Significant Figures

Significant figures are associated with measurements or the result of a calculation using measurements. All measurements have some degree of uncertainty, so when we record a measurement or calculation result using measurements we include all digits known with certainty plus the rightmost digit which has some uncertainty. These are the significant or (meaningful) figures in the measurement or result.

The number of significant figures in a measurement or result is an indication of the precision of the value. To count the number of significant figures in a value, we observe the following rules:

1. All nonzero digits are significant. Thus, 8.15 cm, 0.723 cm, 52.3 cm, and 0.000234 cm all contain three significant figures.
2. Zeros between nonzero digits are significant. There are three significant figures in each of the measurements, 101 cm, 10.5 cm and 1.06 cm.
3. Terminal zeros to the right of the decimal point are significant. There are three significant figures in 0.100 cm, 0.110 cm, and 1.00 cm.
4. Zeros preceding the first nonzero digit are not significant. There are three significant figures in 0.0101 cm, 0.00100 cm and 0.000101 cm.
5. Terminal zeros in a number without an explicit decimal point may or may not be significant. When a measurement is given as 200 cm, you do not know whether one, two, or three significant figures were intended. Any uncertainty can be removed by expressing the number in scientific notation.

Scientific notation is the notation of a number in the form C x 10ⁿ where C is a number with a single nonzero digit to the left of the decimal point and n is an integer, or whole number.

1. When multiplying or dividing measured quantities, you should give as many significant in the answer as there are in the measurement with the least number of significant figures.
2. When adding or subtracting measured quantities, you should give the same number of decimal places in the answer as there are in the measurement with the least number of decimal places.
3. When taking the logarithm of a measurement, you should give as many digits in the mantissa (part to the right of the decimal point) as there are significant figures in the measurement.
4. When taking the antilogarithm (inverse log) of a number, you should give as many significant figures in the answer as there are digits in the mantissa (digits to the right of the decimal point) of the logarithm.

 An exact number is a number that results when you count items or when you define a unit. Exact numbers have no effect on the number of significant figures in a calculation. For example suppose you want the total mass of 7 test tubes when each test tube weighs 5.0 g. The calculation would be 5.0 g x 7 = 35 g. You report the answer to two significant figures because 5.0 g has two significant figures. The number 7 is exact and does not determine the number of significant figures.

Rounding is the procedure of dropping superfluous (nonsignificant) figures in a calculated result and adjusting the last digit reported. If the first dropped digit is 5 or greater, increase the last retained digit by one. Note that rounding does not materially change a value, that is the value after rounding is essentially equal to the value before rounding. Suppose you have the value 6125 g as the answer to some calculation and the rules of significant figures tell you that this answer should be rounded to three significant figures. Many students will say the answer is 613 g, NO! NO! NO!  we cannot just willy nilly throw away digits. 613 g is about 1/10 the value of 6215 this is a material change this is not rounding this is dividing by 10. What we must do is put 6125 into scientific notation and then round the coefficient. 6.125 x 10³ or in computer notation 6.125e3 g we can now round the coefficient and get 6.13e3 g

Chain Calculations

When you have more than one operation in a math problem, you must solve it following the correct order of operations.

Here's the order we use:

1. First, do all operations that lie inside parentheses.
2. Next, do any work with exponents or roots.
3. Working from left to right, do all multiplication and division.
4. Finally, working from left to right, do all addition and subtraction.

Consider the following example.   32.46 - 2.832²  ÷ ( 14.87 - 14.36) 

In a complicated algebraic setup such as this it is best not to round intermediate answers, but you must keep track of the rightmost digit that would be retained (shown here by an underline).