Significant Digits

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Significant Digits
0 1 2 3 4 5 6 7 8 9 . . .
Mr. Gabrielse
2
How Long is the Pencil?
Mr. Gabrielse
3
Use a Ruler
Mr. Gabrielse
4
Cant See?
Mr. Gabrielse
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How Long is the Pencil?
Look Closer
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How Long is the Pencil?
5.8 cm or 5.9 cm ?
5.9 cm
5.8 cm
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How Long is the Pencil?
Between 5.8 cm 5.9 cm
5.9 cm
5.8 cm
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How Long is the Pencil?
At least 5.8 cm Not Quite 5.9 cm
5.9 cm
5.8 cm
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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
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Pick a Number5.80 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
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Pick a Number5.80 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
I pick 5.83 cm because I think the pencil is
closer to 5.80 cm than 5.90 cm.
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Extra Digits
5.837 cm I guessed at the 3 so the 7 is
meaningless.
5.9 cm
5.8 cm
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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
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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

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Identifying Significant Digits
Rule 1 Nonzero digits are always significant.

• Examples
• 45 2
• 19,583.894 8
• .32 2
• 136.7 4

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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.
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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

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Identifying Significant Digits
Rule 3 Middle zeros are always significant.

• Examples
• 7.003 4
• 59,012 5
• 101.02 5
• 604 3

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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

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Your TurnCounting Significant DigitsClasswork
start it, Homework finish it
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Using Significant Digits

• Measure how fast the car travels.

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Example

• Measure the distance 10.21 m

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Example

• Measure the distance 10.21 m

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Example

• Measure the distance 10.21 m
• Measure the time 1.07 s

1.07 s
0.00 s
start
stop
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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

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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.
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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.
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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.
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Math with Significant Digits

• The result can never be more precise than the
  least precise measurement.

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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
wouldnt have more significant digits than 1.07
s.
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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 lt 5, leave X the same

Example Round 345.0 to 2 significant
digits.
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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 lt 5, leave X the same

Example Round 345.0 to 2 significant
digits.
X
Y
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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 lt 5, leave X the same

Example Round 345.0 to 2 significant
digits. 345.0 ? 350
X
Y
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Multiplication Division

• You can never have more significant digits than
  any of your measurements.

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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.

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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.

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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.

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Multiplication Division
(3)
(?)
(2)

• Round the answer so it has the same number of
  significant digits as the least precise
  measurement.

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Multiplication Division
(3)
(2)
(2)

• Round the answer so it has the same number of
  significant digits as the least precise
  measurement.

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Multiplication Division
(3)
(2)
(2)

• Round the answer so it has the same number of
  significant digits as the least precise
  measurement.

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Addition Subtraction
Example 13.05 309.2 3.785
326.035

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

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Addition Subtraction
Example 13.05 309.2 3.785
326.035

• Rule
• The answers 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.
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Addition Subtraction
Example 13.05 309.2 3.785
326.035

• Rule
• The answers doubtful digit is in the same
  decimal place as the measurement with the
  leftmost doubtful digit.

Hint Line up your decimal places.
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Your TurnClasswork Using Significant Digits