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Mr. Gabrielse Significant Digits 0 1 2 3 4 5 6 7 8 9 . . . Mr. Gabrielse
Mr. Gabrielse
Significant Digits
0 1 2 3 4 5 6 7 8 9 . . .Mr. Gabrielse
Mr. Gabrielse
How Long is the Pencil?
Mr. Gabrielse
Mr. Gabrielse
Use a Ruler
Mr. Gabrielse
Mr. Gabrielse
Can’t See?
Mr. Gabrielse
Mr. Gabrielse
How Long is the Pencil?
Look Closer
Mr. Gabrielse
How Long is the Pencil?
5.9 cm
5.8 cm
5.8 cm
or
5.9 cm
?
Mr. Gabrielse
How Long is the Pencil?
5.9 cm
5.8 cm
Between
5.8 cm & 5.9 cm
Mr. Gabrielse
How Long is the Pencil?
5.9 cm
5.8 cm
At least: 5.8 cm
Not Quite: 5.9 cm
Mr. Gabrielse
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.
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
5.8 cmI pick 5.83 cm because I think the pencil is closer to 5.80 cm than 5.90
cm.
Mr. Gabrielse
Extra Digits
5.9 cm
5.8 cm
5.837 cm
I guessed at the 3 so the 7 is
meaningless.
Mr. Gabrielse
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).
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
Examples: 45 [2]19,583.894 [8].32 [2]136.7 [4]
Rule 1: Nonzero digits are always significant.
Mr. Gabrielse
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.
Mr. Gabrielse
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.
Mr. Gabrielse
Identifying Significant Digits
Examples: 7.003 [4]59,012 [5]101.02 [5]604 [3]
Rule 3: Middle zeros are always significant.
Mr. Gabrielse
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.
Mr. Gabrielse
Your Turn
Counting Significant DigitsClasswork: 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 mMeasure the time: 1.07 s
start stop
0.00 s1.07 s
Mr. Gabrielse
speed = distance time
Measure the distance: 10.21 mMeasure the time: 1.07 s
Physicists take data (measurements) and use equations to make predictions.
Mr. Gabrielse
speed = distance = 10.21 m time 1.07 s
Measure the distance: 10.21 mMeasure the time: 1.07 s
Physicists take data (measurements) and use equations to make predictions.
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
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
Mr. Gabrielse
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 1If Y < 5, leave X the
same
Example:
Round 345.0 to 2 significant digits.
Mr. Gabrielse
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 1If Y < 5, leave X the
same
Example:
Round 345.0 to 2 significant digits.
X Y
Mr. Gabrielse
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 1If 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.
Mr. Gabrielse
Multiplication & Division
You can never have more significant digits than any of your measurements.
Mr. Gabrielse
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) = (?)
Mr. Gabrielse
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)
Mr. Gabrielse
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)
Mr. Gabrielse
s
m1.3454545s3.3m4.44
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3)
(2)
(?)
Mr. Gabrielse
s
m1.3454545s3.3m4.44
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3)
(2)
(2)
Mr. Gabrielse
s
m1.3s3.3m4.44
Multiplication & Division
Round the answer so it has the same number of significant digits as the least precise
measurement.
(3)
(2)
(2)
Mr. Gabrielse
Addition & Subtraction
Rule:
You can never have more decimal places than any of your measurements.
Example:
13.05 309.2 + 3.785 326.035
Mr. Gabrielse
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.
Mr. Gabrielse
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.
Mr. Gabrielse
Your TurnClasswork: Using Significant Digits
