Significant Digits 0 1 2 3 4 5 6 7 8 9 . . . 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 Closer

How Long is the Pencil? 5.8 cm or 5.9 cm ? 5.9 cm 5.8 cm

How Long is the Pencil? Between 5.8 cm & 5.9 cm 5.9 cm 5.8 cm

How Long is the Pencil? At least: 5.8 cm Not Quite: 5.9 cm 5.9 cm

Solution: Add a Doubtful Digit 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. 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 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 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

I guessed at the 3 so the 7 is meaningless. Extra Digits 5.837 cm I guessed at the 3 so the 7 is meaningless. 5.9 cm 5.8 cm

Extra Digits 5.837 cm I guessed at the 3 so the 7 is meaningless. Digits after the doubtful digit are insignificant (meaningless). 5.9 cm 5.8 cm

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 Rule 1: Nonzero digits are always significant. Examples: 45 [2] 19,583.894 [8] .32 [2] 136.7 [4]

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 Rule 2: Beginning zeros are never significant. Examples: 0.005,6 [2] 0.078,9 [3] 0.000,001 [1] 0.537,89 [5]

Identifying Significant Digits Rule 3: Middle zeros are always significant. Examples: 7.003 [4] 59,012 [5] 101.02 [5] 604 [3]

Identifying Significant Digits Rule 4: Ending zeros are only significant if there is a decimal point. Examples: 430 [2] 43.0 [3] 0.00200 [3] 0.040050 [5]

Your Turn Counting Significant Digits Classwork: start it, Homework: finish it

Using Significant Digits Measure how fast the car travels.

Measure the distance: 10.21 m Example Measure the distance: 10.21 m

Measure the distance: 10.21 m Example Measure the distance: 10.21 m

Measure the distance: 10.21 m Example Measure the distance: 10.21 m Measure the time: 1.07 s 0.00 s 1.07 s start stop

Measure the distance: 10.21 m 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

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.

Too many 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.

Too many 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 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 so it wouldn’t have more significant digits than 1.07 s.

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.

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 Y 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. 345.0  350 X Y Fill in till the decimal place with zeroes.

Multiplication & Division You can never have more significant digits than any of your measurements.

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.

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.

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.

Multiplication & Division (3) (?) (2) Round the answer so it has the same number of significant digits as the least precise measurement.

Multiplication & Division (3) (2) (2) Round the answer so it has the same number of significant digits as the least precise measurement.

Multiplication & Division (3) (2) (2) Round the answer so it has the same number of significant digits as the least precise measurement.

Addition & Subtraction Example: 13.05 309.2 + 3.785 326.035 Rule: You can never have more decimal places than any of your measurements.

Addition & Subtraction Example: 13.05 309.2 + 3.785 326.035 Rule: The answer’s doubtful digit is in the same decimal place as the measurement with the leftmost doubtful digit. leftmost doubtful digit in the problem Hint: Line up your decimal places.

Addition & Subtraction Example: 13.05 309.2 + 3.785 326.035 Rule: The answer’s doubtful digit is in the same decimal place as the measurement with the leftmost doubtful digit. Hint: Line up your decimal places.

Your Turn Classwork: Using Significant Digits