Serway & Jewett — Physics and Measurement: Problem 29

In a situation in which data are known to three significant digits, we write \(6.379\text{ m} = 6.38\text{ m}\) and \(6.374\text{ m} = 6.37\text{ m}\). When a number ends in \(5\), we arbitrarily choose to write \(6.375\text{ m} = 6.38\text{ m}\). We could equally well write \(6.375\text{ m} = 6.37\text{ m}\), "rounding down" instead of "rounding up," because we would change the number \(6.375\) by equal increments in both cases. Now consider an order-of-magnitude estimate, in which factors of change rather than increments are important. We write \(500\text{ m} \sim 10^3\text{ m}\) because \(500\) differs from \(100\) by a factor of \(5\) while it differs from \(1\text{ }000\) by only a factor of \(2\). We write \(437\text{ m} \sim 10^3\text{ m}\) and \(305\text{ m} \sim 10^2\text{ m}\). What distance differs from \(100\text{ m}\) and from \(1\text{ }000\text{ m}\) by equal factors so that we could equally well choose to represent its order of magnitude as \(\sim 10^2\text{ m}\) or as \(\sim 10^3\text{ m}\)?