Significant Digits

transcript

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

Use a Ruler

Mr. Gabrielse

Can’t See?

Mr. Gabrielse

Look Closer

Mr. Gabrielse

5.9 cm

5.8 cm

5.9 cm

Mr. Gabrielse

5.9 cm

5.8 cm

Between

5.8 cm & 5.9 cm

Mr. Gabrielse

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

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

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

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

Examples: 7.003 [4]59,012 [5]101.02 [5]604 [3]

Rule 3: Middle zeros are always significant.

Mr. Gabrielse

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

Physicists take data (measurements) and use equations to make predictions.

Mr. Gabrielse

speed = distance = 10.21 m time 1.07 s

Use a calculator to make a prediction.

Mr. Gabrielse

speed = 10.21 m = 9.542056075 m 1.07 s s

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

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

Example:

Round 345.0 to 2 significant digits.

Mr. Gabrielse

Rounding Off to X

Example:

Mr. Gabrielse

Rounding Off to X

Example:

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

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

measurement.

(3.45 cm)(4.8 cm)(0.5421cm) = 8.977176 cm3

(3) (2) (4) = (2)

Mr. Gabrielse

measurement.

(3.45 cm)(4.8 cm)(0.5421cm) = 9.000000 cm3

(3) (2) (4) = (2)

Mr. Gabrielse

m1.3454545s3.3m4.44

measurement.

Mr. Gabrielse

m1.3454545s3.3m4.44

measurement.

Mr. Gabrielse

m1.3s3.3m4.44

measurement.

Mr. Gabrielse

Addition & Subtraction

You can never have more decimal places than any of your measurements.

Example:

13.05 309.2 + 3.785 326.035

Mr. Gabrielse

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

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

Significant Digits

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