L3: Scientific Notation



-Convention of Expressing Any Base 10 Number As a Product of a Number Between One and 9, multiplied by 10 to the Power of Some Exponent.


Exponentiation is the replication of multiplication the way multiplication is the replication of addition


100 = 1                                    

101 = 10                                   10-1 = 1/10 = 0.1

102 = 10(10) = 100                    10-2 = 1/[10(10)]=1/100=0.01

103=10(10)(10) = 1000              10-3 = 1/[101(10)(10)] = 1/1000 = 0.001



Advantages of Scientific Notation:         

          1.  Allows Awkwardly Large and Small Numbers to Be Expressed in                       Term of Compact and Easily Written Numbers


          2.  Allows Accurate Representation of  the Number of Significant                                      Figures in a Number, That Is a Measurement’s Precision, the                         “Certainty” of Our Measurements



Convert the following numbers to scientific notation

                   0.00456  and 456.00


A)  0.00456  Step one, since this number is less than 1 you need identify the power of 10 which you can multiply it by and set one digit to the left of the decimal.  Here, we can multiply by 1000 or 103 and force the original number to have one digit to the left of the decimal, but we have changed its value.

0.00456103 = 4.56 =(which is not equal to the original number)


Since we multiplied it by 103 we must also divide by 103 (effectively multiplying the original number by 1 and not changing it's value).  Than we express the factor in the denominator as a power of 10.



Note, once you get the hang of this you can just count the number of digits you need to move the decimal to give the original number one digit to the left of the decimal and multiply that number by 10 to the negative value of those digits.  Here we moved the decimal three positions to the right, so we multiply by 10-3.


B. 456.00, Since this number has a value greater than 9 we must divide it by a factor of 10 to make the first non-zero digit value between 1 and 9.




Note, once you get the hang of this you can just count the number of places you move the decimal to the right (giving a value with one digit to the left of the decimal) and multiply by 10 to the power of that number (2 in this case).



Using Scientific Notation to Express Significant Digits:


How can you express the number 400 to 2 significant digits?

You must use scientific notation.

4.0 x 102


As scientific notation always has a decimal point there is never a problem expressing significant digits.


Multiplication of numbers in Scientific notation


Try plugging the following number into your calculator.  Odds are you will get an error message.



You will often get numbers which are bigger or smaller than a calculator can handle.  You need to treat the power separately from the number






Addition and Subtraction of Numbers in Scientific Notation


Consider the following problem

4.860 x 1012 +  9.7 x 1010 + 3.68x1011  


You first need to express all numbers to the same power so you can line up the decimal point.  It is suggested that you choose the largest power and make everything else a fraction.