Scientific Notation

 Examples of Large and Small Numbers Using Scientific Notation

Example  

Einstein's famous equation says that

        E = mc2

where c is the speed of light 

        c = 300,000,000 m/sec

How much energy is emitted if 2,000 kg of matter is destroyed?

Solution  (Without Scientific Notation)

We compute:  

        E  =  2,000(300,000,000)2  =  180,000,000,000,000,000,000

This is a very awkward number.  

Solution  (With Scientific Notation)

We write

        c  =  300,000,000  =  3.0 x 108        Count the number of digits to the left of the three

and

        m  =  2.0 x 103

So that

        E  =  mc2  =  (2.0 x 103) (3.0 x 108)2  

        = (2.0 x 103) (9.0 x (108)2)       
Distributing the exponent

        = (2.0 x 103) (9.0 x 1016)           Using the Power Rule

        = 18.0 x 1019                             Using the Addition Rule

        = 1.8 x 10 x 1019                      
Putting 18.0 into Scientific Notation

        = 1.8 x 1020                              
Using the Addition Rule

 

In General we count the number of digits to the right of the first digit to determine the exponent.


Exercise

The number of molecules in a mole is

        602,000,000,000,000,000,000,000

Write this number in scientific notation.

        6.02 x 10^23


 

Example:  

The mass of a hydrogen atom is

        .00000000000000000000000167 kg.

what is the mass of 75,000 atoms?

Solution:  

To convert a decimal to scientific notation we count how many moves we must make to bring the decimal to the right of the first nonzero digit.  In our case we need to move the decimal over to the right 24 places hence we can write:

        mass of one atom  =  1.67 x 10-24  

We have that

        Total mass  =  Mass of one atom  x  number of atoms

or

        =  (1.67 x 10-24) (7.5 x 104)  =  12.525 x 10-20  Using the Addition Rule

        = 1.3 x 10 x 10-20   =  1.3 x 10-19                    Addition Rule again


 

Exercise:

A computer can  perform an addition calculation in 3.1 x 10-7 seconds.  How long will it take to perform 4 x 106  (4 million) addition calculations?

        1.24 seconds


Unit Calculations

Example 

How many seconds are there in 70 years?

Solution

                        365.25 days       24 hrs        60 min        60 sec
        70yrs                                                                                                                                                  
                               1 yr               1 day          1 hr            1 min

        =  2,209,032,000 seconds

        =  2.2 x 109  seconds.



Exercise

One Angstrom = 0.0000001 cm.  One light year is 5.86 x 1012 miles.  How many angstroms are there in a light year?

Note:  There are 5,280 feet in a mile and one centimeter is 0.39 inches.

9.52 x 10^24


 

Exercise

What percentage of the US population plays professional baseball?  Assume that there are 30 teams and each team has 40 players.  Also the population of the US is 300,000,000.

4 x 10^-4


 

Exercise

The distance from the earth to the sun is 93,000,000 miles.  The speed of light is 3.0 x 108 meters per second.  How many minutes does it take for a solar flare to reach the earth?

504 minutes


 

Exercise

Estimate how many particles of sand there are on the California coast.

 



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