What is Order of magnitude? — Physics

Definition 1.11 High School Physics · Chapter 1 — Orders of Magnitude: Measuring the Universe

The order of magnitude of a quantity is the power of ten closest to it: write the quantity as $a \times 10^{n}$ with $1 \leq a < 10$ and take $10^{n}$ if $a < 5$ , $10^{n+1}$ if $a \geq 5$ . (The exact boundary convention never matters: an order-of-magnitude statement is only ever meant to within a factor of ten.)

The ladder of the universe, in metres: 42 powers of ten separate the atomic nucleus from the observable universe, and a human stands ten rungs from the atom but twenty-seven from the top. — The ladder of the universe, in metres: $42$ powers of ten separate the atomic nucleus from the observable universe, and a human stands ten rungs from the atom but twenty-seven from the top.

Examples

Example 1.12 (Orders of magnitude)

A human life lasts about $80$ years, i.e. $80 \times 3.16 \times 10^{7}\,\mathrm{s} \approx 2.5 \times 10^{9}\,\mathrm{s}$ : order of magnitude $10^{9}\,\mathrm{s}$ — a few billion seconds, spend them well. A human is $1.7\,\mathrm{m}$ tall: order $10^{0}\,\mathrm{m}$ . Mount Everest, $8848\,\mathrm{m}$ : order $10^{4}\,\mathrm{m}$ , since $8.848 \geq 5$ .

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Definition 1.11 University Physics — Year 1 · Chapter 1 — Units, Dimensions, and Measurement

The order of magnitude of a quantity is the power of ten nearest to it: $6.4 \times 10^{6}\,\mathrm{m}$ (the Earth’s radius) has order of magnitude $10^7$ ; $3 \times 10^{3}\,\mathrm{m}$ has order $10^3$ ; $3.2 \times 10^{3}\,\mathrm{m}$ is closer to $10^3$ than to $10^4$ on a logarithmic scale (the boundary is $\sqrt{10} \approx 3.16$ ). Two quantities are “of the same order” when their ratio is below about $3$ .

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