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:

\(\displaystyle d\) \(=\) \(\displaystyle \ceiling {\dfrac {14} {\log_{10} 2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \ceiling {\dfrac {14} {0 \cdotp 301 \ldots} }\)
\(\displaystyle \) \(=\) \(\displaystyle \ceiling {46 \cdotp 51 \ldots}\)
\(\displaystyle \) \(=\) \(\displaystyle 47\)

$\blacksquare$


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