Recurring Decimal/Examples/0.333...
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Example of Recurring Decimal
The number $\dfrac 1 3$ can be expressed as a terminating decimal:
| \(\ds \dfrac 1 3\) | \(=\) | \(\ds 0 \cdotp 333 \ldots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0 \cdotp \dot 3\) | using recurrence notation |
This would be voiced:
- nought point three recurring
or:
- zero point three recurring
and so on.
Proof
| \(\ds 0 \cdotp 333 \ldots\) | \(=\) | \(\ds 0 \cdotp 333 \ldots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0 + \dfrac 3 {10} + \dfrac 3 {100} + \dfrac 3 {1000} + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \times \sum_{i \mathop \ge 1} \paren {\dfrac 1 {10} }^i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \times \dfrac 1 {1 - \frac 1 {10} }\) | Sum of Infinite Geometric Sequence: Corollary $1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \times \dfrac 1 9\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 3\) |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): decimal
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): decimal