Scientific Notation

I.  Midterm

II.  Examples of Large and Small Numbers Using Scientific Notation

Example  

Einstein's equation says that

E = mc²

where c is the speed of light = 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)

= (2.0 x 10³)(9.0 x 10¹⁶) = 18.0 x 10¹⁹ = 1.8 x 10 x 10¹⁹  = 1.8 x 10²⁰  

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

600,200,000,000,000,000,000,000

Write this number in scientific notation.

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

=  (1.67 x 10^-24)(7.5 x 10⁴) = 12.525 x 10^-20 

= 1.3 x 10 x 10^-20   = 1.3 x 10^-19  

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 calculation?

III.  Unit Calculations

Example:  How many seconds are there in 70 years?

Solution:

70yrs (365.25 days/1 yr)(24 hrs/1 day)(60 mins/1 hr)(60 sec/1 min)  = 2,209,032,000 seconds

= 2.2 x 10⁹  seconds.

Exercise:  

One Angstrom = .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 .39 inches.

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 260,000,000.

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?

Exercise:  

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