Rounding Off and Order of Magnitude

Rounding a Measured or Calculated Value

Once a calculation is done — often on a calculator that displays many more digits than any

measurement actually justifies — the result must be rounded off to a number of digits

consistent with the precision of the original data. The standard rounding rules are:

• If the digit to be dropped is less than 5, the preceding digit is left unchanged (round down).
• If the digit to be dropped is greater than 5 (or exactly 5 followed by further non-zero digits), the preceding digit is increased by one (round up).
• If the digit to be dropped is exactly 5 with nothing following it, the common convention is to round to the nearest even digit — this avoids a small statistical bias that would otherwise creep in if "5" were always rounded up.

Order of Magnitude — Estimating the Power of Ten

For a quick, rough sense of "how big" a quantity is, physicists often care only about its

order of magnitude — the nearest power of ten — rather than its exact value. To find the

order of magnitude of a quantity:

1. Write the quantity in scientific notation, $a \times 10^{b}$, where $1 \le a < 10$.
2. If $a \le 5$, the order of magnitude is $10^{b}$.
3. If $a > 5$, the order of magnitude is $10^{b+1}$ (the coefficient rounds up to the next power of ten).

For example, the radius of a hydrogen atom, about $5.3\times10^{-11}$ m, has order of

magnitude $10^{-10}$ m (since $5.3 > 5$, round the power up); the mass of an electron,

$9.1\times10^{-31}$ kg, has order of magnitude $10^{-30}$ kg by the same rule. …