Rounding and Significant Figures

INTRODUCTION

Often in science, the numbers we deal with come from measurements of physical quantities. The idea of significant figures (“sig figs”) is a method of accounting for error in measurement. The number of significant figures in a physical quantity is the number of digits that are estimated with some reliability.

COUNTING SIG FIGS

The following rules are used to help determine the number of significant figures:

• All non-zero figures are significant (e.g., 46.7 has 3 sig figs).
• Zeros at the beginning of a number are not significant (e.g., 0.0045 has 2 sig figs). 
• Zeros within a number are significant (e.g., 30.6 has 3 sig figs).
• Zeros at the end of a number after the decimal point are significant (e.g., 38.600has 5 sig figs).

For whole numbers without a decimal, zeros at the end may or may not be significant. Scientific notation can be used to make the number of significant figures clear. For example, the number 53200 could have 3, 4, or 5 significant figures. If it’s written in scientific notation as 5.32 x 10⁴, then it has 3 significant figures. However, if it’s written in scientific notation as 5.320 x 10⁴, then it has 4 significant figures.

Example: 

Determine the number of significant figures in each of the following. 

a) \(4.30 \times 10^4\)

b) \(0.003011\)

c) \(3.7\)

Solution:

 a) 3 significant figures–the zero is significant because it is included when the number is written in scientific notation

b) 4 significant figures–the zeros at the beginning are not significant, but the zero in the middle of the number is significant

c) 2 significant figures–both figures are significant

SIG FIG RULES WHEN ADDING/SUBTRACTING/MULTIPLYING/DIVIDING

Now that we know what sig figs are, it is important to know how to use them when performing different operations.

Here are the rules you’ll need:

• When adding or subtracting, the final answer has the same number of decimal places as the number in the question with the least number of decimal places.
• When multiplying or dividing, the final answer has the same number of sig figs as the number in the question with the least number of sig figs.

Often times, you’ll be asked to round to the correct number of sig figs, so we’ll practice this as well in the next examples!

Example: Evaluate \(25 \times 13\).

Solution:

\(25 \times 13 = 325\)

Since both numbers from the question only had 2 significant figures each, the solution can only have 2 significant figures.

Answer: \(3.2 \times 10^2\)

Example: Evaluate \(3.257+27.34+82.1\). 

Solution:

We first add the numbers together:\(3.257 + 27.34 +82.1 = 112.697\)

Now, to figure out the correct number of sig figs to use in the answer, notice that the number with the fewest decimal places in the question is 82.1, which has 1 decimal place… therefore, our final answer should also have 1 decimal place.

 Answer: 112.7

Rounding Rules:

In the previous two examples, the solutions were rounded to the required number of significant figures. The following are the rules to use when determining how to round.

• If the leftmost digit is less than 5, the preceding digit remains the same
• If the leftmost digit dropped is 5 or more, the preceding digit rounds up

Note: If the leftmost digit dropped is 5 and there are no more digits following it, then we round up only if the number preceding the leftmost digit is odd; e.g., 24.45 becomes 24.4 while 24.35 also becomes 24.4. Of course, 24.451 would be rounded up to 24.5, as we’d expect from the above rules, since the leftmost digit that is dropped (5) is followed by other digits.

Example: 

Round 34.5736 to three significant figures.

Solution:

 The first digit being dropped is the 7. We know \(7 > 5\).

\(\therefore\) the preceding digit is rounded up by one.

Answer: 34.6

Example: 

Round 3.0025 to four significant figures.

Solution:

The first digit to be dropped is 5.

The digit preceding 5 is even.

\(\therefore\) the preceding digit is stays the same.

Answer: 3.002

Rounding and Scientific Notation Example:

Multiplication and Scientific Notation Example:

Addition and Scientific Notation Example:

Division and Scientific Notation Example: