Introduction to Significant Figures

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Introduction to Significant Figures

• Scientific Notation

2
Significant Figures

• Scientist use significant figures to determine
  how precise a measurement is
• It tells us how accurate the instrument is we are
  using
• Significant digits in a measurement include all
  of the known digits plus one estimated digit

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An example

• Look at the ruler below
• It reads to 0.1cm (Each line is 0.1cm)
• You can see the arrow is exactly on 13.3 cm
• However, using significant figures, you must
  estimate the next digit
• We always read one place past the smallest
  division
• That would give you 13.30 cm

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Try this one

• Look at this ruler
• The smallest division is .1 so we must read to
  .01
• It is greater than12.8 cm but smaller than 12.9
  cm
• Now estimate the next digit
• 12.85 cm

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The same rules apply with all instruments

• The same rules apply
• Read to the last digit that you know
• Estimate the final digit

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Reading a graduated cylinder

• What is the measurement?
• 53.5 ml
• Since this measures to the ones place we must
  read to the 10ths. It is more than 53 but less
  than 54. It looks like it is about ½ way between
  so we will estimate (guess) the .5

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Rules for Determining the Number of Significant
Figures

• All non-zero digits are significant.
• Zeros located between 2 non-zero digits are
  significant.
• Leading zeros (those at the start of a number)
  are never significant.
• Trailing zeros (those at the end of a number) are
  never significant unless they are preceded by a
  decimal point somewhere in the number.

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Rules for Significant figuresRule 1

• All non zero digits are significant
• Determine the number of significant digits in
  the following

• 274
• 25.632
• 8.987

• 3 Significant Figures
• 5 Significant Digits
• 4 Significant Figures

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Rule 2

• All zeros between significant digits are
  significant
• Determine the number of significant digits in
  the following

3 Significant Figures 5 Significant Digits 4
Significant Figures
504 60002 9.077
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Rule 3

• Zeros in front of a number are NEVER significant.
• All zeros that act as place holders are NOT
  significant (Rule 3 and 4)
• Find the number of sig figs below

0.02 0101 0.000436
133
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Rule 4

• All FINAL zeros are ONLY significant if there is
  a decimal point in the number.
• How many significant digits are in the following
  numbers?

32.0 3 19.000 5 105.0020
7 150 2 200 1
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For example
How many significant digits are in the following
numbers?

• 1
• 3
• 6
• 2
• 1

• 0.0002
• 6.02 x 1023
• 100.000
• 150000
• 800

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Rule 5

• All counted numbers and constants have an
  infinite number of significant digits
• For example
• 1 hour 60 minutes
• 10 mm 1 cm
• 1km 1000 m

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Determine the of significant figures in the
following numbers

• 2
• 6
• 2
• 3
• 4
• 1
• 4

• 0.0073
• 100.020
• 2500
• 7.90 x 10-3
• 670.0
• 0.00001
• 18.84

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• How many significant figures are present in each
  of the following measurements?
• 5.13
• 100.01
• 0.0401
• 0.0050
• 220,000
• 1.90 x 103
• 153.000
• 1.0050

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• 3 sig figs. All non-zero digits are significant.
• 5 sig figs. All non-zero digits are significant.
  All zeros located between two non-zero digits are
  significant.
• 3 sig figs. Leading zeros are not significant.
• 2 sig figs. Leading zeros are not significant.
  Trailing zeros are significant when there is a
  decimal point.
• 2 sig figs. Trailing zeros are not significant
  because there is no decimal point.
• 3 sig figs. Trailing zeros are significant when
  there is a decimal point.
• 6 sig figs. Trailing zeros are significant when
  there is a decimal point.
• 5 sig figs. Zeros between non-zero digits are
  significant. Trailing zeros are significant when
  there is a decimal point.

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Rounding Significant Digits Rule 1

• If the digit to the immediate right of the last
  significant digit is less that 5, drop it and any
  trailing numbers.
• For example, lets say you have the number 43.82
  and you want 3 significant digits
• The last number that you want is the 8 and the
  number to the right of the 8 is a 2
• Therefore, you would not round up the number
  would be 43.8

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Rounding Rule 2

• If the digit to the immediate right of the last
  significant digit is greater that a 5, you round
  up the last significant figure
• Lets say you have the number 234.87 and you want
  4 significant digits
• 234.87 The last number you want is the 8 and the
  number to the right is a 7
• Therefore, you would round up get 234.9

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Try these

• 200.99 (want 3 SF)
• 18.22 (want 2 SF)
• 135.50 (want 3 SF)
• 0.00299 (want 1 SF)
• 98.59 (want 2 SF)

• 201
• 18
• 136
• 0.003
• 99

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Scientific Notation
21

• In general, any number X can be written as the
  product of another number N and a power of ten.
• It's important to remember that 1 lt N lt10. In
  other words, N MUST be at least 1 but less than
  10.
• The general format for a number written in
  scientific notation will be
• N x 10power
• Examples20 2 x 10 2 x 1013500 3.5 x 1000
  3.5 x 1030.0055 5.5 x 0.001 5.5 x 10-3

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Scientific Notation

• Scientific notation is used to express very large
  or very small numbers
• Sometimes it is the only way we can express a
  measurement or derived number to the correct
  number of sig figs.
• We must retain all sig figs when we convert from
  normal to scientific notation

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Converting Numbers gt1

• If the number you start with is greater than 1,
  the exponent will be positive
• Write the number 39923 in scientific notation
• First move the decimal until 1 number is in front
  3.9923
• Now count the number of decimal places that you
  moved (4)
• Since the number you started with was greater
  than 1, the exponent will be positive
• 3.9923 x 104

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Converting Numbers Smaller than 1

• If the number you start with is less than 1, the
  exponent will be negative
• Write the number 0.0052 in scientific notation
• First move the decimal until 1 number is in
  front 5.2
• Now count the number of decimal places that you
  moved (3)
• Since the number you started with was less than
  1, the exponent will be negative
• 5.2 x 10 -3

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Convert the following numbers to scientific
notation

• 99.343
• 4000.1
• 0.000375
• 0.0234
• 94577.1

• 9.9343 x 101
• 4.0001 x 103
• 3.75 x 10-4
• 2.34 x 10-2
• 9.45771 x 104

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Going from Scientific Notation to Ordinary
Notation

• You start with the number and move the decimal
  the same number of spaces as the exponent.
• If the exponent is positive, the number will be
  greater than 1
• If the exponent is negative, the number will be
  less than 1

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Convert the following numbers to normal notation

• 3000000
• 6260000000
• 0.0005
• 0.000000845
• 2250

• 3 x 106
• 6.26x 109
• 5 x 10-4
• 8.45 x 10-7
• 2.25 x 103

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Significant Digits

• Calculations

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Significant Digits in Calculations

• Now that you know how to determine the number of
  significant digits in a measurement
• What happens when you use these numbers to derive
  another (How do you decide what to do when
  adding, subtracting, multiplying, or dividing)

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Rules for Addition and Subtraction

• When you add or subtract measurements, your
  answer can have no more decimal places (places
  past the decimal) than the one with the fewest
  used in the operation

20.4 1.322 83
104.722 is what your calculator will tell
you....
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Addition Subtraction Continued

• Because you are adding, you need to look at the
  number of decimal places
• 20.4 1.322 83 104.722
• (1) (3) (0)
• Since you are adding, your answer must have the
  same number of decimal places as the one with the
  fewest
• The fewest number of decimal places is 0
• Therefore, you answer must be rounded to have 0
  decimal places
• Your answer becomes with rounding rules
• 105

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Addition Subtraction Problems

• 1.23056 67.809
• 23.67 500
• 40.08 32.064
• 22.9898 35.453
• 95.00 75.00

• 69.03956 rounds to 69.040
• - 476.33 rounds to -476
• 72.1440 rounds to 72.14
• 58.4428 rounds to 58.443
• 20 rounds to 20.00

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Rules for Multiplication Division

• When you multiply and divide numbers you look at
  the number of significant digits and decimal
  places
• For example

67.50 x 2.54
171.45 shows up on your calculator
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Multiplication Division

• Because you are multiplying, you need to look at
  the number of significant digits in each number
  not decimal places
• 67.50 x 2.54 171.45
• (4) (3)
• Since you are multiplying, your answer must have
  the same number of significant digits as the one
  with the fewest
• The fewest number of significant digits is 3
• Therefore, you answer must be rounded to have 3
  significant digits
• Your answer rounds to 171

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Solve with the correct number of sig figs

• 10948.845 rounds to 1.09 x 104
• 3.916977 rounds to 3.9170
• 141.954 rounds to 142
• 3.5376 rounds to 3.538
• 30.47123 rounds to 30.47

• 890.15 x 12.3
• 88.132 / 22.500
• (48.12)(2.95)
• 58.30 / 16.48
• 307.15 / 10.08