TITLE: Why are the first few powers of $2^{10}$ a little more than those of 1000?
QUESTION [12 upvotes]: See the complete list here: http://en.wikipedia.org/wiki/Power_of_two#Powers_of_1024.
I'm wondering if there's a mathematical explanation for the relationship or if it's just coincidence.

REPLY [26 votes]: Since $2^{10}=1024$: 
$$2^{10n}=(1000+24)^n=1000^n+24\cdot 1000^{n-1}n+...$$
Thus, as long as $24n$ remains a lot smaller than $1000$, then $2^{10n}$ will be near $1000^n$.

REPLY [13 votes]: A good "explanation" is that $\log_{10} 2 = 0.3010$.
Hence, $\log_{10} 2^{10} = 10 \log_{10} 2 =  3.01$, hence $2^{10}$ is very close to $10^3$.