I often find two numbers roughly "in the same ballpark" if they are within a factor of about $e$ of each other. For example, if I know computers generally cost upward of $\$1000$, then $\$2700$ would probably be the most I would a priori feel is a reasonable upper limit for computer prices.

Is this just a coincidence, or is there some mathematical deep reason why $e$ is inherently the "best" order-of-magnitude exponent?


closed as off-topic by Antonio Vargas, Dan Rust, Norbert, Dominic Michaelis, user61527 Nov 7 '13 at 19:46

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  • $\begingroup$ Interesting observation. $\endgroup$ – Lord Soth Nov 7 '13 at 17:48
  • 4
    $\begingroup$ That is a subjective assessment. Different people have different notions of close or not close. It is easier in certain math operations though. $\endgroup$ – user76844 Nov 7 '13 at 17:49
  • $\begingroup$ $e$ is an amazing number and pops up a lot of places. $\endgroup$ – MasterOfBinary Nov 7 '13 at 17:58

I think that we subconsciously think in a logarithmic way and since the root of 10 is around 3.162, we think of 3 as half way the order of magnitude. So anything under 3,000 we perceive in the order of magnitude of 1,000 and anything over that we perceive as being in the 10,000 order of magnitude.

In other words if 1,000 is 3 in 10 based logarithm and 10,000 is 4 in 10 based logarithm then 3,162 is the 3.5 on this scale and anything under 3,000 is closer to 3 and almost everything over 3,000 is closer to 4.

  • $\begingroup$ Umm, what does the root of 10 have to do with logarithms? $\endgroup$ – ithisa Nov 7 '13 at 18:08
  • $\begingroup$ @user54609 The second paragraph explains it $\endgroup$ – DannyDan Nov 7 '13 at 18:12
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    $\begingroup$ @user54609 Because $\log_{10} \sqrt{10} = 0.5$, i.e., half way on the logarithmic scale. $\endgroup$ – zrbecker Nov 7 '13 at 18:27

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