Scientific Notation
Examples of Large and Small Numbers Using Scientific
Notation
Example
Einstein's famous equation says that
E = mc^{2
}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 10^{8
}Count the number of digits to the
left of the three
and
m = 2.0 x 10^{3
}So that
E = mc^{2}
= (2.0 x 10^{3}) (3.0 x
10^{8})^{2}
= (2.0 x 10^{3}) (9.0 x
(10^{8})^{2}) Distributing
the exponent
= (2.0 x 10^{3}) (9.0 x 10^{16})
Using the Power Rule
= 18.0 x 10^{19}
Using the Addition Rule
= 1.8 x 10 x 10^{19}
Putting 18.0 into Scientific Notation
= 1.8 x 10^{20}
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.
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 10^{4}) = 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 10^{6
} (4 million) addition calculations?
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 10^{9} seconds.
Exercise
One Angstrom = 0.0000001 cm. One light year is
5.86 x 10^{12}
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.
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.
Exercise
The distance from the earth to the sun is 93,000,000 miles. The speed of
light is 3.0 x 10^{8 }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.
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