Significant Digits
0123456789...
Mr. Gabrielse
How Long is the Pencil?
Mr. Gabrielse
Use a Ruler
Mr. Gabrielse
Can’t See?
Mr. Gabrielse
How Long is the Pencil?
Look CloserMr. Gabrielse
How Long is the Pencil?
5.8 cm
or
5.9 cm
5.8 cm
5.9 cm
?
Mr. Gabrielse
How Long is the Pencil?
Between
5.8 cm & 5.9 cm
5.9 cm
5.8 cm
Mr. Gabrielse
How Long is the Pencil?
At least: 5.8 cm
5.9 cm
5.8 cm
Not Quite: 5.9 cm
Mr. Gabrielse
Solution: Add a Doubtful Digit
• Guess an extra
doubtful digit
between 5.80 cm
and 5.90 cm.
5.9 cm
5.8 cm
• Doubtful digits are
always uncertain,
never precise.
• The last digit in a
measurement is
always doubtful.
Mr. Gabrielse
Pick a Number:
5.80 cm, 5. 81 cm, 5.82 cm, 5.83 cm, 5.84 cm, 5.85 cm, 5.86 cm, 5.87 cm, 5.88 cm, 5.89 cm, 5.90 cm
5.9 cm
5.8 cm
Mr. Gabrielse
Pick a Number:
5.80 cm, 5. 81 cm, 5.82 cm, 5.83 cm, 5.84 cm, 5.85 cm, 5.86 cm, 5.87 cm, 5.88 cm, 5.89 cm, 5.90 cm
5.9 cm
I pick 5.83 cm
because I think the
pencil is closer to
5.80 cm than 5.90
cm.
5.8 cm
Mr. Gabrielse
Extra Digits
5.837 cm
I guessed at the 3
so the 7 is
meaningless.
5.9 cm
5.8 cm
Mr. Gabrielse
Extra Digits
5.837 cm
I guessed at the 3
so the 7 is
meaningless.
5.9 cm
5.8 cm
Digits after the
doubtful digit are
insignificant
(meaningless).
Mr. Gabrielse
Example Problem
– Example Problem: What is the average
velocity of a student that walks 4.4 m in 3.3 s?
•
•
•
•
d = 4.4 m
t = 3.3 s
v=d/t
v = 4.4 m / 3.3 s = 1.3 m/s not
1.3333333333333333333 m/s
Mr. Gabrielse
Identifying Significant Digits
Rule 1: Nonzero digits are always
significant.
Examples:
45
19,583.894
.32
136.7
[2]
[8]
[2]
[4]
Mr. Gabrielse
Identifying Significant Digits
Zeros make this interesting!
FYI: 0.000,340,056,100,0
Beginning
Zeros
Middle
Zeros
Ending
Zeros
Mr. Gabrielse
Beginning, middle, and ending zeros are separated by nonzero
digits.
Identifying Significant Digits
Rule 2: Beginning zeros are never
significant.
Examples:
0.005,6
0.078,9
0.000,001
0.537,89
[2]
[3]
[1]
[5]
Mr. Gabrielse
Identifying Significant Digits
Rule 3: Middle zeros are always
significant.
Examples:
7.003
59,012
101.02
604
[4]
[5]
[5]
[3]
Mr. Gabrielse
Identifying Significant Digits
Rule 4: Ending zeros are only significant if
there is a decimal point.
Examples:
430
43.0
0.00200
0.040050
[2]
[3]
[3]
[5]
Mr. Gabrielse
Your Turn
Counting Significant Digits
Classwork: start it, Homework: finish it Mr. Gabrielse
Using Significant Digits
Measure how fast the car travels.
Mr. Gabrielse
Example
Measure the distance: 10.21 m
Mr. Gabrielse
Example
Measure the distance: 10.21 m
Mr. Gabrielse
Example
Measure the distance: 10.21 m
Measure the time: 1.07 s
1.07
0.00 s
start
stop
Mr. Gabrielse
speed = distance
time
Physicists take data (measurements) and use equations to make predictions.
Measure the distance: 10.21 m
Measure the time: 1.07 s
Mr. Gabrielse
speed = distance = 10.21 m
time
1.07 s
Physicists take data (measurements) and use equations to make predictions.
Measure the distance: 10.21 m
Measure the time: 1.07 s
Use a calculator to make a prediction.
Mr. Gabrielse
speed = 10.21 m = 9.542056075 m
1.07 s
s
Physicists take data (measurements) and use equations to make predictions.
Too many significant digits!
We need rules for doing math with
significant digits.
Mr. Gabrielse
speed = 10.21 m = 9.542056075 m
1.07 s
s
Physicists take data (measurements) and use equations to make predictions.
Too many significant digits!
We need rules for doing math with
significant digits.
Mr. Gabrielse
Math with Significant Digits
The result can never be more
precise than the least precise
measurement.
Mr. Gabrielse
speed = 10.21 m = 9.54 m
1.07 s
s
we go over how
to round next
1.07 s was the least precise measurement since it had the least
number of significant digits
The answer had to be rounded to 9.54 ms so it wouldn’t have
more significant digits than 1.07 s.
Mr. Gabrielse
Rounding Off to X
X: the new last
significant digit
Y: the digit after the new
last significant digit
Example:
Round 345.0 to 2 significant
digits.
If Y ≥ 5, increase X by 1
If Y < 5, leave X the
same
Mr. Gabrielse
Rounding Off to X
X: the new last
significant digit
Y: the digit after the new
last significant digit
Example:
Round 345.0 to 2 significant
digits.
X
Y
If Y ≥ 5, increase X by 1
If Y < 5, leave X the
same
Mr. Gabrielse
Rounding Off to X
X: the new last
significant digit
Y: the digit after the new
last significant digit
Example:
Round 345.0 to 2 significant
digits.
X
If Y ≥ 5, increase X by 1
If Y < 5, leave X the
same
Y
345.0  350
Fill in till the decimal place
with zeroes.
Mr. Gabrielse
Multiplication & Division
You can never have more significant digits than
any of your measurements.
Mr. Gabrielse
Multiplication & Division
(3.45 cm)(4.8 cm)(0.5421cm) = 8.977176 cm3
(3)
(2)
(4)
=
(?)
Round the answer so it has the same number of
significant digits as the least precise
measurement.
Mr. Gabrielse
Multiplication & Division
(3.45 cm)(4.8 cm)(0.5421cm) = 8.977176 cm3
(3)
(2)
(4)
=
(2)
Round the answer so it has the same number of
significant digits as the least precise
measurement.
Mr. Gabrielse
Multiplication & Division
(3.45 cm)(4.8 cm)(0.5421cm) = 9.000000 cm3
(3)
(2)
(4)
=
(2)
Round the answer so it has the same number of
significant digits as the least precise
measurement.
Mr. Gabrielse
Multiplication & Division
(3)
(?)
4.44m   1.3454545m
3.3s 
s
(2)
Round the answer so it has the same number of
significant digits as the least precise
measurement.
Mr. Gabrielse
Multiplication & Division
(3)
(2)
4.44m   1.3454545m
3.3s 
s
(2)
Round the answer so it has the same number of
significant digits as the least precise
measurement.
Mr. Gabrielse
Multiplication & Division
(3)
(2)
4.44m  1.3 m
3.3s 
s
(2)
Round the answer so it has the same number of
significant digits as the least precise
measurement.
Mr. Gabrielse
Addition & Subtraction
Rule:
Example:
You can never have
more decimal places
than any of your
measurements.
13.05
309.2
+ 3.785
326.035
Mr. Gabrielse
Addition & Subtraction
Example:
Rule:
The answer’s
doubtful digit is in the
same decimal place
as the measurement
with the leftmost
doubtful digit.
13.05
309.2
+ 3.785
326.035
Hint: Line up your decimal places.
leftmost
doubtful digit
in the problem
Mr. Gabrielse
Addition & Subtraction
Rule:
Example:
The answer’s
doubtful digit is in the
same decimal place
as the measurement
with the leftmost
doubtful digit.
13.05
309.2
+ 3.785
326.035
Hint: Line up your decimal places.
Mr. Gabrielse
Your Turn
Classwork: Using Significant Digits
Mr. Gabrielse