第1章単位と測定

From the discussion so far, it should be clear that to accomplish your goals in any of the various fields within the natural sciences and engineering, a thorough grounding in the laws of physics is necessary. The reason for this is simply that the laws of physics govern everything in the observable universe at all measurable scales of length, mass, and time. Now, that is easy enough to say, but to come to grips with what it really means, we need to get a little bit quantitative. So, before surveying the various scales that physics allows us to explore, let’s first look at the concept of “order of magnitude,” which we use to come to terms with the vast ranges of length, mass, and time that we consider in this text (Figure 1.3).

The order of magnitude of a number is the power of 10 that most closely approximates it. Thus, the order of magnitude refers to the scale (or size) of a value. Each power of 10 represents a different order of magnitude. For example, ${10}^1,{10}^2,{10}^3,$ and so forth, are all different orders of magnitude, as are ${10}^0=1,{10}^{\mathrm{-1}},{10}^{\mathrm{-2}},$ and ${10}^{\mathrm{-3}}.$ To find the order of magnitude of a number, take the base-10 logarithm of the number and round it to the nearest integer, then the order of magnitude of the number is simply the resulting power of 10. For example, the order of magnitude of 800 is 10³ because ${\text{log}}_{10}800\approx 2.903,$ which rounds to 3. Similarly, the order of magnitude of 450 is 10³ because ${\text{log}}_{10}450\approx 2.653,$ which rounds to 3 as well. Thus, we say the numbers 800 and 450 are of the same order of magnitude: 10³. However, the order of magnitude of 250 is 10² because ${\text{log}}_{10}250\approx 2.397,$ which rounds to 2.

An equivalent but quicker way to find the order of magnitude of a number is first to write it in scientific notation and then check to see whether the first factor is greater than or less than $\sqrt{10}={10}^{0.5}\approx 3.$ The idea is that $\sqrt{10}={10}^{0.5}$ is halfway between $1={10}^0$ and $10={10}^1$ on a log base-10 scale. Thus, if the first factor is less than $\sqrt{10},$ then we round it down to 1 and the order of magnitude is simply whatever power of 10 is required to write the number in scientific notation. On the other hand, if the first factor is greater than $\sqrt{10},$ then we round it up to 10 and the order of magnitude is one power of 10 higher than the power needed to write the number in scientific notation. For example, the number 800 can be written in scientific notation as $8\times {10}^2.$ Because 8 is bigger than $\sqrt{10}\approx 3,$ we say the order of magnitude of 800 is ${10}^{2+1}={10}^3.$ The number 450 can be written as $4.5\times {10}^2,$ so its order of magnitude is also 10³ because 4.5 is greater than 3. However, 250 written in scientific notation is $2.5\times {10}^2$ and 2.5 is less than 3, so its order of magnitude is ${10}^2.$

The order of magnitude of a number is designed to be a ballpark estimate for the scale (or size) of its value. It is simply a way of rounding numbers consistently to the nearest (on a logarithmic scale) power of 10. In other words, if numbers differ by one order of magnitude, the one is larger than the other by a multiplicative factor of about ten. This makes doing rough mental math with very big and very small numbers easier. For example, the diameter of a hydrogen atom is on the order of 10⁻¹⁰ m, whereas the diameter of the Sun is on the order of 10⁹ m, so it would take roughly ${10}^9\text{/}{10}^{\mathrm{-10}}={10}^{19}$ hydrogen atoms to stretch across the diameter of the Sun. This is much easier to do in your head than using the more precise values of $1.06\times {10}^{\mathrm{-10}}\text{m}$ for a hydrogen atom diameter and $1.39\times {10}^9\text{m}$ for the Sun’s diameter, to find that it would take $1.31\times {10}^{19}$ hydrogen atoms to stretch across the Sun’s diameter. In addition to being easier, the rough estimate is also nearly as informative as the precise calculation.