Anomalous Cancellation on 2-Digit Numbers/Examples/19 over 95
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Example of Anomalous Cancellation on 2-Digit Numbers
The fraction $\dfrac {19} {95}$ exhibits the phenomenon of anomalous cancellation:
- $\dfrac {19} {95} = \dfrac 1 5$
as can be seen by deleting the $9$ from both numerator and denominator.
This is part of a longer pattern:
- $\dfrac 1 5 = \dfrac {19} {95} = \dfrac {199} {995} = \dfrac {1999} {9995} = \cdots$
Proof
Formally written, we have to show that:
- $\ds \paren {\paren {\sum_{i \mathop = 0}^{n - 1} 9 \times 10^i} + 10^n} \Big / \paren {5 + \paren {\sum_{i \mathop = 1}^n 9 \times 10^i} } = \frac 1 5$ for integers $n > 1$.
So:
\(\ds \) | \(\) | \(\ds \paren {\paren {\sum_{i \mathop = 0}^{n - 1} 9 \times 10^i} + 10^n} \Big / \paren {5 + \paren {\sum_{i \mathop = 1}^n 9 \times 10^i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {\sum_{i \mathop = 0}^{n - 1} \paren {10^{i + 1} - 10^i} } + 10^n} \Big / \paren {5 + \paren {\sum_{i \mathop = 1}^n \paren {10^{i + 1} - 10^i} } }\) | rewriting the $9$'s | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {10^n - 10^0 + 10^n} \Big / \paren {5 + 10^{n + 1} - 10^1}\) | Definition of Telescoping Series | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times 10^n - 1} \Big / \paren {10 \times 10^n - 5}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 5\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16 /64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16/64$