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topic 001

SCIENTIFIC NOTATION
Scientific notation is the way that scientists easily handle very large numbers or very small numbers.
For example, instead of writing 0.0000000056, we write 5.6 x 10-9. So, how does this work?
We can think of 5.6 x 10-9 as the product of two numbers: 5.6 (the digit term) and 10-9 (the
exponential term). Careful, the exponential number is not necessarily raised nor indicated by an "e".

Examples of scientific notation.

               10000 = 1 x 10e4
                                             24327 = 2.4327 x 10e4
               1000 = 1 x 10e3
                                             7354 = 7.354 x 10e3
               100 = 1 x 10e2
                                              482 = 4.82 x 10e2
               10 = 1 x 101
                                                 89 = 8.9 x 101 (not usually done)
                  1 = 100
          1/10 = 0.1 = 1 x 10e-1
                                                   0.32 = 3.2 x 10-1 (not usually done)
          1/100 = 0.01 = 1 x 10e-2
                                                   0.053 = 5.3 x 10-2
           1/1000 = 0.001 = 1 x 10e-3
                                                   0.0078 = 7.8 x 10-3
           1/10000 = 0.0001 = 1 x 10e-4
                                                   0.00044 = 4.4 x 10-4
As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give
the number in long form. A positive exponent shows that the decimal point is shifted that number of
places to the right. A negative exponent shows that the decimal point is shifted that number of
places to the left.


In scientific notation, the digit term indicates the number of significant figures in the number. The
exponential term only places the decimal point. As an example,
                             46600000 = 4.66 x 10e7
This number only has 3 significant figures. The zeros are not significant; they are only holding a place.
As another example,
                              0.00053 = 5.3 x 10e-4
This number has 2 significant figures. The zeros are only place holders.
If this concept is not clear, make certain to tell Dr. Rosenthal

Addition and Subtraction:

     All numbers are converted to the same power of 10, and the digit terms are added or
     subtracted.
     Example: (4.215 x 10e-2) + (3.2 x 10e-4) = (4.215 x 10e-2) + (0.032 x 10e-2) = 4.247 x 10ee-2
     Example: (8.97 x 10e4) - (2.62 x 10e3) = (8.97 x 10e4) - (0.262 x 10e4) = 8.71 x 10e4

Multiplication:

     The digit terms are multiplied in the normal way and the exponents are added. The end result is
     changed so that there is only one nonzero digit to the left of the decimal.
     Example: (3.4 x 10e6)(4.2 x 10e3) = (3.4)(4.2) x 10(6+3) = 14.28 x 109 = 1.4 x 10e10
     (to 2 significant figures)
     Example: (6.73 x 10e-5)(2.91 x 10e2) = (6.73)(2.91) x 10(-5+2) = 19.58 x 10e-3 = 1.96 x 10e-2
     (to 3 significant figures)

Division:

     The digit terms are divided in the normal way and the exponents are subtracted. The quotient
     is changed (if necessary) so that there is only one nonzero digit to the left of the decimal.
     Example: (6.4 x 10e6)/(8.9 x 10e2) = (6.4)/(8.9) x 10(6-2) = 0.719 x 10e4 = 7.2 x 10e3
     (to 2 significant figures)
     Example: (3.2 x 10e3)/(5.7 x 10e-2) = (3.2)/(5.7) x 103-(-2) = 0.561 x 10e5 = 5.6 x 10e4
     (to 2 significant figures)

Powers of Exponentials:

     The digit term is raised to the indicated power and the exponent is multiplied by the number
     that indicates the power.
     Example: (2.4 x 104)3 = (2.4)3 x 10(4x3) = 13.824 x 1012 = 1.4 x 1012
     (to 2 significant figures)
     Example: (6.53 x 10-3)2 = (6.53)2 x 10(-3)x2 = 42.64 x 10-6 = 4.26 x 10-5
     (to 3 significant figures)

     Example:
QUIZ: (assume the e where appropriate)
 Question 1
               Write in scientific notation: 0.000467 and 32000000
 Question 2
               Express 5.43 x 10-3 as a number.
 Question 3
               (4.5 x 10-14) x (5.2 x 103) = ?
 Question 4
               (6.1 x 105)/(1.2 x 10-3) = ?
 Question 5
               (3.74 x 10-3)4 = ?
 Question 6
               The fifth root of 7.20 x 1022 = ?

Answers: (1) 4.67 x 10-4; 3.2 x 10e7 (2)0.00543 (3) 2.3 x 10e-10 (2 significant figures) (4) 5.1 x 10e8
(2 significant figures) (5) 1.96 x 10e-10 (3 significant figures) (6) 3.73 x 10e4 (3 significant figures)

Scientific calculations are frequently handled by expressing quantities in scientific notation. Such
operations require simple manipulation of exponents, usually exponents of 10. When the same base is
used (e.g. 10), the following rules apply:    1.When the operation involves multiplication, add the exponents algebrically.
          example: 103 x 10e4 = 10(3 + 4) = 10e7
          example: 105 x 102 x 10-3 = 10(5 + 2 + (-3)) = 10e4

   2.When the operation involves division, subtract the divisor exponent from the numerator
     exponent.

          example: 105/103 = 10(5 - 3) = 10e2
          example: 107/1012 = 10(7 - 12) = 10e-5
          example: 108/10-3 = 10(8 - (-3)) = 10e11
          example: (106 x 104)/(103 x 102) = 10(6 + 4 - (3 + 2)) = 10e5

   3.When the operation involves powers or roots, multiply the exponent by the power number or
     divide the exponent by the power number, respectively.
          example: (105)3 = 10(5 x 3) = 10e15
          example: (10-7)4 = 10(7 x 4) = 10e28
          example:  = (104)1/2 = 10(4 x 1/2) = 10e2
          example:  = (1020)1/5 = 10(20 x 1/5) = 10e4


QUIZ: Give the correct answer.
 Question 1
            10e3 x 10-e4 = ?
 Question 2
            10e6/10e-3 = ?
 Question 3
            (10e2)4 = ?
 

Answers: (1)10-1 (2) 109 (3) 108

 Credit to: Chem-Math Skills Review, TA&M