Units & Measurements - 03

Significant Figures

Measurements in physics is never absolutely precise—there is an uncertainty associated with every measurement. Among the most important sources of uncertainty are (a) lack of skill by experimenters (b) the limited accuracy of every measuring instrument, and (c) the inability to read an instrument beyond some fraction of the smallest division shown.

A rough idea of the uncertainty in a measurement is deduced by the number of digits used. If you report that a length has a value of 6.2 m, it means that the actual value falls somewhere between 6.15 m and 6.25 m and thus rounds to 6.2 m. This means that in a properly reported measurement, the last digit is the uncertain digit.

The number of digits of accuracy in a number is referred to as the number of significant figures (or sig figs).

Identifying significant digits

There are certain general rules for identifying significant digits. (Whitten et al., 2014, pp. A-6 - A-7)

1. Digits other than zero are significant. [e.g., 41.25 m has 4 sig figs]
2. Zeroes are sometimes significant, and sometimes not.
   • Zeroes at the beginning of a number are not significant, as they are used just to position the decimal point. [e.g., 0.025 m has 2 sig figs. In scientific notation, this can be written as $2.5\times {10}^{-2}$m .
   • Zeroes between nonzero digits are significant. [e.g., 40.1 m has 3 sig figs].
   • Zeroes at the end of a number that contains a decimal point are significant. [e.g., 41.0 m has 3 sig figs, while 441.20 kg has <5. In scientific notation, these can be written respectively as $4.10\times {10}^1\text{m}$ and $4.4120\times {10}^2\text{m}$].
   • Zeroes at the end of a number that does not contain a decimal point may or may not be significant (unless they come from experiment). If we wish to indicate the number of significant figures in such numbers, it is common to use the scientific notation.
     ― e.g., The quantity 52800 km could be having 3, 4, or 5 sig figs―the information is insufficient for decision. If both of the zeroes are used just to position the decimal point (i.e., the number was measured with estimation $\pm 100$ ), the number is $5.28\times {10}^4\text{km}$ (3 sig figs) in scientific notation. If only one of the zeroes is used to position the decimal point (i.e., the number was measured $\pm 10$), the number is $5.280\times {10}^4\text{km}$ (4 sig figs). If the number is $52800\pm 1\text{km}$, it implies $5.2800\times {10}^4\text{km}$ (5 sig figs).
3. Exact numbers, and defined quantities, can be considered as having an unlimited number of significant figures. e.g.:
   1. The rules of significant figures do not apply to (a) the count of 47 people in a hall, or (b) the equivalence: 1 inch = 2.54 centimeters.
   2. In addition, the power of 10 used in scientific notation is an exact number, i.e. the number is ${10}^3$ exact, but the number $1\times {10}^3$ has 1 sig fig.

It actually makes a lot of sense to write numbers derived from measurements in scientific notation, since the notation clearly indicates the number of significant digits in the number.

Arithmetic with significant figures

Rules for Rounding

1. The last digit kept is left unchanged if the next digit being dropped is less than 5.
2. The last digit kept is rounded up if the next digit being dropped is more than 5.
3. If the next digit being dropped is 5, and either it is the last digit or has zeroes following it, then the last digit kept is rounded up if it is odd and left unchanged if it is even. If, however, there are digits following 5 which are other than zero, then the last digit kept is rounded up.
4. If the last digit kept is after the decimal point, then after rounding, the following digits are dropped. However, if it is to the left of the decimal point, the following digits are replaced by zeroes. [e.g. rounding 13472 to three sig figs result in 13500].

Square Roots and Squares

When you take the square root of a number with "n" significant figures, the answer you get will provide "n+1" significant figures. For instance, if you take the square root of 2.00, the actual square root with proper sig figs will be 1.414.

When you square a number, you should lose one sig fig in the answer. For instance, if you squared 2.50, the final answer reported should be 6.2, not 6.25.

Logarithms

We report as many decimal places in the log of the number as there are significant figures in the original number.

Addition and Subtraction

The answer can't have more digits to the right of the decimal point than in either of the original numbers. (Rounding is done as appropriate. Note that the number of significant figures can change during these calculations.) e.g.,

• $37.24+10.3(=47.54)=47.5$
• $21.2352+27.87(=49.1052)=49.11$
• $14.16+3.2(=17.36)=17.4$

Multiplication and Division

The answer contains no more significant figures than the least number of significant figures used in the operation. e.g.,

• $14.0\times 3.0=42$
• $14.0\times 3(=42)=40$
• $14.0÷3(=4.67)=5$

Combined calculations

When a calculation involves several steps, we often show the answer to each step to the correct number of significant figures. But we carry all digits in the calculator to the end of the calculation; then we round the final answer to the appropriate number of significant figures. When carrying out such a calculation, it is safest to carry extra figures through all steps and then to round the final answer appropriately.

Be wary of additions and subtractions. They sometimes increase or reduce the number of significant figures in an answer to greater or fewer than the number in any measured quantity.

During all the intermediate steps, keep track of the minimum number of significant figures among them, as that will decide the number of sig figs in the final result.