Anomalous Cancellation/Examples/3544 over 7531
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Theorem
The fraction $\dfrac {3544} {7531}$ exhibits the phenomenon of anomalous cancellation:
- $\dfrac {3544} {7531} = \dfrac {344} {731}$
as can be seen by deleting the $5$ from both numerator and denominator.
This is part of a longer pattern:
- $\dfrac {344} {731} = \dfrac {3544} {7531} = \dfrac {35544} {75531} = \cdots$
Proof
| \(\ds \frac {355 \cdots 544} {755 \cdots 531}\) | \(=\) | \(\ds \paren {3 \times 10^n + \paren {\sum_{i \mathop = 2}^{n - 1} 5 \times 10^i} + 44} \Big / \paren {7 \times 10^n + \paren {\sum_{i \mathop = 2}^{n - 1} 5 \times 10^i} + 31}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \times 10^n + \paren {500 \times \sum_{i \mathop = 0}^{n - 3} \times 10^i} + 44} \Big / \paren {7 \times 10^n + \paren {500 \times \sum_{i \mathop = 0}^{n - 3} \times 10^i} + 31}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \times 10^n + \paren {500 \times \frac {10^{n - 2} - 1} {10 - 1} } + 44} \Big / \paren {7 \times 10^n + \paren {500 \times \frac {10^{n - 2} - 1} {10 - 1} } + 31}\) | Sum of Geometric Sequence | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {3 \times 10^n \times \paren {10 - 1} + 500 \times \paren {10^{n - 2} - 1} + 44 \times \paren {10 - 1} } {7 \times 10^n \times \paren {10 - 1} + 500 \times \paren {10^{n - 2} - 1} + 31 \times \paren {10 - 1} }\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {27 \times 10^n + 500 \times 10^{n - 2} - 104} {63 \times 10^n + 500 \times 10^{n - 2} - 221}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2700 \times 10^{n - 2} + 500 \times 10^{n - 2} - 104} {6300 \times 10^{n - 2} + 500 \times 10^{n - 2} - 221}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {3200 \times 10^{n - 2} - 104} {6800 \times 10^{n - 2} - 221}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {8 \times \paren {400 \times 10^{n - 2} - 13} } {17 \times \paren {400 \times 10^{n - 2} - 13} }\) | factoring | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 8 {17}\) | factoring | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {8 \times 43} {17 \times 43}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {344} {731}\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16 /64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16/64$