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

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## 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:

- $A := \left({0, 7}\right)$
- $B := \left({7, 6}\right)$
- $C := \left({6, 9}\right)$
- $D := \left({9, 2}\right)$
- $E := \left({2, 3}\right)$
- $F := \left({3, 0}\right)$

## Sources

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

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