Significant figures
Rules of Significant figures
Non zero numbers are always significant
Zeros between nonzero numbers are always significant
Zeros before nonzero numbers are never significant.
Zeros to the right of a number are significant ONLY if there is a decimal point
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Math with Significant figures
Addition and Subtraction
The number of decimal places in the answer should be equal to the number of decimal places in the least precise number used in the question
example:
12.10 + 3.021 + 2.3 = 17.421
but the least precise number is 2.3 ( has 1 decimal place ) so answer should have 1 decimal place = 7.4
Steps:
1) Count the number of significant figures in the decimal portion of each number in the problem. (The digits to the left of the decimal place are not used to determine the number of decimal places in the final answer.)
2) Add or subtract in the normal fashion.
3) Round the answer to the LEAST number of places in the decimal portion of any number in the problem.
2) Add or subtract in the normal fashion.
3) Round the answer to the LEAST number of places in the decimal portion of any number in the problem.
Addition and Subtraction
The LEAST number of significant figures in any number of the problem( the least precise) determines the number of significant figures in the answer.
Example #1: 12.10 X 3.0 = 36.3 = 36
4 s.f. 2 s.f. 2 s.f.
Example #2: 2.5 x 3.42.
The answer to this problem would be 8.6 (which was rounded from the calculator reading of 8.55). Why?
2.5 has two significant figures while 3.42 has three. Two significant figures is less precise than three, so the answer has two significant figures.
Example #3: How many significant figures will the answer to 3.10 x 4.520 have?
You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundedth's place is not recognized as significant when, in fact, it is. 3.10 has three significant figures.
Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.
Another common error is for the student to think that 14 and 14.0 are the same thing. THEY ARE NOT. 14.0 is ten times more precise than 14. The two numbers have the same value, but they convey different meanings about how trustworthy they are.
Four is also an incorrect answer given by some ChemTeam students. It is too many significant figures. One possible reason for this answer lies in the number 4.520. This number has four significant figures while 3.10 has three. Somehow, the student (YOU!) maybe got the idea that it is the GREATEST number of significant figures in the problem that dictates the answer. It is the LEAST.
Sometimes student will answer this with five. Most likely you responded with this answer because it says 14.012 on your calculator. This answer would have been correct in your math class because mathematics does not have the significant figure concept.
Example #4: 2.33 x 6.085 x 2.1. How many significant figures in the answer?
Answer - two.
Which number decides this?
Answer - the 2.1.
Why?
It has the least number of significant figures in the problem. It is, therefore, the least precise measurement.
Example #5: (4.52 x 10¯4) ÷ (3.980 x 10¯6).
How many significant figures in the answer?
Answer - three.
Which number decides this?
Answer - the 4.52 x 10¯4.
Why?
It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. Notice it is the 4.52 portion that plays the role of determining significant figures; the exponential portion plays no role.
WARNING: the rules for add/subtract are different from multiply/divide. A very common student error is to swap the two sets of rules. Another common error is to use just one rule for both types of operations.
1) 3.0800
2) 0.00418
3) 7.09 x 10¯5
4) 91,600
5) 0.003005
6) 3.200 x 109
7) 250
8) 780,000,000
9) 0.0101
10) 0.00800
11) 3.461728 + 14.91 + 0.980001 + 5.2631
12) 23.1 + 4.77 + 125.39 + 3.581
13) 22.101 - 0.9307
14) 0.04216 - 0.0004134
15) 564,321 - 264,321
16) (3.4617 x 107) ÷ (5.61 x 10¯4)
17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)]. Watch your order of operations on this problem.
18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)]
19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]
1) 3.0800 - five significant figures. All the rules are illustrated by this problem. Rule one - the 3 and the 8. Rule Two - the zero between the 3 and 8. Rule three - the two trailing zeros after the 8.
2) 0.00418 - three significant figures: the 4, the 1, and the 8. This is a typical type of problem where the student errs by giving five significant figures as the answer.
3) 7.09 x 10¯5 - three significant figures. When a number is written in scientific notation, only significant figures are placed into the numerical portion. If this number were taken out of scientific notation, it would be 0.0000709.
4) 91,600 - three significant figures. The last two zeros are not considered to be significant (at least normally). Suppose you had information that showed the zero in the tens place to be significant. How would you show it to be different from the zero in the ones place, which is not significant? The answer is scientific notation. Here is how it would be written: 9.160 x 104. This CLEARLY indicates the presence of four significant figures.
5) 0.003005- four significant figures. No matter how many zeros there are between two significant figures, all the zeros are to be considered significant. A number like 70.000001 would have 8 significant figures.
6) 3.200 x 109 - four significant figures. Notice the use of scientific notation to indicate that there are two zeros which should be significant. If this number were to be written without scientific notation (3,200,000,000) the significance of those two zeros would be lost and you would - wrongly - say that there were only two significant figures.
7) 2
8) 2
9) 3
10) 3
11) 3.461728 + 14.91 + 0.980001 + 5.2631
12) 23.1 + 4.77 + 125.39 + 3.581
In each of these two problems, examine the decimal portion only. Find the number with the LEAST number of digits in the decimal portion. In problem 1 it is the 14.91 and in problem 2 it is 23.1.
That means problem 1 will have its answer rounded to the 0.01 place and problem 2 will have its answer rounded to the 0.1 place. The correct answers are 24.61 and 156.8.
13) 22.101 - 0.9307
The answer is 21.170. The first value in the problem, with three significant places to the right of the decimal point, dictates how many significant places to the right of the decimal point in the answer.
14) 0.04216 - 0.0004134
The answer is 0.04175.
15) 564,321 - 264,321
This problem is somewhat artifical. The correct answer is 300,000, BUT all of the significant figures are retained. The most correct way to write the answer would be 3.00000 x 105.
16) (3.4617 x 107) ÷ (5.61 x 10¯4)
The calculator shows 6.1706 x 1010 which then rounds to 6.17 x 1010 - three significant figures. The value which dictates this is in boldface.
17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)]
The calculator shows 1.3398 x 101 which then rounds to 13.40 - four significant figures. In this problem pay attention to order of operations, since division is not commutative. There are two ways to do this problem using the calculator: 1) multiply the last two numbers, put the result in memory, multiply the first two, then divide that by what is in memory or 2) multiply the first two numbers then do two divisions.
18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)]
The calculator shows 4.625 x 10¯22, which then rounds to 4.62 x 10¯22 - three significant figures. Notice the use of the rounding with five rule.
19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]
The calculator shows 2280.3972, which rounds off to 2280, three significant figures. In scientific notation, this answer would be 2.28 x 103.
Note this last use of scientific notation to indicate significant figures where otherwise you might not realize they were significant. For example, 2300 looks like it has only two significant figures, but you know (from the problem) it really has three. How do you show this. One way is to use scientific notation like this: 2.30 x 103. Now the 2.30 portion clearly has three significant figures.
Example #1: 12.10 X 3.0 = 36.3 = 36
4 s.f. 2 s.f. 2 s.f.
Example #2: 2.5 x 3.42.
The answer to this problem would be 8.6 (which was rounded from the calculator reading of 8.55). Why?
2.5 has two significant figures while 3.42 has three. Two significant figures is less precise than three, so the answer has two significant figures.
Example #3: How many significant figures will the answer to 3.10 x 4.520 have?
You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundedth's place is not recognized as significant when, in fact, it is. 3.10 has three significant figures.
Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.
Another common error is for the student to think that 14 and 14.0 are the same thing. THEY ARE NOT. 14.0 is ten times more precise than 14. The two numbers have the same value, but they convey different meanings about how trustworthy they are.
Four is also an incorrect answer given by some ChemTeam students. It is too many significant figures. One possible reason for this answer lies in the number 4.520. This number has four significant figures while 3.10 has three. Somehow, the student (YOU!) maybe got the idea that it is the GREATEST number of significant figures in the problem that dictates the answer. It is the LEAST.
Sometimes student will answer this with five. Most likely you responded with this answer because it says 14.012 on your calculator. This answer would have been correct in your math class because mathematics does not have the significant figure concept.
Example #4: 2.33 x 6.085 x 2.1. How many significant figures in the answer?
Answer - two.
Which number decides this?
Answer - the 2.1.
Why?
It has the least number of significant figures in the problem. It is, therefore, the least precise measurement.
Example #5: (4.52 x 10¯4) ÷ (3.980 x 10¯6).
How many significant figures in the answer?
Answer - three.
Which number decides this?
Answer - the 4.52 x 10¯4.
Why?
It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. Notice it is the 4.52 portion that plays the role of determining significant figures; the exponential portion plays no role.
WARNING: the rules for add/subtract are different from multiply/divide. A very common student error is to swap the two sets of rules. Another common error is to use just one rule for both types of operations.
Practice Problems
1) 3.0800
2) 0.00418
3) 7.09 x 10¯5
4) 91,600
5) 0.003005
6) 3.200 x 109
7) 250
8) 780,000,000
9) 0.0101
10) 0.00800
11) 3.461728 + 14.91 + 0.980001 + 5.2631
12) 23.1 + 4.77 + 125.39 + 3.581
13) 22.101 - 0.9307
14) 0.04216 - 0.0004134
15) 564,321 - 264,321
16) (3.4617 x 107) ÷ (5.61 x 10¯4)
17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)]. Watch your order of operations on this problem.
18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)]
19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]
1) 3.0800 - five significant figures. All the rules are illustrated by this problem. Rule one - the 3 and the 8. Rule Two - the zero between the 3 and 8. Rule three - the two trailing zeros after the 8.
2) 0.00418 - three significant figures: the 4, the 1, and the 8. This is a typical type of problem where the student errs by giving five significant figures as the answer.
3) 7.09 x 10¯5 - three significant figures. When a number is written in scientific notation, only significant figures are placed into the numerical portion. If this number were taken out of scientific notation, it would be 0.0000709.
4) 91,600 - three significant figures. The last two zeros are not considered to be significant (at least normally). Suppose you had information that showed the zero in the tens place to be significant. How would you show it to be different from the zero in the ones place, which is not significant? The answer is scientific notation. Here is how it would be written: 9.160 x 104. This CLEARLY indicates the presence of four significant figures.
5) 0.003005- four significant figures. No matter how many zeros there are between two significant figures, all the zeros are to be considered significant. A number like 70.000001 would have 8 significant figures.
6) 3.200 x 109 - four significant figures. Notice the use of scientific notation to indicate that there are two zeros which should be significant. If this number were to be written without scientific notation (3,200,000,000) the significance of those two zeros would be lost and you would - wrongly - say that there were only two significant figures.
7) 2
8) 2
9) 3
10) 3
11) 3.461728 + 14.91 + 0.980001 + 5.2631
12) 23.1 + 4.77 + 125.39 + 3.581
In each of these two problems, examine the decimal portion only. Find the number with the LEAST number of digits in the decimal portion. In problem 1 it is the 14.91 and in problem 2 it is 23.1.
That means problem 1 will have its answer rounded to the 0.01 place and problem 2 will have its answer rounded to the 0.1 place. The correct answers are 24.61 and 156.8.
13) 22.101 - 0.9307
The answer is 21.170. The first value in the problem, with three significant places to the right of the decimal point, dictates how many significant places to the right of the decimal point in the answer.
14) 0.04216 - 0.0004134
The answer is 0.04175.
15) 564,321 - 264,321
This problem is somewhat artifical. The correct answer is 300,000, BUT all of the significant figures are retained. The most correct way to write the answer would be 3.00000 x 105.
16) (3.4617 x 107) ÷ (5.61 x 10¯4)
The calculator shows 6.1706 x 1010 which then rounds to 6.17 x 1010 - three significant figures. The value which dictates this is in boldface.
17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)]
The calculator shows 1.3398 x 101 which then rounds to 13.40 - four significant figures. In this problem pay attention to order of operations, since division is not commutative. There are two ways to do this problem using the calculator: 1) multiply the last two numbers, put the result in memory, multiply the first two, then divide that by what is in memory or 2) multiply the first two numbers then do two divisions.
18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)]
The calculator shows 4.625 x 10¯22, which then rounds to 4.62 x 10¯22 - three significant figures. Notice the use of the rounding with five rule.
19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]
The calculator shows 2280.3972, which rounds off to 2280, three significant figures. In scientific notation, this answer would be 2.28 x 103.
Note this last use of scientific notation to indicate significant figures where otherwise you might not realize they were significant. For example, 2300 looks like it has only two significant figures, but you know (from the problem) it really has three. How do you show this. One way is to use scientific notation like this: 2.30 x 103. Now the 2.30 portion clearly has three significant figures.
