MATH for SCIENCE Scientific Notation

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MATH for SCIENCE Scientific Notation

• Scientists
• A. Deal with
• Some very large numbers
• Some extremely small numbers
• These numbers can be quite cumbersome to work
  with. To
• make it easier scientists frequently use
  Scientific Notation.
• B. Scientific Notation
• A numerical shorthand frequently used for writing
  very large and extremely small numbers.
• C. Converting Decimal format to Scientific
  Notation format
• Scientific Notation sets up numbers with
• a. Only the leading, non-zero digit/number to the
  left of the decimal point
• in the units place.
• b. All the remaining numbers are placed to the
  right of the decimal point.
• c. Then, that number is multiplied by 10n.

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• d. The power/exponent n will correspond
  to
• 1. the number of places.
• 2. the direction the decimal point was
  moved.
• e. The power n is
• 1. positive () when the original number is
  greater than 1
• 2. negative (-) when the original number is less
  than 1.
• f. For numbers greater than 1
• 1. count the number of places the decimal point
  was
• moved to the left until you have only one
  non-zero
• number/digit to the left of the decimal point.
• 2. that number becomes the power/exponent that
  goes to
• the upper right of the 10n.

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• g. Examples
• Moving the Decimal Pt. Answer
• i. 98765 9.8765 9.8765 x 104
• 4 3 2 1
• ii. 123 1.23 1.23 x 102
• 2 1
• iii. 4680 4.680 4.680 x 103
• 3 2 1

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h. For numbers less than 1

• i. count the number of places the
  decimal point was moved to the right until you
  have only one non-zero number/digit to the left
  of the decimal point.
• ii. count the number of places the
  decimal point was moved to the right until you
  have only one non-zero number/digit to the left
  of the decimal point.
• iii. Examples
• Moving the Decimal Pt. Answer
• 0.00012 0.0001.2 1.2 x 10 -4
• 1 2 3 4
• 0.0000000345 0.00000003.45 3.45 x 10 -8
• 1
  2 3 4 5 6 7 8
• 0.067 0.06.7 6.7 x 10 -2
• 1
  2

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D. Converting Scientific Notation format to
Decimal format

• 1. For numbers with 10n
• a. Move the decimal point to the right to
  make the number bigger (greater than 1).
• b. When you move the decimal point and
  there are no
• numbers left, fill the counting loops in with
  zeros.
• 2. Examples
• Moving the Decimal Pt. Answer
• 7.43 x 105 7.43000. 743,000.
• 1 2 3 4 5
• 2.153 x 102 2.15.3 215.3
• 1 2
• 6.8 x 104 6.8000. 68,000.
• 1 2 3 4

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• 3. For numbers with 10-n
• Move the decimal point to the left
  to
• make the number smaller (less than
  1).
• 4. Examples
• Moving the Decimal Pt. Answer
• 3.75 x 10 -2 03.75 0.0375
• 2 1
• 8.4 x 10 -5 .00008.4 0.000084
• 5 4 3 2 1
• 1.26 x 10 -3 .001.26 0.00126
• 3 2 1

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II. Computations with Scientific
Notation When multiplying or dividing
with two or more numbers in
Scientific Notation format, the process is done
in two stages.

• A. Multiplication
• 1. Stage 1 has 2 steps
• a. Step 1 Multiply the two leading
  numbers together.
• b. Step 2 Multiply the base 10
  numbers together.
• (Remember, this means you just add the
  powers/exponents.)
• c. Example
• (2.5 x 103) (5.0 x 102)
• (2.5 x 5.0) (103 x 102)
• 12.5 x 105

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• 2. Stage 2 has 2 steps
• These two steps are determined by which format,
  decimal or Scientific
• Notation, is required for the answer.
• Decimal Format Scientific Notation
  Format
• Step 3 Move the decimal point the number Step
  3 Take the decimally formatted first
• of places and the direction
  indicated number and change it to
• by the x 10n exponent.
  Scientific Notation.
• Step 4 Fill in the blank loops/spaces with
  Step 4 Multiply the number from step 3
• zeros. with the base 10 number from step
• 12.5 x 105 12.5 x 105
• 12.50000. (1.25 x 101) (105)
• 1 2 3 4 5
• 1,250,000. 1.25 x 106

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B. Examples

• 1. (3.3 x 10 -2) (4.5 x 105)
• (3.3 x 4.5) (10 -2 x 105)
• 14.85 x 103
• Decimal Format Scientific Notation Format
• 14.85 x 103 14.85 x 103
• 14.850. (1.485 x 101) (103)
• 1 2 3
• 14,850. 1.485 x 104
• 2. (8.2 x 10-3) (3.6 x 10-2)
• (8.2 x 3.6) (10-3 x 10-2)
• 29.52 x 10-5
• Decimal Format Scientific Notation Format
• 29.52 x 10-5 29.52 x 10-5
• .00029.52 (2.952 x 101) (10-5)
• 5 4 3 2 1
• 0.0002952 2.952 x 10-4

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• 3. (6.95 x 104) (2.3 x 10-7)
• (6.95 x 2.3) (104 x 10-7)
• 15.985 x 10-3
• Decimal Format Scientific Notation
  Format
• 15.985 x 10-3 15.985 x 10-3
• .015.985 (1.5985 x 101) (10-3)
• 3 2 1
• 0.015985 1.5985 x 10-2

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C. Division

• 1. Stage 1 has 2 steps
• Step 1 Divide the two leading numbers, then
• Step 2 Divide the base 10 numbers
• (Remember this means you just subtract the
  exponents/powers.)
• 2. Stage 2 Convert the result of stage 1 to
  either or both decimal format /or
  Scientific Notation.
• D. Examples
• 1. 96.24 x 10-3 ? 96.24 x 10-3 ? 80.2 x 10-3
  (-5) 80.2 x 102 8.02 x 103 or 8020
• 1.2 x 10-5 1.2 10-5
• 2. 8.2 x 105 ? 8.2 x 105 ? 1.2 x 103 or
  1,200
• 6.0 x 102 6.0 102
• 3. 1.92 x 104 ? 1.92 x 104 ? 0.3048 x 107
  (3.048 x 10-1) (107) 3.048 x 106
• 6.3 x 10-3 6.3 10-3
  or 3,048,000

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• E. Addition Subtraction
• 1. To add or subtract any number in Scientific
  Notation, each number MUST
• a. Be converted back to decimal format.
• b. Line up the decimal point.
• c. Then, add or subtract the numbers.
• F. Examples
• 1. 1.4 x 103 3.0516 x 104 9.723 x 102
• 1.4 x 103 1400.
• 3.0516 x 104 30516.
• 9.723 x 102 972.3
• 32,888.3 3.28883 x 104
• 2. 4.0125 x 103 - 6.375 x 102
• 4.0125 x 103 4012.5
• 6.375 x 102 - 637.5
• 3375.0 3.3750 x 103

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• 3. 1.3842 x 102 4.965 x 101 8.6 x 10-2
• 1.3842 x 102 138.42
• 4.965 x 101 49.65
• 8.6 x 10-2 .086
• 188.156 1.88156 x 102
• 4. 7.385 x 10-2 - 8.126 x 10-3
• 7.385 x 10-2 0.07386
• 8.126 x 10-3 - 0.008126
• 0.065724 6.5724 x 10-2