Uncertainty in Measurement
There is no such thing as an exact
measurement. There is always some
uncertainty or error involved.
That's got to be
24 and one-half
inches....give or
take a bit.
Calculations are affected, too!
In physics, we plug these measurements into
equations and calculate answers.
 x 35 m
=
=?
 t 22 . 4 s
How exact is the answer we obtain?
It cannot be any more certain or exact than the least
certain measurement that went into the calculation.
The Weakest Link
A chain is no stronger than its weakest link.
Our most uncertain
measurement is the
weakest link in the
chain of events that
leads to our answer.
The answer can
be no more
certain than that!
Keeping Track Of It All
So how do we keep track of the degree of
uncertainty in each of our measurement?
The rules of
significant digits
were created to
help us with this
task!
Keeping Track Of It All
The following procedure is used when you
have to multiply or divide measurements.
1. Using the rules, count how many significant
digits there are in each measurement you will
use in your calculation.
2. Find the measurement with the fewest significant
digits. Let's say it's 3.
3. That means the answer to your calculation must
be rounded to 3 significant digits.
The measurement with the fewest significant
digits is the weakest link. There is no way your
answer can be any more precise than that!
The Rules of Significant Digits
1. All non-zero digits are always significant.
2. Zeros at the beginning of a number are not
significant. For example: (0.023 meter) has two
significant digits. (The “2” and the “3”)
3. Zeros at the end of a number are not significant if
the number has no decimal point. For example:
(37600 feet) has 3 significant digits.
(The “3”, the “7”, and the “6”.)
The Rules of Significant Digits
4. Final zeros are significant if the number has
a decimal point. (290. cm) has 3 significant
digits. (3.560 kilometers) has 4 significant
digits.
5. Zeros in between non-zero digits are always
significant. (304 liters) has 3 significant
digits.
6. When a number is written in scientific
notation, all digits before the times sign are
significant. (3.0 X 108 m/s) has 2 significant
digits.
Some Examples
How many significant digits are in the
following number? 1683
Go to next page for
answer.
Some Examples
How many significant digits are in the
following number? 1683
There are 4 significant
digits.
This was an easy one! There are no zeros
in the number. All non-zero digits are
automatically significant.
Some Examples
How many significant digits are in the
following number? 265.80
Go to next page for
answer.
Some Examples
How many significant digits are in the
following number? 265.80
There are 5 significant
digits.
Final zeros are significant when the number has a
decimal point in it. In this case, the zero does not
have to be there as a place holder. It is there to
show that someone went to the trouble of
measuring to the hundredths decimal place.
Some Examples
How many significant digits are in the
following number? 0.0084
Go to next page for
answer.
Some Examples
How many significant digits are in the
following number? 0.0084
There are only 2
significant digits in this
number.
Leading zeros are never significant. They are only
place holders.
Some Examples
How many significant digits are in the
following number? 1500
Go to next page for
answer.
Some Examples
How many significant digits are in the
following number? 1500
There are only 2
significant digits.
Ending zeros are not significant when there is
no decimal point in the number.
Some Examples
How many significant digits are in the
following number? 15070
Go to next page for
answer.
Some Examples
How many significant digits are in the
following number? 15070
There are 4 significant
digits and they are
shown in red.
The first zero is significant because it is found in
between two non-zero digits (which are always
significant). The ending zero is not significant because
there is no decimal point in this number.
Significant Digits in Equations
Now let's try solving some equations. The
significant digits rules will help us round off our
answer to correctly express the amount of
uncertainty we have in it.
1.
2.
3.
Recall the procedure:
Count how many significant digits there are in each
measurement you will use in your calculation.
Find the measurement with the fewest significant digits.
Let's say it's 3.
Round your answer to have no more significant digits
than that measurement with the fewest. In this case it's 3.
Sample Problem
What is the area of a rectangle of length 13.9 m
and width 9.6 m?
A=l×w= 13. 9 m  9. 6 m 
2
The answer has not
A = 133. 44 m
yet been rounded off!
The measurement, 9.6 m has the smallest number of
significant digits (2). Therefore we must round off the
answer to 130 m2.
Sample Problem
An automobile travels 472 meters in 17 seconds.
What is its average velocity?
 x 472 m
v =
=
=27.7647 m/s
17 s
t
The measurement with the smallest number of
significant digits is 17 s, which has 2. Therefore, the
answer must be rounded to 28 m/s.
Rules for Addition and Subtraction
What do we do when we are required to add or
subtract measurements that have been expressed
with different degrees of precision?
Instead of being
concerned with
how many
significant digits
are in each
measurement,
we now look at
place value.
Rules for Addition and Subtraction
Suppose we have to add the following
lengths together:
measured to the hundredths place
measured to the tenths place
measured to the units place
this is the answer before rounding
Since the least precise measurement is to the
units place, the answer must be rounded to the
units place. The answer is 32 cm.
Sample Problem
A man is required by law to place a fence around
his backyard swimming pool. The area he needs to
enclose is 13.6 meters by 7.14 meters. What total
length of fencing material does he need to buy?
To find the perimeter of a rectangle we must add up the
length of all four sides of the reactangle.
P rect =lwlw
Sample Problem
P rect =lwlw
P rect =13.6 m 7.14 m 13.6 m 7.14 m
P rect =41.48 m
Although the width was measured to the
hundredth of a meter, the length was only
measured to the tenth of a meter. Therefore the
answer must be rounded to the nearest tenth of a
meter. The answer becomes 41.5 meters.
Sample Problem
A motorist accelerates his car from 24 m/s to
26.8 m/s while traveling in a straight line. By
how much did his velocity change?
 v=v f −v i
 v=26.8 m/s −24 m/s
 v=2.8 m/s
The answer must be rounded to 3 m/s because the least
precise measurement was the initial velocity which was
only measured to the nearest whole number.
Additional Guidelines
Coefficients in equations are considered to be exact
numbers. They do not impose any limitations of
their own on your use of significant digits.
2
f
2
o
v =v 2 a  x
Likewise, exponents are of no concern, unless of
course there is a variable in the exponent into which
you must substitute a measurement.
q=q o e
t
− RC
Additional Guidelines
In multi-part problems where the answer to the first
calculation is used in the second, you generally don't
round off to the correct number of significant digits
until you get to the very end of the final calculation.
 v 30.6 m/s
2
a=
=
=6.375 m/s
4.8 s
t
2
F =ma=15.4 kg6.375 m/s =98.175 N
F =98 N
If the acceleration has been rounded off
to 6.4 m/s2, the final answer would have
been 99 newtons rather than 98 N.
Additional Guidelines
Quantities that are spelled out or are used as a
reference or a basis for comparison are generally
taken to be exact and do not limit the number of
significant digits in your final answer.
For example: All of the following expressions shown in
green are taken to be exact values.
An object is dropped from rest. Find its velocity at the
end of the first four seconds of free fall.
There are 60 seconds in a minute.
A string vibrates at 430 Hz. Find the time it takes to
complete one vibration.