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Explain why the order of magnitude is sometimes not the same as the exponent in scientific notation. It is because of the units?

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The reason is the exact definition of order of magnitude. It is defined as the logarithm (base 10) rounded to the nearest whole number (see wikipedia page). Thus, all numbers between $\sqrt{10}\cdot 10^{m-1}$ and $\sqrt{10}\cdot 10^{m}$ have $m$ as order of magnitude, even though, for example, $4\cdot 10^{m-1}$ has $m-1$ as exponent of $10$ in scientific notation ($4 > \sqrt{10}=3.1622...$).

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  • $\begingroup$ I know what you're trying to say, but the example isn't well written. In particular, $0.5>\sqrt{10}$ is obviously false. $\endgroup$ – David Hammen Feb 13 at 14:03
  • $\begingroup$ @DavidHammen Thanks! Sometimes one is thinking something and writing something else! $\endgroup$ – GiorgioP Feb 13 at 14:30
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$n \,e \,m$ is just usually thought of as $n \times 10^m$, coming from the early age of computers/pocket calculators, it's a historical way of writing the scientific notation. On the contrary, $n \times e^m$ is just that, it is never written as $n \,e \,m$.

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