Anomalous Cancellation on 2-Digit Numbers/Examples/26 over 65
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Example of Anomalous Cancellation on 2-Digit Numbers
The fraction $\dfrac {26} {65}$ exhibits the phenomenon of anomalous cancellation:
- $\dfrac {26} {65} = \dfrac 2 5$
as can be seen by deleting the $6$ from both numerator and denominator.
This is part of a longer pattern:
- $\dfrac 2 5 = \dfrac {26} {65} = \dfrac {266} {665} = \dfrac {2666} {6665} = \cdots$
Proof
\(\ds \frac {266 \cdots 66} {666 \cdots 65}\) | \(=\) | \(\ds \paren {2 \times 10^n + \paren {\sum_{i \mathop = 0}^{n - 1} 6 \times 10^i} } \Big / \paren {\paren {\sum_{i \mathop = 1}^n 6 \times 10^i} + 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times 10^n + 6 \times \paren {\frac {10^n - 1} {10 - 1} } } \Big / \paren {6 \times 10 \times \paren {\frac {10^n - 1} {10 - 1} } + 5}\) | Sum of Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 \times \paren {10 - 1} 10^n + 6 \times \paren {10^n - 1} } {60 \times \paren {10^n - 1} + 5 \paren {10 - 1} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {18 \times 10^n + 6 \times 10^n - 6} {60 \times 10^n - 60 + 45}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {24 \times 10^n - 6} {60 \times 10^n - 15}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 \times \paren {12 \times 10^n - 3} } {5 \times \paren {12 \times 10^n - 3} }\) | factoring | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 2 5\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16 / 64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16 / 64$