Significant Figures

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Significant Figures

• Michelle H-duke
• Department of Chemistry/Biochemistry

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Definitions

• Quantitative observations use the ability to
  count and usually involve scientific instruments.
• Qualitative observations do not use numbers.
• Base Units are specific units used for each
  physical quantity. It is important that you
  write the symbol correctly (m, g, l, , etc.)
• Derived units are those that are multiples of
  base units such as millimeter (mm).

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Significant Figures

• Significant figures (often referred to as sig
  figs) are used to tell how precisely the
  measurements were taken. They are also an
  indication of the precision of the instrument or
  measuring device used in a particular
  measurement. There are rules to be learned about
  how to determine the number of sig figs.
• It is the minimum number of digits needed to
  write the given value without loss of accuracy
  appropriate to the measurement or measuring
  device.

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• Scientific notation is used most often in the
  sciences to express the degree of accuracy. It is
  less commonly used in the maths and soft
  sciences.
• Example 145.5 has four sig figs. 1.455 X 102
  but if you wrote it as 1.4550 X 102 it implies
  that you know the value after the 5 which is not
  the case in 145.5.
• 1.4550 X 102 has five sig figs.
• If the degree of accuracy to be reported is less
  than the number, rounding is required.

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• Rules
• All non-zero numbers are significant
• -Example 1, 2, 3.
• Zeros are sometimes significant.
• a. Zeros between significant numbers are
  significant.
• -Example 2505 has 4 sig fig.
• b. Zeros at the end of a number to the right
  of a decimal point are significant.
• -Example 10.020450 has 8 sig fig.
• c. Zeros in numbers to the right of a decimal
  not preceded by a whole number are not
  significant.
• -Example 0.0025 has 2 sig fig.
• The numbers used in conversion factors are not
  included and are called exact numbers
• -Example 1 inch 2.54 cm. Thus
  in calc. 2.54 is not considered in the final
  answer.

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Rounding Off Look at the digits beyond the last
place desired -if the number is greater than 5
round to the next highest number -42.47354 to 3
sig figs 42.5 -if the number is less than 5
the leave the number the same, considered to be
rounding down -42.47354 to 4 sig figs
42.47 -In special cases where the number is
exactly halfway, round to the nearest even
digit. -42.47354 to 5 sig figs 42.474
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Scientific/Exponential Notation

• 450000 can be written as 4.5 x 105
• 0.0000045 is the same as 4.5 x 10-6
• Problems
• 1. Express 7250000 in scientific notation.
• 25005 in scientific notation
• 625.25 in scientific notation
• 2. Expand 5.55 x 10-8.

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Math Operations

• Multiply Multiply the bases and add the
  exponents
• Ex 2.5 x 102 x 1.2 x 105 3.0 x 107
• 2.5 x 102 x 1.2 x 10-5 3.0 x 10-3
• Division Divide the bases and subtract the
  exponents
• 2.5 x 102 / 1.2 x 105 2.08 x 10-3 ---gt 2.1 x
  10-3
• 2.5 x 102 x 1.2 x 10-5 2.08 x 107--gt2.1 X 107

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• Addition and subtraction
• Convert all of the numbers to the same power of
  10
• Add/Subtract the numbers together and use the
  common exponent
• Example
• 255 3125 45025
• 255 x 102 31.25 x 102 450.25 x 102
• 736.50 x 102 ? 7.37 x 104 (justified)

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Sig Figs. In Math Operations

• Addition and Subtraction
• Complete all of the addition or subtraction as
  you normally would. Retain numbers in answer
  until the final answer. The final answer should
  contain the same number of digits to the right of
  the decimal point as the measurement with the
  least number of digits to the right of the
  decimal.
• Example 5.345
• 6.2227
• - 0.25
• 11.3177 ? 11.32 (units)

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• Multiplication and Division
• When multiplying and dividing numbers, the final
  answer should contain no more significant figures
  than the fewest number of significant figures in
  any of the numbers being multiplied or divided in
  the calculation.
• Example 175 x 3126 x 25 1367625 (units)
• There are only 2 sig figs in the least so the
  answer should be reported as 1400000

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Note When using scientific notation, the power
of 10 has no influence on the figures to be
retained. Example 3.26 x 10-5 x 1.78 5.80 x
10-5 4.3179 x 1012 X 3.6 x10-19 1.6 x10-6
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Note units should always be included as it gives
information of the magnitude as well as the
degree of accuracy of the measurement. When
doing conversions, the conversion factor is not
considered when determining significant
figures. Example How many centimeters are in
82.00 inches? 1 inch 2.54 cm 1 in/2.54 cm
82.00/x x 208 cm Since 82.00 has 4 sig fig.
Your answer should be reported as 208.0 cm.
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SI Prefixes