Significant Figures

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Significant Figures
A tutorial adapted from www.highschoolchem.com
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What are significant figures?

• Significant figures are a way of expressing
  precision in measurement.  In a measurement in
  lab, the digits which you can read for certain,
  plus one uncertain digit, are significant.

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• For instance, on the graduated cylinder shown to
  the left, you will notice that the solution is
  somewhere between 25 mL and 30 mL.  The first
  digit is certain.  We know it has to be 2.  When
  reading a measurement, always read  the certain
  plus one uncertain.  Take a good guess at the
  uncertain digit.  Probably 8.  We would report
  the volume of liquid in this graduated cylinder
  to be 28 mL.  These would be the significant
  digits.  You wouldn't want to report any more. 
  It would be foolish to try to get any more digits
  in this answer.  We just can't be sure!
• If the graduated cylinder had markings every mL
  instead of every 5, we could get even more
  specific.  You would certainly know the first two
  digits of the measurement.  The uncertain digit
  would be whatever you "guess" to be the fraction
  of liquid to be between the two markings.

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Determining Significant Figures in a Measurement

• There are a few basic rules to remember when
  counting the number of significant figures in a
  measurement.
• All non-zero numbers ARE significant. 
• The number 33.2 contains THREE significant
  figures because all of the digits present are
  non-zero.
• Zeros between two significant digits ARE
  significant. 
• 2051 has FOUR significant figures. Since the zero
  is between a 2 and a 5, it's significant.
• Leading zeros are NEVER significant. 
• They're nothing more than "place holders".  For
  instance, 0.54 has only TWO significant figures
  because the zero is leading and a place holder. 
  0.0032 also has TWO significant figures. 
• Trailing zeros to the right of a decimal ARE
  significant. 
• There are FOUR significant digits in 92.00
  because the zeros are trailing to the right of 
  the decimal.  Remember, they must be there
  because the person measuring this value must have
  been able to read these numbers from the
  apparatus.

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• Trailing zeros in a whole number with the decimal
  shown ARE significant. 
• Placing a decimal at the end of a number is
  usually not done.  By convention, however, this
  decimal will indicate a significant zero.  For
  instance, 540. indicates that the (trailing) zero
  IS significant.  There are a total of THREE
  significant digits in this number.
• Trailing zeros in a whole number with no decimal
  shown are NOT significant. 
• Writing just 540 indicates that the zero is NOT
  significant and there are only TWO significant
  figures in this value.
• Exact numbers have an INFINITE number of
  significant figures. 
• Numbers that are definitions or exact have an
  infinite number of significant figures.  For
  example1 meter 1000 millimeters.  1 meter
  equals 1000.0000000... millimeters as
  1.0000000...meters equals 1000 millimeters.  Both
  are definitions and therefore have infinite
  significant figures.

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Practice

• 13.06 mL
• 0.0450 g
• 1.20 kg
• 10 lbs
• 10.0 seconds
• 1.820 L
• 5902.05 mg
• 1010.2060 g

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Using significant figures in mathematical
calculations

• The expression "a chain is only as strong as its
  weakest link" explains why significant figures
  need to be considered when calculating. 
  Remember, significant figures represent the
  accuracy of a measurement.  When manipulating
  these measurements by adding, subtracting,
  multiplying, or dividing, your final answer
  cannot be more accurate than the numbers you
  started with.  Your answer can only be as
  accurate as the measurements you start with.

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Multiplication and Division

• When multiplying or dividing, count how many
  significant figures are in each measurement. 
  Your final answer should contain the same number
  of significant figures as which ever starting
  value has the least. 
• For example, when you take 2.045 cm X 1.3 cm,
  your two measurements have 4 significant figures
  and 2 significant figures.  Since the LEAST
  number of significant figures is 2, your final
  answer can only have 2 significant figures.

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Addition and Subtraction

• When adding or subtracting, you must count
  decimal places instead of significant figures. 
  Your final answer should contain the same number
  of decimal places as which ever starting value
  has the least.
• For example, 1.994 16.3 18.294 which rounds
  to 18.3 using 1 decimal place

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Practice

• 15.04 / 3.1
• 188.20 92.334 1.0008
• 345.04 g - 227.1 g
• 2.11 X 0.0006
• 0.891 X 200. X 13.8