Is it correct to say that 9.0 is one order of magnitude smaller than 10.0?
Has anyone a link/source about confronting order of magnitudes, apart from wikipedia?
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I suppose the OP is looking for some general rule to be used when you want to say "A is N orders of magnitude bigger (smaller) than B". In that case, consider $$N = || \log_b(A/B)) ||$$ (where $ || \dots || $ is taken to mean round to the nearest integer, and negative values just mean chose "A is smaller than B", but the magnitude retains the same significance. Here $b$ the the base you are speaking in (10 generally, but it is sometimes useful in computer science circles to speak of binary orders of magnitude). In this case $\log_{10}(10/9) = 0.045 \approx 0$ so 9 and 10 are of the same order of magnitude as one would naively expect. You can manage this rule without having to extract logarithms by noting that $0.5 = \log_{10}(R)$ implies $R = \sqrt{10} \approx 3.16$. Just count the number of digits difference in the long-hand written form and add one is the ratio of the leading values is at least 3.2. That is
Final note: Don't obsess over this! Orders of magnitude are useful because then let you make quick and reasonable accurate guesses, and guesses are not subject to precise rules. For instance $\pi$ is close enough to $\sqrt{10}$ that it's OK to treat it as the same order of magnitude as either 1 or 10. |
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