Introduction%20to%20Significant%20Figures%20

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Introduction to Significant Figures

• Scientific Notation

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Scientific Method
Logical approach to solving problems by
observing
collecting data
formulating a hypotheses
testing Hypotheses
Formulating theories
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Significant Figures

• Scientist use significant figures to determine
  how precise a measurement is

• Significant digits in a measurement include
• all of the known digits plus one estimated
  digit

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For example

• Look at the ruler below
• Each line is 0.1cm
• You can read that the arrow is on 13.3 cm
• However, using significant figures, you must
  estimate the next digit
• That would give you 13.30 cm

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Lets try this one

• Look at the ruler below
• What can you read before you estimate?
• 12.8 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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Lets try graduated cylinders

• Look at the graduated cylinder below
• What can you read with confidence?
• 56 ml
• Now estimate the last digit
• 56.0 ml

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One more graduated cylinder

• Look at the cylinder below
• What is the measurement?
• 53.5 ml

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Rules for Significant figuresRule 1

• All non zero digits are ALWAYS significant
• How many significant digits are in the following
  numbers?

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

• 274
• 25.632
• 8.987

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Rule 2

• All zeros between significant digits are ALWAYS
  significant
• How many significant digits are in the following
  numbers?

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

• All FINAL zeros to the right of the decimal ARE
  significant
• How many significant digits are in the following
  numbers?

3 Significant Figures 5 Significant Digits 7
Significant Figures
32.0 19.000 105.0020
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Rule 4

• All zeros that act as place holders are NOT
  significant
• Another way to say this is zeros are only
  significant if they are between significant
  digits OR are the very final thing at the end of
  a decimal

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For example
How many significant digits are in the following
numbers?

• 1 Significant Digit

1 Significant Digit

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

3 Significant Digits
6 Significant Digits
2 Significant Digits
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Rule 5

• All counting numbers and constants have an
  infinite number of significant digits
• For example
• 1 hour 60 minutes
• 12 inches 1 foot
• 24 hours 1 day

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How many significant digits are in the following
numbers?

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

• 2 Significant Digits
• 6 Significant Digits
• 2 Significant Digits
• 3 Significant Digits
• 4 Significant Digits
• 1 Significant Digit
• 4 Significant Digits

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Rules Rounding Significant DigitsRule 1

• If the digit to the immediate right of the last
  significant digit is less than 5, do not round up
  the last significant digit.
• 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 43.82
• 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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Rounding Rule 3

• If the number to the immediate right of the last
  significant is a 5, and that 5 is followed by a
  non zero digit, round up
• 78.657 (you want 3 significant digits)
• The number you want is the 6
• The 6 is followed by a 5 and the 5 is followed by
  a non zero number
• Therefore, you round up
• 78.7

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Rounding Rule 4

• If the number to the immediate right of the last
  significant is a 5, and that 5 is followed by a
  zero, you look at the last significant digit and
  make it even.
• 2.5350 (want 3 significant digits)
• The number to the right of the digit you want is
  a 5 followed by a 0
• Therefore you want the final digit to be even
• 2.54

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Say you have this number

• 2.5250 (want 3 significant digits)
• The number to the right of the digit you want is
  a 5 followed by a 0
• Therefore you want the final digit to be even and
  it already is
• 2.52

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Lets try these examples

• 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

• Scientific notation is used to express very large
  or very small numbers
• It consists of a number between 1 10 followed
  by x 10 to an exponent
• The exponent can be determined by the number of
  decimal places you have to move to get only 1
  number in front of the decimal

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Large Numbers

• 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 at x 10 3.9923 x 10
• 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 10 4

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Small Numbers

• 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 at x 10 5.2 x 10
• 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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Scientific Notation Examples
Place the following numbers in 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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Going to Ordinary Notation Examples
Place the following numbers in ordinary 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 you know how to determine the number of
  significant digits in a number
• 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 must have the same number of decimal
  places as the one with the fewest
• For example

20.4
1.322
83
104.722
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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
• 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 ? 69.040
• - 476.33 ? -500
• 72.1440 ? 72.14
• 58.4428 ? 58.443
• 20 ? 20.00

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Rules for Multiplication Division

• When you multiply and divide numbers you look at
  the TOTAL number of significant digits NOT just
  decimal places
• For example

67.50 x 2.54
171.45
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Multiplication Division

• Because you are multiplying, you need to look at
  the total number of significant digits not just
  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 becomes
• 171

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Multiplication Division Problems

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

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

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More Significant Digit Problems

• 18.36 g / 14.20 cm3
• 105.40 C 23.20 C
• 324.5 mi / 5.5 hr
• 21.8 C 204.2 C
• 460 m / 5 sec

• 1.293 g/cm3
• 82.20 C
• 59 mi / hr
• 226.0 C
• 90 or 9 x 101 m/sec

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1. How many significant digits are in each of
the following? a. 12.5 b. .00230 c.
.01000 d. 100.025 e. 100.000

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3
4
6
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2. Round each of the following to three sig
figs. a. 125.365 b. .3002536458 c. 455.5
d. 278.96 e. 9.96543
.300
456
279
9.97
125
3. Change each of the following to scientific
notation. a. 100.00 b. .0020 c.
1000000 d. .02500 e. 70
360.4
360.4
1.0000 x 102
2.0 x 10-3
1 x 106
2.500 x 10-2
7 x 101
4. Add the following using sig figs.
a. 110.1 b. 78.59681 250.326 10.
89
360.4
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5. Subtract each of the following using sig
figs . a. 125.63 b. 56.056 25.364
25.4
100.27
30.7

• Multiply each of the following using sig figs.
• a. 200.00 x 30.0 b. 25.11 x 5.0

130
6.00 x 103