Calculations with Significant Figures

Significant Figures (Sig Figs)

How do we account for uncertainty in measurement?

Significant Figures and How To Determine Them

Why do we need significant figures?

For some reason, teachers never really tell students why significant figures are important. Significant figures are important because they tell us how good the data we are using are. (Grammar note: the word “data” is plural for “datum.)

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For example, consider the following three numbers:

100 grams

100. grams

100.00 grams

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■ The first number has only one significant figure (namely, the “1” in the beginning). Because this digit is in the “hundreds” place, this measurement is only accurate to the nearest 100 grams (i.e., the value of what we’re measuring is closer to 100 grams than it is to 200 grams or 0 grams).

■ The second number has three significant figures (the decimal makes all three digits significant, as we’ll discuss later). Because the last significant figure is in the “ones” place, the measurement is accurate to the nearest gram (i.e., the value of what we’re measuring is closer to 100 grams than it is to 101 grams or 99 grams).

■ The third number has five significant figures (as we’ll talk about later). Because the last significant figure is in the “hundredths” place, the measurement can be considered to be accurate to the nearest 0.01 grams (i.e., the value of what we’re measuring is closer to 100.00 grams than it is to 100.01 or 99.99 grams).

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In short, when you plug these three numbers into your calculator, there’s no difference in how the calculator will manipulate them – your calculator neither knows nor cares about how good the numbers it’s working with are. However, to you, the scientist, these three numbers tell you whether or not your data is good enough to pay attention to.​

​Significant Figures (or Sig. Fig.) Rules

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These are the 'official' and confusing rules: 

Rule 1:  All non-zero numbers are significant.

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Rule 2:  Any zeros that are between two non-zeros are significant.

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Rule 3:  Any zeros before all of the non-zero digits are insignificant

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Rule 4:  Any zeros after all of the non-zero digits are significant, only if there is a decimal point

​Here is the easy way to determine sig. figs. :

​Pacific

​Ocean

​Atlantic

​Ocean

​Once you have familiarized yourself with what you're looking at, continue to next slide...

​The Atlantic / Pacific Rule

Watch the video to learn about the easy "Atlantic / Pacific Rule" for ​determining sig figs.

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​NOTE: don't get too caught up in the video info from 1:00 to 4:04 (that is the time when the confusing rules are being discussed). The info after 4:04 is the key stuff about the Atlantic / Pacific Rule.

Multiple Choice

0.0097 has how many sig figs?

• 2

• 3

• 4

• 5

Multiple Choice

How many sig figs are in 1010?

• 1

• 2

• 3

• 4

Multiple Choice

Identify the number of sig figs in 0.0810

• 2

• 3

• 4

• 5

​Manipulating Significant Figures

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Addition or Subtraction –

When adding or subtracting measurements with decimals, the answer must have the same number of decimals (digits to the right of the decimal point) as there are in the measurement having the fewest decimals.

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Ex., 25.1 g + 2.03 g = 27.13 g (this is what a calculator would display)

The proper way to represent the answer is 27.1 g because the measure of 25.1 g has the fewest decimals (one).

Multiple Choice

12.03 + 43.5 = ?

Express your answer to the correct place value.

• 55.530

• 55.53

• 55.5

• 56

​Manipulating Significant Figures

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Multiplication and Division –

For multiplication and division, the answer can have no more significant figures than are in the measurement with the fewest significant figures.

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Ex., 3.050 g / 8.47 mL = 0.360094451 g/mL, (this is what a calculator would display)

The proper way to represent the answer is 0.360 g/mL because the measure of 8.47 mL has the fewest sig figs (three), and 0.360 has three sig figs.

Multiple Choice

$$7.80\div0.091=\ ?$$   Round your answer to the correct number of sig figs.

• 85.71429

• 85.71

• 85.7

• 86

​Why do we need to round this way?

​In science we work with measured values, which are inherently uncertain no matter how sensitive our measuring tools are. Performing calculations with measured values doesn't take the uncertainty away, so we must round our answers to accurately reflect the uncertainty in the measurements.