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Significant Figures and Rounding

Significant figures are the digits used to represent the precision of a measured or

calculated quantity. Below is a summary of the rules for significant figures.

Rules:

1. For addition or subtraction, the answer should contain the same number of decimal places as the term with the fewest decimal places.

Example:

161.032

5.6 ( Number with fewest digits after the decimal

+ 32.4524

199.0844 ( 199.1 (1 digit to the right of the decimal)

2. For multiplication or division, the answer should contain no more significant figures

than the term with the fewest significant figures.

Example:

152.06 x 0.24 = 36.49 ( 36 (2 significant digits)

Contains fewest number

of significant figures

3. In multi-step calculations, you may round at each step or round at the end only.

4. Exact numbers, such as integers, are treated as if they had infinitely many significant figures.

5. Round 5's to the nearest even number when all the subsequent digits are zero. If

the 5 is followed by one or more non-zero digits, round up.

1. Significant figures

How many significant figures are there in the following numbers?

a. 12.3456

Answer: 6 significant figures (All the digits are significant.)

b. 0.005030

Answer: 4 significant figures (The zeros preceding the first nonzero digit are place holders and are used to locate the decimal place.)

c. 6000

Answer: Confusing. This number has 1 significant figure as given (i.e., without a decimal after the last zero. Another way to express this number is in scientific notation; e.g., 6.000 x 103 has 4 significant figures, and 6 x 103 has 1 significant figure.

Note: In this class, this will be made clear on exams by using scientific notation or a decimal (i.e., 6000.).

2. The difference between significant figures and decimal places

Addition

5.05 x 10-3 + 1.008734

5.05 x 10-3 has 3 significant figures but if you write it out (0.00505) it has 5

decimal places. This could affect how many places you are allowed to report in your

answer.

1.008734

+ 0.00505

1.013784

We are allowed to report only to the 5th decimal place so the correct answer is

1.01378

Multiplication

(5.05 x 10-3) x (1.008734) = 0.005094107

Since we are doing multiplication, we use the number with the least number of

significant figures, so the correct answer is 0.00509 or 5.09 x 10-3

3. The correct method of rounding for addition:

Carry out arithmetical operations and THEN round the final answer to the correct

# of significant figures instead of rounding the input data first.

Correct: 15 m incorrect: 15 m ( 15 m

+ 6.6 m 6.6 m ( 7m

+ 12.6 m 12.6 m ( 13 m

34.2 ( 34 m 35 m

4. Rounding

Rounding is the procedure of dropping non-significant digits in a calculation result

and adjusting the last digit reported. Look at the leftmost digit to be dropped:

1. If digit is greater than 5, add 1 to the last digit to be retained and drop all the digits farther to the right. Thus, rounding 1.2161 to 3 significant figures gives 1.22

2. If this digit is less than 5, simply drop it and all digits farther to the right.

Rounding 1.2143 to 3 significant figures gives 1.21

3. If the first non-significant number to the right is 5 and it is followed by non-zero digits, add 1 to the last digit to be retained and drop all the digits farther to the right.

Rounding 1.2256 to 3 sig figs gives 1.23 (round up because the 5 is followed

by a non-zero digit).

Examples of rounding to the correct number of significant figures with a 5 as

first non-significant figure

Round 4.7475 to 4 significant figures: 4.7475 is 4.748 because the first non-

significant digit is 5, and we round the last significant figure up to 8 to make it

even.

Round 4.7465 to 4 significant figures: 4.7465 is 4.747

According to these rules, 1.59 rounded to 1 significant figure is 2

2.59 rounded to 1 significant figure is 3

1.5 rounded to 1 significant figure is 2

5. Ways to avoid round-off error

If you are solving a problem where you add or subtract in one step and multiply or divide in another step, you must follow the rules for determining the number of significant figures before you move on to an operation with another set of rules. For multiplication and division, use the value with the least number of significant figures to determine the number of significant figures in your answer. For addition and subtraction, the number with the least number of decimal places puts a limit on the number of significant figures you are allowed to legitimately report in the answer.

It is wise to keep more significant figures than needed, keeping track of the number of significant figures or least number of decimal places that you are allowed before changing operations. Then, when you get a final answer, examine how many significant figures are allowed and do your final rounding.

This will help you avoid "round-off error."

6. Example of a multiple-step problem where you can "lose" sig figs by doing an operation. NOTE: For the purpose of this example, we are assuming that none of the values given are "exact numbers.”

The mass of 19F is 18.99840 u. How much mass is converted to energy when a 19F atom is assembled from its constituent protons, neutrons, and electrons?

19F ( 9 p+ + 9 e- + 10 n

integer; has ( SF

⎢

9(1.00728 u) + 10(1.00867 u) + 9(5.48580 x 10-4 u) =

⎡ ⎡ ⎡

6 SF 6 SF 6 SF

5 decimal 5 decimal 9 decimal

places places places

9.06552 u + 10.0867 u + 4.93722 x 10-3 u =

⎡ ⎡ ⎡

still 6 SF 6 SF 6 SF

5 decimal 4 decimal 8 decimal

places places places

9.06552

+ 10.0867

0.00493722

19.15715722

We are only allowed 4 decimal places because 10.0867 had only 4 decimal places.

Therefore, we need to round. The digit to right of 1 is 5, which is followed by non-

zero digits so we round up.

The rounded answer is19.1572 u

19.1572 u (sum of p+, n, and e-)

- 18.99840 u (mass of 19F given)

0.15880 u allowed 4 decimal places, so 0.1588 u is mass loss

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