1.3.B Significant Figures
The Scientific Method starts with making observations = precise and accurate measurements
• 1.3.3. Significant Figures (Significant Digits)
• 1.3.4. Round Off Error
Measurement and Precision • Measurement tools (instruments) typically have
an operational range (maximum and minimum) and limited precision (degree of exactness)
• Example: Spring scales – To what precision can these measure? – General rule: round to the nearest marking shown;
do not interpolate between markings – However if the distance between markings is large,
you may be able to interpolate one more digit – Can these scales be used to estimate the weight of a
single sheet of paper? 5 million sheets?
• The precision of any measurement is limited, and is indicated by using “significant digits”
Significant Figures (Significant Digits)
93 = 93.0
Is this statement true?
In Mathematics yes, but in Physics no. The number of significant digits used indicates the precision to which a measurement has been made.
93 means 93 ± 0.5
93.0 means 93.0 ± 0.05
Significant Figures (Significant Digits)
Significant Figures: the number of digits in a quantity that are known with certainty (reliability). This is important to record!
e.g. you measure with a ruler 25.2 cm ± 0.05 cm
25.2 3 significant figures 0.0345, 568
12.009 5 significant figures 5437600, 23.000
4300 2 significant figures 3500, 6.2, 0.060
0.005 1 significant figure 600, 0.1, 0.009
Rules for Significant Figures
• All nonzero digits are significant.
• Zeroes between nonzero digits are significant
1.001 has 4 sig figs.
• Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point.
0.001 oC has 1 sig fig.
• Trailing zeros on whole numbers are not significant.
27000 has 2 sig figs.
• Trailing zeros to the right of a decimal point are significant.
1.0 has 2 sig figs.
0.060 has 2 sig figs.
Exercise: Significant Figures
How many significant figures? 0.0305 3 significant figures 5437600 5 significant figures 0.070 2 significant figures 5.0 2 significant figures 600 1 significant figure 5.01 3 significant figures 0.9 1 significant figure
Mathematics of Significant Figures
• The number of significant figures after multiplication or division is the number of significant figures in the least known quantity:
23.41 × 4.1 = 95.981 = 96 (round, don’t truncate)
• The number of decimal places after addition and subtraction is equal to the smallest number of decimal places in any of the input values:
23.41 m + 4.1 m = 27.51 m = 27.5 m
Mathematics of Significant Figures
• Note: calculators do not keep track of significant digits!
0.200 + 0.300 = 0.5
• Keep in mind if any if the numbers are exact!
23.41 × 4 (exact) = 93.64
• Use conversion factors with at least one more significant figure than the measurements:
5.21 miles × (1.6 km/mi) = 8.336 km = 8.3 km
5.21 miles × (1.609 km/mi) = 8.38289 km = 8.38 km
Exercises
• 4.34 + 2.1 = 6.44 =
= 6.4
• 3.14159 / 2 = 1.570795 =
= 2
• 3.14159 / 2.0 = 1.570795 =
= 1.6
• 100 – 4.67 = 95.33 =
= 100
Round-off Error
The last digit in a calculated number may vary
depending on how it is calculated, due to sequential
rounding off of insignificant digits.
1.4 + 0.54 + 0.046 = ?
If add left-to-right and round during the procedure:
1.94 + 0.046 = 1.9 + 0.046 = 1.946 = 1.9
If add right-to-left and round during the procedure:
1.4 + 0.586 = 1.4 + 0.59 = 1.99 = 2.0
Conclusion: Rounding in the middle of a calculation
increases the imprecision (“round-off error”).
Solution: Only round at the very end!
1.94 + 0.046 = 1.986 = 2.0
Scientific Notation Exponential Notation, with one digit to the left of the decimal
3.1 × 106
Number between 1.0 and 9.999…. 10-based Exponent
Correct: -2 × 104, 5.1, 4.33 × 10-8
Incorrect: 10.1 × 103, 330, 0.05, -0.1 × 10-2
Fix! Also: -2e+04, 5.1e+0, 4.33e-8
Scientific Notation The advantage of scientific notation is that significant digits can be directly expressed.
“Yesterday 2500 people attended the concert.”
2500 is vague due to trailing zeroes. Generally 2500 is considered to have 2 significant digits. Or was 2500 the exact count? Better:
2.5 × 103 has 2 significant digits.
2.500 × 103 has 4 significant digits.
Exponential Notation
Example
Exponential multiplication and division
On Precise Measurements
“I often say that when you can measure something and express it in numbers, you know something about it. When you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind. It may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the stage of science, whatever it may be.”
—Lord Kelvin
1824-1907
