Significant Figures

The Idea Behind Significant Figures

Every measurement you make in a laboratory carries some uncertainty. The source of this uncertainty is twofold: the limitations of the instrument you are using, and the skill of the person making the measurement. Consider a simple example. You weigh an object on a platform balance and get a reading of 9.4 g. That same object, when placed on a much more sensitive analytical balance, reads 9.4213 g. Which number is "correct"? Both are correct within the limits of the instruments used. The first measurement tells you the mass is between 9.3 g and 9.5 g. The second tells you it is between 9.4212 g and 9.4214 g. The second measurement is more precise, but both are valid.

The digits that carry meaningful information about a measurement are called significant figures. They are all the digits that are known with certainty, plus the first digit that is uncertain. In the platform balance reading of 9.4 g, the '9' is certain and the '4' is uncertain — so there are two significant figures. In the analytical balance reading of 9.4213 g, the '9', '4', '2', and '1' are certain, and the '3' is uncertain — so there are five significant figures.

Rules for Determining Significant Figures

The textbook states these as numbered rules (1)–(5); here they are, in the book's own order, each with an example.

(1) All non-zero digits are significant.

For example, 285 cm has three significant figures and 0.25 mL has two.

(2) Zeros preceding the first non-zero digit are not significant.

Such zeros only indicate the position of the decimal point. Thus 0.03 has one significant figure and 0.0052 has two.

(3) Zeros between two non-zero digits are significant.

Thus 2.005 has four significant figures.

(4) Zeros at the end or right of a number are significant, provided they are on the right side of the decimal point.

For example, 0.200 g has three significant figures. But if there is no decimal point, the terminal zeros are not significant: 100 has only one significant figure, while 100. has three and 100.0 has four. Such numbers are better represented in scientific notation — 100 can be expressed as $1 \times 10^2$ (one significant figure), $1.0 \times 10^2$ (two) or $1.00 \times 10^2$ (three).

(5) Counting the numbers of objects — for example 2 balls or 20 eggs — gives exact numbers with infinite significant figures, since these can be represented by writing an infinite number of zeros after placing a decimal: 2 = 2.000000… and 20 = 20.000000…

In numbers written in scientific notation, all digits are significant: $4.01 \times 10^2$ has three significant figures and $8.256 \times 10^{-3}$ has four.

Precision and Accuracy

These two terms are often confused, but they mean different things.

Precision refers to how close repeated measurements of the same quantity are to each other. If you measure the same object five times and get 2.01 g, 2.00 g, 2.02 g, 1.99 g, and 2.01 g, your measurements are precise — they cluster tightly together.

Accuracy refers to how close a measurement is to the true value. If the true mass is 2.00 g, and your measurements average to 2.00 g, your measurements are accurate.

The textbook illustrates this with a clear example involving three students.

The Three Students

Student A's measurements are close to each other (precise) but far from the true value of 2.00 g (not accurate). Student B's measurements are far from each other (not precise) and also far from the true value (not accurate). Student C's measurements are close to each other (precise) and close to the true value (accurate).

Operations with Significant Figures

When you perform calculations with measured quantities, the result cannot be more precise than the least precise measurement you started with. The textbook gives specific rules for addition/subtraction and for multiplication/division.

Addition and Subtraction

The rule is about decimal places, not significant figures.

Consider this addition:

The number 18.0 has only one digit after the decimal point. Therefore, the result must be rounded to one digit after the decimal point. The correct answer is 31.1.

Multiplication and Division

The rule here is about the total number of significant figures.

Consider this multiplication:

The number 2.5 has two significant figures. The number 1.25 has three significant figures. The measurement with the fewest significant figures is 2.5, with two. Therefore, the result must be rounded to two significant figures. The correct answer is 3.1.

Rounding Off Numbers

When you limit a result to the required number of significant figures, you must round it properly. The textbook gives three specific rules for rounding.

Rule 1: If the rightmost digit to be removed is more than 5, increase the preceding digit by one. …