Suppose you were now asked to state the TOTAL length of the ruler diagrammed? A reasonable value, to 3 significant figures, might be 5 .28 cm. This measurement could also be stated as 52800 µm, 52 .8 mm, 0 .528 dm, 0 .0528 m, 0 .0000528 km ( lesson on metric conversions). Because the value cannot become more accurate by converting it to other units, each of these new representations must also have only three significant figures. The question arises, when is zero a significant figure?
A zero is said to be significant if:
(1) it is between two nonzero digits 
3001 m, 30.001 m 
4 SD, 5 SD 
(2) it is at the end of a decimal expression 
0.00310 km 
3 SD 
(3) it is required when expressing the number in scientific notation 
3.10 x 10^{6} m 
3 SD 
Otherwise a zero is considered to be only a placeholder.
150,000 m 
all four zeros are placeholders 
1.5 x 10^{5} m 
0.0015 km 
all three zeros are placeholders 
1.5 x 10^{3} km 
150. Gm 
no zeros are placeholders (note the deliberate inclusion of the decimal) 
1.50 x 10^{11} m 
When multiplying or dividing two measurements, your answer should be rounded off so that it only has accurate as many significant digits as your least accurate original value. When adding or subtracting two measurements, first convert them to the same unit of measurement, then line up the decimals. Your final answer should be rounded off so that it only has as many decimal places as your least accurate original value.
Numerical constants (π, e, ½) do not have significant digits.
For example, the volume of sphere is calculated with the formula V = ^{4}/_{3} πr^{3}.
Using this formula, a sphere with a measured diameter of 24 cm would have a volume equal to
^{4}/_{3} π(12)^{3} = ^{4}/_{3} π(1728) = 2304π cm^{3}
Since 12 only had two significant digits, your final value for the sphere's volume should only have 2 SD.
This means that a calculated value of V = 7238.229 cm^{3} should be expressed in final form as
7200 cm^{3} = 7.2 x 10^{3} cm^{3}
Scientific Notation. Express your value so that it has one digit to the left of the decimal and all other significant digits to the right of the decimal. It should then be multiplied by an appropriate power of 10.
(1) When the absolute value of the original number is greater than one, then moving the decimal point will require the resulting number to be multiplied by 10 raised to a positive exponent.
(2) When the absolute value of the original number is less than one, then moving the decimal point will require the resulting number to be multiplied by 10 raised to a negative exponent.



where the decimal was moved ....

0.066 g 
6.6 x 10^{2 }g 
0.066<1 
two decimal places to the right 
0's are placeholders 
200.0 g 
2.000 x 10^{2 }g 
200.0>1 
two decimal places to the left 
all three zeros are significant 
0.543 g 
5.43 x 10^{1 }g 
0.543<1 
one decimal place to the right 
0 is a placeholder 
1600 g 
1.6 x 10^{3 }g 
1600>1 
three decimal places to the left 
both zeros are placeholders 
75.2 g 
7.52 x 10^{1 }g 
75.2>1 
one decimal place to the left 
     
