Definition:Anomalous Cancellation

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Definition

Let $r = \dfrac a b$ be a fraction where $a$ and $b$ are integers expressed in conventional decimal notation.

Anomalous cancellation is a phenomenon whereby deleting (that is, cancelling) common digits from the numerator $a$ and the denominator $b$ of $r$, the value of $r$ the fraction does not change.


Examples on $2$-Digit Numbers

$16 / 64$

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


$19 / 95$

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


$26 / 65$

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


$49 / 98$

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


Examples

$3544 / 7531$

$\dfrac {344} {731} = \dfrac {3544} {7531} = \dfrac {35544} {75531} = \cdots$


$143 185 / 17 018 560$

$\dfrac {1435} {170 \, 560} = \dfrac {143 \, 185} {17 \, 018 \, 560} = \dfrac {14 \, 318 \, 185} {1 \, 701 \, 818 \, 560} = \cdots$


Variants

$37 + 13 / 37 + 24$

$\dfrac {37 + 13} {37 + 24} = \dfrac {37^3 + 13^3} {37^3 + 24^3}$


$3 + 25 + 38 / 7 + 20 + 39$

$\dfrac {3 + 25 + 38} {7 + 20 + 39} = \dfrac {3^4 + 25^4 + 38^4} {7^4 + 20^4 + 39^4}$


Also see


Historical Note

The phenomenon of anomalous cancellation appears frequently in volumes and articles on recreational mathematics.

A popular technique is to introduce it as an amusingly incorrect piece of classroom work by a particularly shiftless pupil with an entertainingly alliterative cognomen, for example "Irving the idiot", or "Dennis the dunce" as used by David Wells in his $1986$ work Curious and Interesting Numbers.


Sources