Significant Figures and Scientific Notation

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Significant Figures and Scientific Notation

• Math in the Science Classroom

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Scientific Notation

• Scientific notation is the way that scientists
  easily handle very large numbers or very small
  numbers.
• For example, instead of writing 0.0000000056, we
  write 5.6 x 10-9. So, how does this work?

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Scientific Notation
10000 1 x 104 24327 2.4327 x 104
1000 1 x 103 7354 7.354 x 103
100 1 x 102 482 4.82 x 102
10 1 x 101 89 8.9 x 101 (not usually done)
1 100
1/10 0.1 1 x 10-1 0.32 3.2 x 10-1 (not usually done)
1/100 0.01 1 x 10-2 0.053 5.3 x 10-2
1/1000 0.001 1 x 10-3 0.0078 7.8 x 10-3
1/10000 0.0001 1 x 10-4 0.00044 4.4 x 10-4

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Scientific Notation

• A positive exponent shows that the decimal point
  is shifted that number of places to the right.
• 251 x 104 2,510,000
• A negative exponent shows that the decimal point
  is shifted that number of places to the left.
• 251 x 10-4 .0251

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Scientific Notation Practice

1. Write in scientific notation 0.000467
2. Write in scientific notation 32000000
3. Express 5.43 x 10-3 as a number.
4. Express 6.34 x 109 as a number.

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Significant Figures

• There are two kinds of numbers in the world
• exact
• example There are exactly 12 eggs in a dozen.
• example Most people have exactly 10 fingers and
  10 toes.
• inexact numbers
• example any measurement.If I quickly measure
  the width of a piece of notebook paper, I might
  get 220 mm (2 significant figures). If I am more
  precise, I might get 216 mm (3 significant
  figures). An even more precise measurement would
  be 215.6 mm (4 significant figures).

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PRECISION VS ACCURACY

• Accuracy refers to how closely a measured value
  agrees with the correct value.
• Precision refers to how closely individual
  measurements agree with each other.

accurate(the average is accurate)not precise
precisenot accurate
accurateandprecise
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Significant Figures

• The number of significant figures is the number
  of digits believed to be correct by the person
  doing the measuring. It includes one estimated
  digit.
• So, does the concept of significant figures deal
  with precision or accuracy?

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Significant Figures

• Consider the beaker pictured below

• The smallest division is 10 mL, so we can read
  the volume to 1/10 of 10 mL or 1 mL.
• We measure 47 mL
• So, How many significant figures does our volume
  of 47?
• Answer - 2! The "4" we know for sure plus the "7"
  we had to estimate.

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Significant Figures

• Now consider a graduated cylinder.

• First, note that the surface of the liquid is
  curved. This is called the meniscus.
• The smallest division of this graduated cylinder
  is 1 mL.
• We measure 36.5mL
• How many significant figures does our answer
  have? 3! The "3" and the "6" we know for sure and
  the "5" we had to estimate a little.

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Significant Figures

• Rules for Working with Significant Figures
• Leading zeros are never significant.
• Imbedded zeros are always significant.
• Trailing zeros are significant only if the
  decimal point is specified. Hint Change the
  number to scientific notation. It is easier to
  see.

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Significant Figures

• Example Number of Significant
  Figures Scientific Notation
• 0.00682
• 3 6.82 x 10-3
• 1.072
• 4 1.072 (x 100)
• 300
• 1 3 x 102
• 300.
• 3 3.00 x 102

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Significant Figures