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# Lesson 1.5: Uncertainty in Measurement

← Lesson 1.4: Lesson 1.6: →
• 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? • 1.25 g
• 1.250 g
• 0.0125 g
• 1.25 x 103 g
• 125 000 g
• 1.25 x 105 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 cm3?

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