$.

There's also the Pascal's triangle series:

$\frac{1}{999} = 0.001\,001\,001\,...$
$\frac{1}{999^2} = 0.001\,002\,003\,...$
$\frac{1}{999^3} = 0.001\,003\,006\,...$

My favourite, though, is this monster:

$\frac{1\,001\,000}{999^3} = 0.001\,004\,009\,016\,...$, giving the square numbers! (This is the same as $\frac{10^6}{999^3} + \frac{10^3}{999^3}$, which adds the third diagonal of Pascal's triangle (the triangular numbers) to itself, offset by one -- making the triangles into squares!

Someone with enough patience ought to be able to generate the cubes or higher powers with a bit of work.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


2 comments on “Decimal curiosities

  • Mark Ritchings



    with help from Wolfram Alpha

    • Colin

      Oo, good work! Any thoughts on where those numbers come from (or is that left as an exercise?)

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