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390 Mastering Technical Mathematics
(3.045 10 5 ) (6.853 10 7 ) 304,500 0.0000006853
304,499.9999993147
3.044999999993147 10 5
SIGNIFICANT FIGURES
HOW ACCURATE ARE YOU?
The number of significant figures (or digits) in an expression indicates the degree of
accuracy to which you know a numerical value, or to which you have measured, or can
measure, a quantity.
When you do multiplication, division, or exponentiation using scientific notation, the
number of significant figures in the final calculation result cannot “legally” be greater
than the number of significant figures in the least exact expression. Consider the two
numbers x 2.453 10 4 and y 7.2 10 7 . The following is a perfectly valid statement
if the numerical values are exact:
xy (2.453 10 4 ) (7.2 10 7 )
(2.453 · 7.2) 10 11
17.6616 10 11
1.76616 10 12
But if x and y represent measured quantities, as is nearly always the case in experimental
science and engineering, the above statement needs qualification. You must pay
close attention to how much accuracy you claim.
When you see a product or quotient containing quantities expressed in scientific notation,
count the number of single digits in the coefficients of each number. Then take the
smallest number of digits. This is the number of significant figures you can claim in the
final answer or solution.
In the above example, there are four single digits in the coefficient of x and two single
digits in the coefficient of y. So you must round off the answer, which appears to contain
six significant figures, to two significant figures. It is important to use rounding,
and not truncation, as follows:
xy (2.453 10 4 ) (7.2 10 7 )
1.76616 10 12
≈ 1.8 10 12
In situations of this sort, if you insist on being “mathematically rigorous,” you can use
approximate-equality symbols (the “squiggly” ones) throughout, because you are
always dealing with approximate values. But most folks are content to use ordinary