significant digits (physics)
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Significant Digits
0 1 2 3 4 5 6 7 8 9 . . .
How Long is the Pencil?
Use a Ruler
Can’t See?
How Long is the Pencil?
Look Closer
How Long is the Pencil?
5.9 cm
5.8 cm
5.8 cm
or
5.9 cm
?
How Long is the Pencil?
5.9 cm
5.8 cm
Between
5.8 cm & 5.9 cm
How Long is the Pencil?
5.9 cm
5.8 cm
At least: 5.8 cm
Not Quite: 5.9 cm
Solution: Add a Doubtful Digit
5.9 cm
5.8 cm
• Guess an extra doubtful digit between 5.80 cm and 5.90 cm.
• Doubtful digits are always uncertain, never precise.
• The last digit in a measurement is always doubtful.
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
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 cmI pick 5.83 cm because I think the pencil is closer to 5.80 cm than 5.90
cm.
Extra Digits
5.9 cm
5.8 cm
5.837 cm
I guessed at the 3 so the 7 is
meaningless.
Extra Digits
5.9 cm
5.8 cm
5.837 cm
I guessed at the 3 so the 7 is
meaningless.
Digits after the doubtful digit are
insignificant (meaningless).
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
Identifying Significant Digits
Examples:
45 [2]
19,583.894 [8]
.32 [2]
136.7 [4]
Rule 1: Nonzero digits are always significant.
Identifying Significant Digits
Zeros make this interesting!
FYI: 0.000,340,056,100,0
Beginning Zeros
Middle Zeros
Ending Zeros
Beginning, middle, and ending zeros are separated by nonzero digits.
Identifying Significant Digits
Examples:
0.005,6 [2]
0.078,9 [3]
0.000,001 [1]
0.537,89 [5]
Rule 2: Beginning zeros are never significant.
Identifying Significant Digits
Examples:
7.003 [4]
59,012 [5]
101.02 [5]
604 [3]
Rule 3: Middle zeros are always significant.
Identifying Significant Digits
Examples:
430 [2]
43.0 [3]
0.00200 [3]
0.040050 [5]
Rule 4: Ending zeros are only significant if there is a decimal point.
Your Turn
Counting Significant DigitsClasswork: start it, Homework: finish it
Using Significant Digits
Measure how fast the car travels.
Example
Measure the distance: 10.21 m
Example
Measure the distance: 10.21 m
Example
Measure the distance: 10.21 m
Measure the time: 1.07 s
start stop
0.00 s1.07 s
speed = distance time
Measure the distance: 10.21 m
Measure the time: 1.07 s
Physicists take data (measurements) and use equations to make predictions.
speed = distance = 10.21 m time 1.07 s
Measure the distance: 10.21 m
Measure the time: 1.07 s
Physicists take data (measurements) and use equations to make predictions.
Use a calculator to make a prediction.
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.
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.
Math with Significant Digits
The result can never be more precise than the least precise
measurement.
speed = 10.21 m = 9.54 m 1.07 s s
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 so it wouldn’t have
more significant digits than 1.07 s.sm
we go over how to round next
Rounding Off to X
X: the new last significant digit
Y: the digit after the new last significant digit
If Y ≥ 5, increase X by 1
If Y < 5, leave X the same
Example:
Round 345.0 to 2 significant digits.
Rounding Off to X
X: the new last significant digit
Y: the digit after the new last significant digit
If Y ≥ 5, increase X by 1
If Y < 5, leave X the same
Example:
Round 345.0 to 2 significant digits.
X Y
Rounding Off to X
X: the new last significant digit
Y: the digit after the new last significant digit
If Y ≥ 5, increase X by 1
If Y < 5, leave X the same
X Y
Example:
Round 345.0 to 2 significant digits.
345.0 350
Fill in till the decimal place with zeroes.
Multiplication & Division
You can never have more significant digits than any of your measurements.
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3.45 cm)(4.8 cm)(0.5421cm) = 8.977176 cm3
(3) (2) (4) = (?)
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3.45 cm)(4.8 cm)(0.5421cm) = 8.977176 cm3
(3) (2) (4) = (2)
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3.45 cm)(4.8 cm)(0.5421cm) = 9.000000 cm3
(3) (2) (4) = (2)
s
m1.3454545
s3.3
m4.44
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3)
(2)
(?)
s
m1.3454545
s3.3
m4.44
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3)
(2)
(2)
s
m1.3
s3.3
m4.44
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3)
(2)
(2)
Addition & Subtraction
Rule:
You can never have more decimal places than any of your measurements.
Example:
13.05
309.2
+ 3.785
326.035
Addition & Subtraction
Rule:
The answer’s doubtful digit is in the same decimal place as the measurement with the leftmost doubtful digit.
Example:
13.05
309.2
+ 3.785
326.035
leftmost
doubtful digit
in the problem
Hint: Line up your decimal places.
Addition & Subtraction
Rule:
The answer’s doubtful digit is in the same decimal place as the measurement with the leftmost doubtful digit.
Example:
13.05
309.2
+ 3.785
326.035
Hint: Line up your decimal places.
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