Number of Binary Digits in Power of 10/Example/1000

Theorem

When expressed in binary notation, the number of digits in $1000$ is $10$.

Proof

Let $m$ be the number of digits in $1000$.

From Number of Binary Digits in Power of 10:

$m = \ceiling {3 \log_2 10}$

From Logarithm Base 2 of 10:

$\log_2 10 \approx 3 \cdotp 32192 \, 8 \ldots$

and so:

$m \approx 9 \cdotp 96$

Hence the result.

The actual number is:

$1000_{10} = 1 \, 111 \, 101 \, 100_2$

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3 \cdotp 321 \, 928 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 32192 \, 8 \ldots$