Points Defined by Adjacent Pairs of Digits of Reciprocal of 13 lie on Hyperbola

Theorem

Consider the digits that form the recurring part of the reciprocal of $13$:

$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3$

Take the digits in ordered pairs, and treat them as coordinates of a Cartesian plane.

It will be found that they all lie on a hyperbola:

Proof

Let the points be labelled to simplify:

$A := \left({0, 7}\right)$

$B := \left({7, 6}\right)$

$C := \left({6, 9}\right)$

$D := \left({9, 2}\right)$

$E := \left({2, 3}\right)$

$F := \left({3, 0}\right)$

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Sources

• Oct. 1987: Edward Kitchen: Problem 1248: A Curious Property of 1/7: Solution (Math. Mag. Vol. 60, no. 4: p. 245)  www.jstor.org/stable/2689350

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $142,857$