The term significant figures refers to the number of important single digits (0 through 9 inclusive) in the coefficient of an expression in scientific notation . The number of significant figures in an expression indicates the confidence or precision with which an engineer or scientist states a quantity.
The table shows several examples of numbers written in standard decimal notation (first column) and in scientific notation (second column). The third column shows the number of signficant figures in the corresponding expression in the second column.
Decimal expression | Scientific notation | Sig. figs. |
1,222,000.00 | 1.222 x 10 ^{6} | ^{4} |
1.22200000 x 10 ^{6} | ^{9} | |
0.00003450000 | 3.45 x 10 ^{-5} | ^{3} |
3.450000 x 10 ^{-5} | ^{7} | |
-9,876,543,210 | -9.87654 x 10 ^{9} | ^{6} |
-9.876543210 x 10 ^{9} | 10 | |
-0.0000000100 | -1 x 10 ^{-8} | ^{1} |
-1.00 x 10 ^{-8} | ^{3} |
Significant figures are arrived at by rounding off an expression after a calculation is executed. In any calculation, the number of significant figures in the solution must be equal to, or less than, the number of significant figures in the least precise expression or element. Consider the following product:
2.56 x 10 ^{67} x -8.33 x 10 ^{-54}
To obtain the product of these two numbers, the coefficients are multiplied, and the powers of 10 are added. This produces the following result:
2.56 x (-8.33) x 10 ^{67+(-54)}
= 2.56 x (-8.33) x 10 ^{67-54}
= -21.3248 x 10 ^{13}
The proper form of common scientific notation requires that the absolute value of the coefficient be larger than 1 and less than 10. Thus, the coefficient in the above expression should be divided by 10 and the power of 10 increased by one, giving:
-2.13248 x 10 ^{14}
Because both multiplicands in the original product are specified to only three significant figures, a scientist or engineer will round off the final expression to three significant figures as well, yielding:
-2.13 x 10 ^{14}
as the product.
Also see order of magnitude and scientific notation .
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