Anomalous Cancellation on 2-Digit Numbers/Examples/16 over 64
Jump to navigation
Jump to search
Example of Anomalous Cancellation on 2-Digit Numbers
The fraction $\dfrac {16} {64}$ exhibits the phenomenon of anomalous cancellation:
- $\dfrac {16} {64} = \dfrac 1 4$
as can be seen by deleting the $6$ from both numerator and denominator.
This is part of a longer pattern:
- $\dfrac 1 4 = \dfrac {16} {64} = \dfrac {166} {664} = \dfrac {1666} {6664} = \cdots$
Proof
\(\ds \frac {166 \cdots 66} {666 \cdots 64}\) | \(=\) | \(\ds \paren {10^n + \paren {\sum_{i \mathop = 0}^{n - 1} 6 \times 10^i} } \Big / \paren {\paren {\sum_{i \mathop = 1}^n 6 \times 10^i} + 4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {10^n + 6 \times \paren {\frac {10^n - 1} {10 - 1} } } \Big / \paren {6 \times 10 \times \paren {\frac {10^n - 1} {10 - 1} } + 4}\) | Sum of Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {10 - 1} 10^n + 6 \times \paren {10^n - 1} } {60 \times \paren {10^n - 1} + 4 \paren {10 - 1} }\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {9 \times 10^n + 6 \times 10^n - 6} {60 \times 10^n - 60 + 36}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {15 \times 10^n - 6} {60 \times 10^n - 24}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {15 \times 10^n - 6} {4 \times \paren {15 \times 10^n - 6} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 4\) |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16 /64$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16/64$