PhysicsLAB Resource Lesson
Significant Figures and Scientific Notation

Whereas when counting objects, everyone would count the same number, and have an equally exact answer; when making measurements, there is no such thing as an EXACT measurement. Instead objects are measured to an accepted level of precision based on the limitations of the measuring apparatus. A digit is said to be a significant figure if it is either known with certainty or if it is the first estimated digit in a measurement.  
 
For example, suppose that there are three slips of paper on a desk. No matter which student counts them, they will all tell you the same answer, "There are three pieces of paper." However, if each student then measures the length of each slip of paper, there will most likely be a difference in their answers. One student might report the length as 8.20 cm, another as 8.19 cm and yet another as 8.22 cm. In each case, each answer has three significant figures. All three student agree that the slip of paper is greater than 8 cm long. All three would round off their answers to 8.2 cm. But they have each estimated a final digit. All three answers, within the limitations of their rulers, should be considered accept.
 
Let's look at some examples.
 
Refer to the following information for the next six questions.

meter stick segment
 
If each of the numbers (1, 2, 3, 4, 5) represent centimeters, then what is the reading for each of the specified locations? Note that in each of these measurements there should be two (2) certain digits and a third estimated digit.
 
 

 

 

 

 

 

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 non-zero 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 106 m 3 SD
 
 
Otherwise a zero is considered to be only a placeholder.
 
150,000 m all four zeros are placeholders 1.5 x 105 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 1011 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 πr3.
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 cm3
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 cm3 should be expressed in final form as
7200 cm3 = 7.2 x 103 cm3
 
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 102 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 103 g |1600|>1 three decimal places to the left both zeros are placeholders
75.2 g 7.52 x 101 g |75.2|>1 one decimal place to the left - - - - -
 
 
Refer to the following information for the next five questions.

State each of the following numbers in scientific notation with the correct number of significant digits.
 126 sec 

 126000 m/sec 

 0.0126 m/sec2 

 12060 m 

 0.001260 m/sec 





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