Significant figures in computations, multiplication, division, addition and subtraction with examples

Dr. Walt Volland revised July 5, 2012 all rights reserved,

MULTIPLICATION and DIVISION All measurements have a limited number of significant figures. The number of significant digits or figures depends on the quality of the calibrations on the tool used in the measurement. When a measured value is multiplied by another the result has a limited number of significant figures.  The result will only be as good as the less well known number .  For example if 31.3 is multiplied by 2.5 the result is 78.25.   This needs to be rounded off to 78. with two significant figures. The reason for this is that 2.5 has only two sf and the result can't be any better than the worse value used in the calculation. This idea matches the concept that a chain can only be as strong as its' weakest link. Another example is what happens when we multiply 21. cm by 42.3 cm. The result will be limited to two significant figures because the 21. has only two sf.  The number 42.3 cm has three significant figures.  The multiplication gives 888.3 cm cm  = 888.3 cm2.  This result needs to be rounded off to two significant figures to 890 cm2.  Here the "0" is a place holder. . EXAMPLE What is the result  when we multiply  24. inches times 519.44 inches.  24.1 inches x 519.44 inches = 124518.504 inches2 The answer needs to be rounded off to three significant figures to give 125000 inches2 where the zeros are decimal place holders. EXAMPLE DIVISION How many pieces of wire can be made when 349 cm is cut into pieces that are 0.00240 cm? 349 cm ( 1 / 0.0240 cm)  = 14541.66 pieces Rounded to 14500 pieces The answer needs to be rounded off to three significant figures to give 14500 pieces. Both values have three significant figures. The trailing zeros before the decimal point are place holders and not significant digits. ADDITION AND SUBTRACTION When adding or subtracting quantities the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of  decimal places. EXAMPLE What is the sum of the following quantities?  34.1 cm, 0.693 cm, 2.74 cm, 129.16 cm The total is 166.693 cm which must be rounded off to one decimal place, 166.7 cm, which reflects the lowest number of decimal places in the set of measurements.   One 'easy' way to deal with these types of problems is to arrange them in column form.   3 4 .1           .6 9 3     2 .7 4         .1 6   <-------- This quantity determines the number of decimal places in the result.       .