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Scientific Calc
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Scientific Calc

Scientific Notation and Standard Form

Formula, practice problems

Scientific notation is just a short hand way of expressing gigantic numbers like 1,300,000 or incredibly small numbers like 0.0000000000045. Also known as exponential form, scientific notation has been one of the oldest mathematical approaches. It is favored by many practicioners. If numbers are too big or too small to be simply calculated, people reffer to scientific notation to handle these circumstances. This method is used by engineers, mathematicians, scientists.
    An example of scientific notation is 1.3 ×106 which is just a different way of expressing the standard notation of the number 1,300,000. Standard notation is the normal way of writing numbers.
    Key Vocabulary
  • mantissa = this is the integer or first digit in any Scientific Notation. For example in 1.3 ×106, the mantissa is the "1"
    Other examples:
  • 1.2 ×1014 the positive exponent indicates a large number
  • 4.22×1011
  • 7.89 × 10-21 the negative exponent indicates a small number
 General Formula of Scientific Notation  
The general from of a number in scientific notation is

a ×10n where  1 ≤ a ≤ 10 and n is an integer.  In other words the number  that we'll call "a"  is is multiplied by 10, raised to some exponent n. This number "a" must be no smaller than 1 and no larger than 10. To illustrate this definition examine the following:
1.4 ×104 is a proper example of scientific notation because
  • 1.4, which is "a" in this example, is not smaller than 1 and not larger than 10 so it's ok.
  • 10's exponent is the integer 4.

  • .9 ×104 is a NOT proper example because
  • .9 which is "a" in this example, is smaller than 1 which is not allowed in scientific notation

  • 3.34 ×10½ is a NOT proper example because
  • 10's exponent is not an integer.

  • 4.34 ×10-55 is a proper example because
  • 4.34, which is "a" in this example, is not smaller than 1 and not larger than 10
  • 10's exponent is the integer -55. Integers can be negative

  • Scientific Notation Standard Form
    1.23 ×102 123
    1.23 × 103 1,230
    1.23 ×104 12,300
    1.23 × 105 123,000
    1.23 ×106
    1,230,000

    Convert Scientific Notation to standard form

    In the following sentences, convert from scientific notation to standard form. Practice Problems
      Answer
    Scientific Notation Standard Form
    1.303 •105
    9.43 •104
    3.423 •107
    3.23 •106
    6.003 •109

    Convert standard to Scientific Notation

    Practice Problems
    In the following sentences, convert from standard form to scientific notation.
      Answer
    Standard Form Scientific Notation
    19,300
    200,000
    3,013,000,000
    12,000,000,000
    130,000,000,000,000,000,000,000

    In the following sentences, state whether or not the given number is in Scientific Notation and explain your answer. 

      Answer
    Scientific Notation ??? Standard Form
    1) 13 •105
    2) 1.3 •105
    3) 3.423 •1090909090
    4) 3.23 •10-6
    5) 931 •10-9