- Numbers that result from counting or are based on definitions are exact.
- Numbers that result from measurements are inexact.
- Numbers from measurements are recorded in such a way that the last digit is uncertain.
- If a number has leading zeros they are not significant.
- For example, 0.023 g has two significant figures, the 2 which is certain, and the 3 which is uncertain but significant.
- If a number does not have a decimal the trailing zeros are not significant.
- For example, 2300 g has two significant figures, the 2 which is certain, and the 3 which is uncertain but significant.

##
Example 1: How would you record the length of the red line? How many significant figures does the measurement contain?

##
Example 2: How would you record the length of the red line? How many significant figures does the measurement contain?

##
Example 3: How would you record the length of the red line? How many significant figures does the measurement contain?

##
Example 4: How would you record the volume of the liquid used? How many significant figures does the measurement contain?

##
Example 5: How many significant figures are in the following?

- 1.25 g
- 1.250 g
- 0.0125 g
- 1.25 x 10
^{3} g
- 125 000 g
- 1.25 x 10
^{5} g

## Significant Figures in Calculations

There are two rules, one for multiplication and another for addition.
Focus on the rule for multiplication – this is what we use most of the time.

## The rule for multiplication

The result of your calculation should be no more precise than the measurement with the least number of significant figures.

##
Example 6: What is the density of an object that has a mass of 2.46 g and a volume of 2.0 ml?

$$
\begin{align*}
density &= \frac{mass}{volume}\\\\
density &= \frac{2.46 \, g}{2.0 \, ml} \\\\
& = 1.23 \, g/ml \\\\
& = 1.2 \, g/ml
\end{align*}
$$

##
Example 7: What is the average velocity of an object that covers 125.0 m in 3.2 s?

$$
\begin{align*}
velocity &= \frac{distance}{time}\\\\
velocity &= \frac{125.0 \, m}{3.2 \, s} \\\\
& = 39.0625 \, m/s \\\\
& = 39 \, m/s
\end{align*}
$$

##
Example 8: What is the density of an object that has a mass of 0.124 g and a volume of 0.100 cm^{3}?

$$
\begin{align*}
density &= \frac{mass}{volume}\\\\
density &= \frac{0.124 \, g}{0.100 \, cm^3} \\\\
& = 1.24 \, g/cm^3
\end{align*}
$$

## The rule for addition

The result of your calculation should have the same number of decimal places as the least precise measurement.

##
Example 9: What is the sum of three measures made with different rulers: 12.44 cm + 1.686 cm + 15.1 cm?