Scientific Notation, Exponents and Significant Figures

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Scientific Notation, Exponents and Significant
Figures
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Scientific Notation

• Scientific Notation It is notation used to
  express very large or very small numbers using
  powers of 10.
• It is written as a number multiplied by 10x
• Example 1000 1 x 103 (10 x 10 x 10)
• Here the 1000 is in standard notation
• 1 x 103 is scientific notation

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Scientific Notation

• How do we express these terms?
• For Scientific Notation
• The number is expressed as gt1 and lt10 and then
  multiplied by a power of 10.
• Example 523,000 5.23 x 105

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Scientific Notation
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Scientific Notation

• So, how do we change from standard notation to
  scientific notation?
• Move the decimal point to create number that is
  between 1 and 10
• Example 7,231,967 7.231967 x 106
• 0.00003433 3.433 x 10-5

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Scientific Notation

• Rules
• The decimal place should end up at to the right
  of the first nonzero digit.
• The total number of spaces moved becomes the
  exponent of 10 in the scientific notation.
• If the given number is greater than 1, the
  exponent is positive.
• If the given number is less than 1 (but gt0), the
  exponent is negative.

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Scientific Notation

• Practice writing these
• 17 mL
• 153 kg
• 24883.5 km
• 2000 miles
• 0.4502 g
• 0.00063401 m

• in scientific notation.

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Scientific Notation

• You can also use this information to write a
  number in standard notation.
• Example 2.3445 x 103 g 2344.5 g
• 2.21 x 10-7 m 0.000000221 m

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Scientific Notation

• Practice Write the follow
• 6.423 x 103 g
• 1.002 x 10-6 m
• 5.0023 x 1010 m
• 3.3 x 10-9 sec

• ing in standard notation
• 6423 g
• 0.000001002 m
• 50,023,000,000 m
• 0.0000000033 sec

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Rules of Exponents

• Remember that exponents, especially with powers
  of 10, help count zeros.
• It is easier to see keep track of zeros with and
  exponent like 106 than with the standard notation
  of 1,000,000.
• Using the rules of exponents, you can multiply
  and divide exponents easily.

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Exponents

• Rules of Exponents
• (10m)(10n) 10mn
• (10m)n 10mn
• 10m/10n 10m-n
• 10-m 1/10m
• 100 1

• Example
• (102)(103) 105
• (103)2 106
• 106/102 104
• 10-8 1/108

1001000 100,000
(1000)2 1,000,000
1,000,000 10,000 100
1 0.00000001 100,000,000
1 x 10-8
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Scientific Notation Exponents
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Scientific Notation Exponents

• Practice
• Convert numbers or
• 1000 g
• 5.3 x 103 m
• 4.5 x 10-6 m
• 1.7 x 10-3g
• 22000 seconds

• exponents to prefix.
• 1 x 103 g or 1 kg
• 5.3 km
• 4.5 µm
• 1.7 mg
• 22 kiloseconds

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Scientific Notation Exponents

• 2.4 mg
• 2 km
• 1.6 Mm (megameter)
• 15 msec (milliseconds)
• 253 km

• 2.4 x 10-3 g
• 2 x 103 m
• 1.6 x 106 m
• (1.5 x101) x 10-3 sec
• or 1.5 x 10-2 sec
• (2.53 x 102) x 103 m
• Or 2.53 x 105 m

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Exponents

• How many milligrams are in a kilogram?
• 1 kg 1000 g 103 g x 1 mg
• 10-3g
• 106
  mg

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Exponents

• How many picograms in a microgram?
• 1 µg (1 x 10-6 g)(1 pg )
• 1x10-12 g
• 1/(1 x 10-6) pg
• 106 pg 1,000,000 pg

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Significant Figures

• With scientific measurements, you want to know
  accuracy, precision and certainty.
• Accuracy How close a measurement is to an
  accepted value
• Precision How close a measurement is to other
  measurements of the same thing.
• Certainty Degree of confidence of a
  measurement. The last digit to the right is
  usually an uncertain digit.

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Significant Figures

• So for any measured value, well record all of
  the certain digits plus an uncertain digit.
• All together, they are the significant figures of
  the measurement.

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Significant Figure RULES

• All non zero digits (1,2,3,4,5,6,7,8, and 9) are
  significant.
• Final zeros to the right of the decimal point are
  significant.
• Zeros between two significant digits are
  significant.
• Zeros used for spacing the decimal point are not
  significant.
• For numbers in scientific notation, all of the
  digits before the x 10x are significant.

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How Many Significant Figures?

• Measurement
• 135.3
• 4.6025
• 200,035
• 0.0000300
• 2.0000300
• 0.002
• 4.44 x 103
• 2.0 x 10-2
• 10.00
• 10
• 102,000

• of Sig Figs
• 4 sig figs
• 5 sig figs
• 6 sig figs
• 3 sig figs
• 8 sig figs
• 1 sig fig
• 3 sig figs
• 2 sig figs
• 4 sig figs
• 1 sig fig
• 3 sig figs

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Significant Figures

• Multiplying and Dividing with Sig Figs
• When multiplying or dividing measurements, the
  answer must have the same number of sig figs as
  the measurement with the fewest sig figs.
• Example 22 feet x 9 feet 198 square feetbut
• Since 9 feet only has 1 sig fig the correct
  answer is
• 200 ft2

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Calculate the Area in square blocks
3
2
1
0
1
2
3
4
5
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Significant Figures

• Calculation
• 2.86 m x 1.824 m
• 460 miles/
• 8 hours
• 98.50 in x
• 1.82 in

• Calcd Answer
• 5.21664 m2
• 57.5 mi/hr
• 179.27 in2

Ans w/sig figs 5.22 m2 60 mi/hr 179 in2
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Significant Figures

• Calculation
• 2.100 m x 0.0030 m
• 10.00 g /
• 5.000 L
• 4.610 ft x
• 1.7 ft

• Calcd Answer
• 0.0063 m2
• 2 g/L
• 7.837 ft2

Ans w/sig figs 0.0063 m2 2.000 g/L 7.8 ft2
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Significant Figures

• Defined numbers part of a definition and is not
  measured. So, defined numbers (unit conversion
  factors) do not limit the sig figs in an answer.
• Also, counting numbers do not limit sig figs.
• Example You cut a 24 ft piece of wood into 4
  pieces. Each is 24 ft/4 6.0 ft/piece.

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Significant Figures

• Addition and Subtraction
• The sig figs with addition and subtraction are
  handled differently than with x and /.
• The answer cannot have more certainty than the
  least certain measurement.
• This means the answer must have the same number
  of sig figs to the right of the decimal as the
  measurement with the fewest sig figs to the right
  of the decimal place.

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Significant Figures

• Example
• 4.271 g (3 sig figs to right of decimal)
• 2 g (0 sig figs to right of decimal)
• 10.0 g (1 sig fig to right of decimal)
• 16.271 g is calculated answer but
• since 2 g has no sig figs to right of decimal
  the final answer is 16 g.