You ask your friend how much cash he has on him. Is it different if he says around $80 versus around $76? He sounds surer of his estimate at $76, right?
A piece of mail weighs 14 g on your kitchen scale. You weigh the same piece of mail with a lab balance and get 14.46 g. Which scale gives the mass with greater certainty? The lab scale.
The higher the certainty of a measurement, the more accurate or precise it is. We usually use the words accuracy and precision interchangeably, but they have different meanings in the context of scientific measurements. Accuracy is how close a measurement is to the correct value while precision is how close the measurements are to each other.
The proper way of reporting a measured quantity is for the value to reflect the certainty of the measuring method or device. A lab scale which is capable of displaying two decimal places determines masses with greater certainty than a kitchen scale that does not show any decimal place. Another way of expressing the certainty of a value is the number is significant figures, or sig figs, for short. In the example above, $70 and $78 both have no decimal place, but $78 is more certain because it has more significant figures than $70.
Counting the significant figures in a number is one of the first topics covered in a Chemistry or Physics class. Many students can get stumped by sig figs, and worse, teachers dock off points for wrong sig figs in homework, exams and lab reports. You need to figure out the deal with sig figs as soon as possible, and we’re here to help!
How to Count Significant Figures
Rule 1 (for non-zeroes):
All non-zero digits are significant. This rule should be simple, if the digit is not zero, count it as significant.
Exercise 1: The number 643 has how many significant figures?
643 All digits are non-zero, all digits are significant.
The number 643 has 3 significant figures.
Exercise 2: The number 8.9 has how many significant figures?
8.9 All digits are non-zero, all digits are significant.
The number 8.9 has 2 significant figures.
This rule is simple enough. What students usually find confusing are the zeroes because they may or may not be significant. Let’s focus on them now.
Rule 2 (for sandwiched zeroes):
Zeroes sandwiched between two significant digits are significant.
Exercise 3: The number 1005 has how many significant figures?
1005 The non-zeroes are significant.
1005 The sandwiched zeroes are significant.
The number 1005 has 4 significant figures.
Exercise 4. The number 9.01 has how many significant figures?
9.01 The non-zeroes are significant.
9.01 The sandwiched zero is significant.
The number 9.01 has 3 significant figures.
This rule should not be confusing either. If it’s a sandwiched zero, there is no reason to doubt that it is significant. What most students find difficult to remember the rules for the leading zeroes (like in 0.0056) or trailing zeroes (like in 1.20 or 750). Let’s move on to those.
Rule 3 (for zeroes at the beginning):
Zeroes at the beginning of a number are never significant.
Exercise 5: The number 0.432 has how many significant figures?
0.432 The non-zeroes are significant.
0.0432 The leading zeroes are not significant.
The number 0.432 has 3 significant figures.
Exercise 6: The number 0.0000006 has how many significant figures?
0.0000006 The non-zero is significant.
0.0000006 The leading zeroes are not significant.
The number 0.0000006 has 1 significant figure.
Exercise 7: The number 0.02108 has how many significant figures?
0.02108 The non-zeroes are significant.
0.02108 The sandwiched zero is significant.
0.02108 The leading zeroes are not significant
Answer: The number 0.02108 has 4 significant figures.
Exercise 8: The number 0.00801008 has how many significant figures?
0.00801008 The non-zeroes are significant.
0.00801008 The sandwiched zeroes are significant.
0.00801008 The leading zeroes are not significant.
Answer: The number 0.00801008 has 6 significant figures.
All of the rules we have discussed so far are definitive. Non-zeroes are always significant. Sandwiched zeroes are always significant. Zeroes at the beginning are never significant.
Zeroes at the end, which is covered in the last guideline, may or may not be significant.
Rule 4 (for zeroes at the end):
Zeroes at the end of a number are significant if there is a decimal point.
Exercise 9: The number 3.80 has how many significant figures?
3.80 The non-zeroes are significant.
3.80 The trailing zero is significant (decimal point present).
The number 3.80 has 3 significant figures.
Exercise 10: The number 250 has how many significant figures?
250 The non-zeroes are significant.
250 The trailing zero is not significant (no decimal point).
The number 250 has 2 significant figures.
Exercise 11: The number 4080 has how many significant figures?
4080 The non-zeroes are significant.
4080 The sandwiched zero is significant.
4080 The trailing zero is not significant (no decimal point).
The number 4080 has 3 significant figures.
Exercise 12: The number 2.050 has how many significant figures?
2.050 The non-zeroes are significant.
2.050 The sandwiched zero is significant.
2.050 The trailing zero is significant (decimal point present).
The number 2.050 has 4 significant figures.
Exercise 13: The number 0.01200 has how many significant figures?
0.01200 The non-zeroes are significant.
0.01200 The beginning zeroes are not significant.
0.01200 The ending zeroes are significant (decimal point present).
The number 0.01200 has 4 significant figures.
Numbers in Scientific Notation
For scientific notation, count the sig figs for the coefficient. In counting the sig figs for the number 8.06 x 10-3, consider only the coefficient 8.06 when counting, so there are 3 significant figures. The number 2.30 x 105 has 3 significant figures because its coefficient 2.30 has 3 sig figs.
Exercise 14: The number 6.030 x 10-6 has how many significant figures?
We’ll count the number significant digits in the coefficient.
6.030 The non-zeroes are significant.
6.030 The sandwiched zero is significant.
6.030 The trailing zero is significant (decimal point present).
The number 6.030 x 10-6 has 4 significant figures.
Exact Numbers
Numbers that are obtained by measurement are called inexact numbers since there is always a degree of uncertainty about the measured value. The mass of the piece of mail measured by a lab scale at 14.46 g comes with an error of ±0.01, and its value can be anywhere between 14.45 g to 14.47 g.
Exact numbers are numbers that are known with complete certainty. Values obtained by counting are exact numbers; for example, there are 12 books in that shelf or there are 159 pages in this book. Conversion factors are also exact numbers; there are exactly 12 inches in a foot or there are exactly 2.54 centimeters in 1 inch.
Exact numbers are considered to have an infinite number of significant figures.
How to Use Significant Figures in Calculations
When adding or subtracting numbers, the answer must be expressed with the same number of decimal places as the participating number with the least number of decimal places.
When multiplying or dividing numbers, the answer must be expressed with the same number of significant figures as the participating number with the least number of significant figures.
Exercise 15: 2.187 + 10.2 = ?
The calculator gives the answer 12.387. We’re adding here, so the rule of least number of decimal places applies. One number has 3 decimal places, and the other has 1 decimal place. The answer should, therefore, have 1 decimal place, so the answer is 12.4.
Exercise 16: 178.1 – 2.08 + 15 = ?
The calculation involves addition/subtraction, so we are applying the rule of least number of decimal places. The three participating numbers, 178.1, 2.08 and 15, have 1, 2 and 0 decimal places, respectively, so the answer should have no decimal place. The calculator’s answer is 191.02, but we’re rounding it off to 191.
Exercise 17:(0.081)(1090)/31.0= ?
Here, the calculation involves multiplication and division, so the answer has the same sig figs as the number with the least number of sig figs. The numbers 0.081, 1090 and 31.0 have 2, 4 and 3 sig figs, respectively. The answer should have only 2 sig figs. The calculator’s answer is 2.848064516. We need to round off the answer to 2.8.
Exercise 18: What is the perimeter of a square with a side measuring 10.5 inches?
The answer is calculated by multiplying 10.5 inches by 4. The number 10.5 has 3 significant figures. The number 4 is an exact number; you count, and not measure, that there are 4 sides to a square. The number of sides is thus considered to have an infinite number of sig figs and should not limit the certainty of the perimeter. The answer should have the same number of sig figs as the length of the side. The answer is 42 inches, but since the answer should follow the number of sig figs as 10.5 inches, the perimeter is reported as 42.0 inches.
When the calculation is a mix of addition/subtraction and multiplication/division, the rules are applied stepwise in the order in which the mathematical operations are performed.
All digits of intermediate answers are carried through in the calculation while noting how many significant figures or decimal places they really should have. This minimizes deviations from the final answer that occurs when intermediate answers are estimated. Rounding off must only be done for the final answer.
Exercise 19: (201.2 – 3.24)/1.25= ?
This calculation involves subtraction and division, so we are using both rules,applied in the same order as the operations performed. Subtracting 3.24 from 201.2, we get 197.96, but note that this answer should have 1 decimal place, and to remember, the numbers that are significant are marked by bolding. We then proceed dividing 197.96 by 1.25. Note that we should calculate using all the digits of the intermediate answer to minimize estimation errors; however, we keep in mind that it has 1 decimal place and 4 sig figs. The answer is 158.368, but we round off the answer based on the number of sig figs because the last step is division. We are dividing a number with 4 sig figs by a number with 3 sig figs, so the answer is rounded to 3 sig figs at 158.
Exercise 20: 6.305 + 0.25 x 5.10 = ?
The correct order for this calculation is to first multiply 0.25 by 5.10, then add the answer to 6.305. The product of 0.25 and 5.21 is 1.275, with this intermediate answer following the least number of sig figs rule at 2 sig figs. We then proceed to the next calculation step, keeping all digits of 1.275, noting that it has 2 sig figs. Since this intermediate answer is to be used for addition next, we also note that it has 1 decimal place and adding it to a number with 3 decimal places, the final answer should have 1 decimal place. The sum of 1.275 and 6.305 is 7.58, but we round it off to 7.6.
So, there we have it – what you need to know about significant figures. Always mind your sig figs by counting them correctly and reporting your calculation results properly. It’s time to give an end to those pesky sig fig error deductions. Good luck!
Let’s put everything into practice. Try this Chemistry practice question:
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