# Anomalous Cancellation on 2-Digit Numbers

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## Contents

## Theorem

There are exactly four anomalously cancelling vulgar fractions having two-digit numerator and denominator when expressed in base $10$ notation:

### $16 / 64$

- $\dfrac 1 4 = \dfrac {16} {64} = \dfrac {166} {664} = \dfrac {1666} {6664} = \cdots$

### $19 / 95$

- $\dfrac 1 5 = \dfrac {19} {95} = \dfrac {199} {995} = \dfrac {1999} {9995} = \cdots$

### $26 / 65$

- $\dfrac 2 5 = \dfrac {26} {65} = \dfrac {266} {665} = \dfrac {2666} {6665} = \cdots$

### $49 / 98$

- $\dfrac 4 8 = \dfrac {49} {98} = \dfrac {499} {998} = \dfrac {4999} {9998} = \cdots$

## Proof

## Historical Note

According to David Wells in his $1986$ work *Curious and Interesting Numbers*, this result, as well as some others, was demonstrated by Alfred Moessner in Volumes 19 and 20 of *Scripta Mathematica*, but it has proven difficult to find an archived copy to consult directly.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $16 /64$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $16/64$