 # Significant Figures Chemistry Tutorial

## Key Concepts

• Significant figures, or significant digits, establish the value of a number.
• Zeros shown merely to locate a decimal point are NOT significant figures

 NOT significant 0.00 148 3 significant figures

• Zeros located to the right of another number after a decimal point are significant

 NOT significant 0.00 1480000 7 significant figures

• The last significant figure on the right is the one which is somewhat uncertain (even if it is 0)

 certain 0.0014 8 uncertain

• An exact number, such as the number of objects counted, can be considered to have an infinite number of zeros after the decimal point, all of which are significant

3 atoms = 3.0 atoms = 3.00 atoms = 3.000 atoms etc

• It is impossible to tell how many significant figures are in a large number with zeros to the left of the decimal point without expressing the number in scientific notation

1,250 could be expressed as :

1.25 × 103 (3 significant figures, "5" is uncertain)

or as

1.250 × 103 (4 significant figures, "0" is uncertain)

• To find the number of significant figures in a given number:

⚛ count all the digits starting at the first non-zero digit on the left

⚛ for a number written in scientific notation count only the digits in the coefficient

• When adding or subtracting numbers, the number of digits to the right of the decimal point in the result should be the same as the number of digits to the right of the decimal point in the number with the fewest digits to the right of the decimal point

 1.23 + 0.123 = 1.35 (2 digits) (3 digits) (2 digits) 1.23 - 0.123 = 1.11

• When multiplying or dividing numbers, the number of significant figures in the result is the same as the least number of significant figures in any of the multiplied or divided terms

 1.234 × 0.123 = 0.152 (4 sig fig) (3 sig fig) (3 sig fig) 1.234 ÷ 0.123 = 10.0

• While performing calculations use at least one more significant digit than is required until the final answer is obtained, then round the answer up or down to achieve the correct number of significant figures.

Rounding off to the last significant figure at each step of a multi-step calculation can introduce rounding errors in the final calculation.

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## Finding the Number of Significant Figures in a Given Number

The number of significant figures in a given number is a shortcut used to tell the reader the precision with which you know this number.
If you use a ruler with gradations in millimetres to measure the length of something a bit more than 3 mm long, then you are certain of the "3", but you can only guess the fraction of a millimetre after that, say 3.2 mm. The "2" is somewhat uncertain, your friend might argue that the measurement is really 3.1 mm, and someone else might measure it as 3.3 mm. We include this uncertain number in the number of significant figures, but, our readers know that it is uncertain. If our ruler had even finer divisions, say tenths of a millimetre, then we might be certain that our object is a bit more than 3.2 mm long, so we guess the next number, 3.27, and the "7" becomes the uncertain number in a number expressed to 3 significant figures.

The general rules for determining the number of significant figures in a given number are as follows:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.
2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.
• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.
• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

### Finding the Number of Significant Figures Worked Examples

Determine the number of significant figures in 5 mL

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

Not applicable : this is a measurement of volume

2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.

 5 most significant

• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

 5 most significantleast significant

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.

Not applicable : no decimal point

• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

1 significant figure

Determine the number of significant figures in 5.2 g

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

Not applicable : this is a measurement of mass

2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.

 5 .2 most significant

• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

Not applicable : decimal point is present

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.

 least significant 5 . 2 most significant

• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

2 significant figures

Determine the number of significant figures in 5.0 kg

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

Not applicable : this is a measurement of mass

2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.

 5 .0 most significant

• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

Not applicable : decimal point is present

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.

 least significant 5 . 0 most significant

• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

2 significant figures

Determine the number of significant figures in 5.000 L

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

Not applicable : this is a measurement of volume

2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.

 5 .000 most significant

• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

Not applicable : decimal point is present

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.

 least significant 5 .00 0 most significant

• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

 most significant least significant 5 .00 0 1, 2,3 ,4

4 significant figures

Determine the number of significant figures in 0.005 m

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

Not applicable : this is a measurement of length

2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.

 0.00 5 most significant

• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

Not applicable : decimal point is present

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.

 0.00 5 most significantleast significant

• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

1 significant figure

Determine the number of significant figures in 5 football players

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

This a counted number of people so it is an exact number.

Infinite number of significant figures

2. For an inexact number such as a measurement:

Not applicable: this is an exact number

• The leftmost digit which is not a zero is the most significant digit.
• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.
• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

Determine the number of significant figures in 500 mm

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

Not applicable : this is a measurement of length

2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.

 5 00 most significant

• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

The rightmost number is a zero so we cannot determine the number of significant figures.

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.
• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

Determine the number of significant figures in 5.00 × 103 g

Solution:

1. An exact number (a counted number of items) is considered to have an infinite number of significant figures.

Not applicable : this is a measurement of mass

2. For an inexact number such as a measurement:

• The leftmost digit which is not a zero is the most significant digit.

 5 .00 × 103 most significant

• If the number does not have a decimal point then the rightmost digit is the least significant digit unless it is a zero.

If the rightmost number is a zero it is not possible to determine the number of significant figures and the number should be expressed in scientific notation.

Not applicable : decimal point is present

• If the number does have a decimal point, the rightmost digit is the least significant digit, even if it is a zero.

 most significant least significant 5 .0 0 × 103

• Every digit between the least significant digit and the most significant digit should be counted as a significant digit.

 most significant least significant 5 .0 0 × 103 1 ,2 ,3

3 significant figures

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## Determining the Number of Significant Figures in the Result of an Addition or Subtraction Calculation

When you add two numbers together you are also adding together their uncertainty.
1234.5 + 0.0006
The larger number has 5 significant figures with the last digit being uncertain.
The smaller number has only 1 significant figure which is itself somewhat uncertain.
If I write the answer as 1234.5006 the only part of the number which is certain is 1234. The "5" is already uncertain, so the "6" has no effect on the level of certainty and uncertainty in the final result so the answer should be expressed as 1234.5
A similar result would be found for the substraction of two numbers.
Therefore, when numbers are added or subtracted the number of significant figures in the final answer is based on the number in the calculation that had the fewest digits after the decimal point.

To determine the number of significant figures after an addition or subtraction calculation:

• Determine the number of digits after the decimal point in each number to be added or subtracted.
• Select the fewest number of digits after the decimal point to apply to the final result.
• Perform the calculation
• Round the result up or down in order to achieve the correct number of digits after the decimal point.

### Worked Example of Significant Figures After Addition Calculations

Question 1: 12.47 g + 7 g = ? g

Solution:

• Determine the number of digits after the decimal point in each number to be added or subtracted.

12.47 has 2 digits after the decimal point

7 has no digits after the decimal point

• Select the fewest number of digits after the decimal point to apply to the final result.

Final result will have no digits after the decimal point

• Perform the calculation

12.47 + 7 = 19.47

• Round the result up or down in order to achieve the correct number of digits after the decimal point.

19.47 rounded down to 19 (no digits after the decimal point)

Question 2: 12300 g + 1.23 g = ? g

Solution:

• Determine the number of digits after the decimal point in each number to be added or subtracted.

12300 has 0 digits after the decimal point, but we do not know how certain the 3 or the zeroes are!

1.23 has 2 digits after the decimal point

• Select the fewest number of digits after the decimal point to apply to the final result.

Final result will have no digits after the decimal point

• Perform the calculation

12300 + 1.23 = 12301.23

• Round the result up or down in order to achieve the correct number of digits after the decimal point.

Assuming the zeroes were significant: 12301.23 rounded down to 12301 (no digits after the decimal point)

Assuming the zeroes were NOT significant: 12301.23 rounded down to 12300 (no digits after the decimal point)

Best Answer: 12300 g (assuming the zeroes were NOT significant)

### Worked Example of Significant Figures After Subtraction Calculation

Question 1: 32.56 mm - 4.9 mm = ? mm

Solution:

• Determine the number of digits after the decimal point in each number to be added or subtracted.

32.56 has 2 digits after the decimal point

4.9 has 1 digit after the decimal point

• Select the fewest number of digits after the decimal point to apply to the final result.

Final result will have 1 digit after the decimal point

• Perform the calculation

32.56 - 4.9 = 27.66

• Round the result up or down in order to achieve the correct number of digits after the decimal point.

27.66 rounded up to 27.7 (1 digit after the decimal point)

Question 2: 13.7 mL - 1.3 mL = ? mL

Solution:

• Determine the number of digits after the decimal point in each number to be added or subtracted.

13.7 has 1 digit after the decimal point

1.3 has 1 digit after the decimal point

• Select the fewest number of digits after the decimal point to apply to the final result.

Final result will have 1 digit after the decimal point

• Perform the calculation

13.7 - 1.3 = 12.4

• Round the result up or down in order to achieve the correct number of digits after the decimal point.

12.4 (does not need rounding up or down because it has 1 digit after the decimal point)

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## Determining the Number of Significant Figures in the Result of a Multiplication or Division Calculation

When you multiply two numbers you are also multiplying their uncertainties.

Consider a common calculation in chemistry where you multiply a volume in litres by the concentration of solution in moles per litre:
0.02518 L × 1.03 mol L-1
There is uncertainity in the "8" of 0.02518 L (could it be 0.02517 or 0.02519?)
There is uncertainty in the "3" of 1.03 mol L-1 (could it be 1.02 or 1.04?)
Let's multiply the 4 combinations of "high" and "low" values to see what we get

 0.02517 × 1.02 = 0.0256734 lowest value 0.02517 × 1.04 = 0.0261768 0.02519 × 1.02 = 0.0256938 0.02519 × 1.04 = 0.0261976 highest value

How many significant figures should we use?
In the above calculations it seems only the fist non-zero number is certain!
But just how uncertain is the next digit after that? Should it be a "5" or a "6" ?
If we rounded off all four of the numbers they would all round out to 0.026, so maybe this second digit is also quite certain.
That means the third digit is determining the value of the second digit and is therefore the least certain and least significant digit so we will use 3 significant numbers in our final calculation:

0.02518 × 1.03 = 0.0259

You will notice that we have used the same number of significant figures as were present in the number with fewest significant figures, that is, 1.03 has 3 significant figures compared to 0.02518 which has 4 significant figures.

In general, when we multiply or divide numbers we use the same number of significant figures in the result as were present in the number with fewest significant figures.

To determine the number of significant figures in the result of a multiplication or division calculation:

• Count the number of significant figures in each number.
• Choose the least number of significant figures to apply to the result of the calculation.
• Perform the calculation.
• Round the result up or down to the correct number of significant figures.

### Worked Examples of Number of Significant Figures After a Multiplication Calculation

Question 1: 4.1 × 103 g mol-1 × 8.635 × 102 mol = ? g

Solution:

• Count the number of significant figures in each number.

4.1 × 103 has 2 significant figures

8.635 × 102 has 4 significant figures

• Choose the least number of significant figures to apply to the result of the calculation.

2 significant figures

• Perform the calculation.

4.1 × 103 × 8.635 × 102 = 3540350 = 3.540350 × 106

• Round the result up or down to the correct number of significant figures.

3.540350 × 106 rounded down to 2 significant figures is 3.5 × 106

Question 2: 2.00 mol L-1 × 10.14 L = ? mol

Solution:

• Count the number of significant figures in each number.

2.00 has 3 significant figures

10.14 has 4 significant figures

• Choose the least number of significant figures to apply to the result of the calculation.

3 significant figures

• Perform the calculation.

2.00 × 10.14 = 20.28

• Round the result up or down to the correct number of significant figures.

20.28 is rounded up to 20.3 so that the answer has 3 significant figures.

### Worked Examples of Number of Significant Figures After a Division Calculation

Question 1: 1.473 g ÷ 2.6 g mol-1 = ? mol

Solution:

• Count the number of significant figures in each number.

1.473 has 4 significant figures

2.6 has 2 significant figures

• Choose the least number of significant figures to apply to the result of the calculation.

2 significant figures

• Perform the calculation.

1.473 ÷ 2.6 = 0.566538461 = 5.66538461 × 10-1

• Round the result up or down to the correct number of significant figures.

5.66538461 × 10-1 is rounded up to 5.7 × 10-1 so that the answer has 2 significant figures.

Answer: 5.7 × 10-1 mol or 0.57 mol

Question 2: 29.5 g ÷ 13.1 L = ? g L-1

Solution:

• Count the number of significant figures in each number.

29.5 has 3 significant figures

13.1 has 3 significant figures

• Choose the least number of significant figures to apply to the result of the calculation.

3 significant figures

• Perform the calculation.

29.5 ÷ 13.1 = 2.251908397

• Round the result up or down to the correct number of significant figures. `

2.251908397 is rounded down to 2.25 to give an answer with 3 significant figures.