Lecture 1.2: Significant Figures

Lecture 1.2: Significant Figures

By Joshua Baltzell

Uncertainty in Data

How do we know the values we're given?

Let's say you are doing an experiment involving a piece of metal. You have a scale that measures the mass of the metal at 5.1 g. Your friend borrows the metal and uses a different scale to measure the mass. They find a mass of 5.082 g.

Did the metal lose mass?​

What happened to cause the mass of the metal to be different between the two scales?

Open Ended

Explain what you believe caused the mass of the metal to be different between the two scales.

Precision with Instruments

Instruments we use in labs all have a measurement of precision associated with them. In our example, the piece of metal never changed mass. Instead, the scales had different levels of precision in how they measured. When dealing with measured quantities, the last digit is considered uncertain due to precision.​

Why would the last digit be considered uncertain with measured quantities?

Application of Example

The two scales had to report data to you based on a piece of metal. Let's pretend that we know the exact mass of the piece of metal to be 5.082144... g.

The first scale has a precision of 0.0 g, while the second has a precision of 0.000 g.​ This is seen in how many digits we can measure on the screen.

Now can you explain why the scales reported their numbers?

​Theoretical Value = 5.082144... g​

Scale's Precision = 0.000 g

Rounding = 5.082 g​

Scale 2

Theoretical Value = 5.082144... g​

Scale's Precision = 0.0 g

Rounding = 5.1 g​

​

Scale 1

How the scales rounded

Explain the statement: Measured quantities assume that the last digit is uncertain.

Open Ended

In your own words, explain what is meant by "measured quantities are generally reported in such a way that only the last digit is uncertain".

Significant Figures

Significant Figures are digits that tell us information about the precision of a number. The greater the number of significant digits, the greater the certainty of the measured quantity.

2.2 has two significant figures

2.4056 has five significant figures​

Subject | Subject

Rules for Significant Figures

Not all digits are significant. The rules for significant figures are shown below:

1. Zeroes between significant digits are always significant.

2. Zeroes at the beginning of a number are never significant.

3. Zeroes at the end of a number are significant only when the number contains a decimal point.​

If there is a decimal point...

Start counting on the left (Pacific Ocean)

​

If there is not a decimal point...​

Start counting on the right (Atlantic Ocean)

The US Method

​

Fill in the Blanks

Identify the number of significant figures...

52.3

Fill in the Blanks

Identify the number of significant figures...

52.00

Fill in the Blanks

Identify the number of significant figures...

0.0025

Fill in the Blanks

Identify the number of significant figures...

53.001

Math with Significant Digits

Core Idea

When performing calculations, it's important to use the appropriate number of significant digits in your answer. Your answer can only contain the same precision as the data points used to calculate it.

Can your answer have higher precision than the data points that are measured?

Addition and Subtraction

When performing addition and subtraction, you need to round your answer based on the digits that follow the decimal point. Your final answer can only be as precise as your least precise number.​ Always solve first, then round.

Subject | Subject

Some text here about the topic of discussion

20.42

1.322

+ 83.1

104.842​

Answer: 104.8​

​The ​red shows excess decimal points based on the number with the lowest precision.

Fill in the Blanks

Answer the question using the correct precision:

2.54 + 5.416

Fill in the Blanks

Answer the question using the correct precision:

12.6 - 5.84

Fill in the Blanks

Answer the question using the correct precision:

6.58 + 7.9862 - 2.145

Multiplication and Division

Multiplication and division depends on the significant digits in the entire measured quantity, not just the digits after the decimal point. ​Similarly, your final answer can only be as precise as your least precise number. Remember to solve first, then round your answer.

Subject | Subject

Some text here about the topic of discussion

​6.221 (4 S.F.)

x 5.2 (2 S.F.)

32.3492​

Answer: 32​

Fill in the Blanks

Answer the question using the correct precision:

2.43 x 0.2

Fill in the Blanks

Answer the question using the correct precision:

5.600 x 2.000

Mixed Mathematics

Sometimes you will need to switch between addition/subtraction and multiplication/division. When this occurs, it's important to round between the shifts.

Fill in the Blanks

Solve the following problem with the correct number of significant figures...

$\frac{32.451}{7.37}+11.42319$  

Fill in the Blanks

Solve the following problem with the correct number of significant figures...

$47.3-2.1776\cdot3.853$