Significant Figures
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Precision in Measurements
Significant Figures
When you make a measurement in a laboratory, there is
always some uncertainty that comes with that
measurement.
 By reporting the number with a particular number of
digits, you show how precise your measurement was.
 For example – 2 people determine the mass of a block of
iron. One is able to report it as 23 grams. The other is able
to tell us that it is 23.965 grams. Which number was more
precise?

Significant Figures and
Precision
The 2nd report of 23.965 grams is more precise
because it has more digits shown.
 The number of digits that you can use depends
on what you are measuring with.
 Some instruments are very precise (and
therefore let us report more digits) where as
some instruments are not precise (and
therefore let us report fewer digits)

Reporting Precision




How precise your number is depends entirely on what
instrument you use.
For example, think of using a ruler to measure a piece of string.
The more markings on our ruler, the more precise our
measurements can be.
What are the measurements from the following rulers?
decimetres
centimetres
millimetres
Precision and Significance
So we know that the more digits that are
present, the more precise our number is.
 This doesn’t mean that we can just add
more digits to make it more precise.
 We have to use rules to determine which
numbers are significant and how many
digits we can use

Significant Figures

In a reported number, there are rules to use to
tell which numbers are significant and which
aren’t.
All numbers that are not ZERO are significant. (1-9)
1.

2.
Example – the number 421 has 3 significant digits because it
has a 4, a 2, and a 1
Zeros that are in between significant figures are also
significant.


Example – in the number 4021, the zero is significant
because it is in between a 4 and a 2. This number has 4
significant figures.
In the number 0421, the zero is not significant. This number
has 3 significant figures.
Examples
How many significant
figures in each number?
1.
967
5. 0967.05
2.
9.67
6. 0032.004
3.
90.67
7. 0709.08
4.
9.607
Significant Figures
3.
Zeros after (to the right of) a decimal
point and after (to the right of a number)
are significant.


4.
Example – in the number 42.0, the zero is
significant. This number has 3 significant figures.
In the number 42.10000, all of the zeros are
significant. This number has 7 significant figures
A zero before a decimal is not significant

Example – in the number 0.421, the zero is not
significant. This number has 3 significant figures.
Examples
How many sig figs?
1.
3.20
5. 00.67
2.
60.0320
6. 0.4080
3.
06.200
7. 0.400000100
4.
0.5607
8. 0.010
Significant Figures
5. Zeros to the right of a decimal but before the
number are not significant

Example – In the number 0.000421, all of the zeros are not
significant. This number has 3 significant figures.
6. Zeros after the number but before the decimal
can be confusing as they might be significant and
they might not be significant.


Example – In the number 421000, it could be read as
having 3, 4, 5, or even 6 significant digits. It is impossible
to tell.
Numbers such as this should be written in scientific
notation to avoid confusion
Examples
How many sig figs?
Rewrite these numbers into proper
scientific notation.
1.
0.0097
5. 07.00420
2.
0.0200
6. 0.90007
3.
4300
7. 0.003420
4.
52.0067
8. 0.001
Adding and Subtracting
Rules
As well as having rules for reporting the number
of significant figures, we also have rules to tell us
how to add and subtract using significant
figures.
 When adding and subtracting using significant
figures, your answer should have the same
number of decimal places as the number with the
least number of decimal places.

Example
3.461728
14.91
0.980001
+ 5.263
24.614729
If you are adding (or
subtracting) 2 or more
numbers, first look at the
number of decimal places
for each.
 Next, add or subtract
normally.
 Finally, round off your
number to the same
number of decimals as the
lowest number of decimals
used at the start.

→ 6 numbers after decimal
→ 2 numbers after decimal
→ 6 numbers after decimal
→ 3 numbers after decimal
Round off to 2 decimal places
because 14.91 has the least number of
decimal places.
24.61
The answer is 24.61
Examples
What is the solution? How
many sig figs does the
solution have?
1.
3.298 + 0.14536
5. 3.298 - 0.14536
2.
645.95 + 273.7
6. 645.95 - 273.7
3.
50.72334 + 13.214
7. 50.72334 - 13.214
4.
2.00 + 1.0300
8. 2.00 - 1.0300
Multiplying and Dividing Rules

When you multiply and divide, your final
solution should have the same number of
significant figures as the number used to
calculate the solution with the lowest number
of significant figures.
Example
First, determine how
many significant figures
are in the numbers
provided.
 Next, multiply or divide
normally.
 Finally, round off your
solution so that it has the
same number of
significant figures as the
number with the smallest
amount of significant
figures.

3.6 → 2 significant figures
7.63 → 3 significant figures
0.245 → 3 significant figures
x 4.671 → 4 significant figures
31.43424186
Because 3.6 has the least
significant figures (2), our
answer must have the same
number of significant figures
(2).
31
The answer is 31.
Examples
What is the solution? How
many sig figs does it have
1.
13.2 x 4.l
5) 13.2 ÷ 4.1
2.
6.5 x 9.0321
6) 6.5 ÷ 9.0321
3.
0.0023 x 611.59
7) 0.0023 ÷ 611.59
4.
94.60 x 0.0900
8) 94.60 ÷ 0.0900
BEDMAS Rules

When you must evaluate an expression with both addition and
multiplication, the rules of BEDMAS still apply, but you must
determine the correct number of significant digits at every step along
the way to the final answer

Although you need to keep track of the sig figs along the way, you
must not round off your any number until you reach your final
answer!
BEDMAS Examples
1)
150.0 x 14.5 - 1.0032
2)
750 ÷ 2.0 + 8
3)
4)
(14.32 + 12.5) x (2.77 ÷ 0.9910)
(105.20 - 92.5) x 5500.3