Number of Bits for Decimal Integer/Examples/14 Digits
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Example of Number of Bits for Decimal Integer
A positive integer $n \in \Z_{>0}$ which has $14$ digits when expressed in decimal notation may require $47$ bits to represent in binary.
Proof
Let $n$ have $m$ digits.
Let $d$ be the number of bits that may be needed to represent $n$.
From Number of Bits for Decimal Integer:
| \(\ds d\) | \(=\) | \(\ds \ceiling {\dfrac {14} {\log_{10} 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \ceiling {\dfrac {14} {0 \cdotp 301 \ldots} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \ceiling {46 \cdotp 51 \ldots}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 47\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: Exercise $19$