Scientific Notation
S
10-5.
By Dr. Ma. Louise De Las Peñas
cientific notation is used most of the time when dealing with very large and very small
numbers. Scientific notation is an expression written as the product of a number whose
absolute value is between 1 and 10, including 1, and an integer power of 10.
For instance, in scientific notation, -5 000 000 is written as -5 × 106 and .000034 is 3.4 ×
One will find it very common to find numbers expressed in scientific notation in areas such
as astronomy, which deals with very large distances and huge masses; environmental
science, which deals with global populations and resources; biology and
chemistry, which deal with very small cells and molecules; economics
that deals with very large amounts of money and financial data; and
most other areas of science.
Here is a very large number: the number of one-celled organisms
that will sustain a whale for a few hours—it is 400 000 000 000 000!
A very small number for example, is the shortest wavelength of visible
light—about .0000004 meter!
In this epages article, we will look at some real-world situations where numbers are
expressed in scientific notation. In most cases, it is easier to handle computation involving very
large or very small numbers when these numbers are expressed in scientific notation.
Example 1. Common table salt consists of atoms of the elements sodium (Na) and chlorine (Cl),
arranged with alternate atoms at the corners of a cube. The distance between neighboring atoms
is 0.00000000000412 meters. What is the distance in scientific notation?
Solution:
0.00000000000412 = 4.12 × 10-12 (move the decimal to the right 12 places).
The distance between neighboring atoms in scientific notation is 4.12 × 10-12 meters.
Example 2. In 1985, the US national health care expenditure was 428.2 billion dollars. By 1995,
this figure had risen by a factor of 2.3; that is, it more than doubled in only 10 years. What was
the expenditure in 1995?
Solution:
Using scientific notation, we may write the 1985 health care expenditure amount in dollars
as
428.2 billion = 428.2 × 109
= (4.282 × 102) × 109
= 4.282 × 1011
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Now, to determine the expenditure in 1995, we multiply the result above by
2.3, as follows:
( 4.282 × 1011 ) × 2.3 = (2.3 × 4.282) × 1011
= 9.849 × 1011
Thus, the 1995 expenditure was nearly 984.9 × 1011 billion dollars.
Example 3. The population in the world in 1994 was estimated to be 5 642 151 000.
If the world population doubles, what will the population be? How about if the
world population triples?
Solution:
The population in the world in 1994 expressed in scientific notation is
5 642 151 000 = 5.642151 × 109.
If the world population doubles, we have
(5.642151 × 109) × 2 = 11.284302 × 109
= 1.1284302 × 1010
so that there would be about 1.1284302 × 1010 number of
people.
On the other hand, if the world population triples, we
obtain:
(5.642151 × 109) × 3 = 16.926453 × 109
= 1.6926453 × 1010
and there would be about 1.6926453 × 1010 number of people.
When performing calculations with the aid of a scientific calculator or a
graphics calculator, the technological tool may return results automatically in
scientific notation form.
For example, here are some screen dumps of the problem given above:
Now, let us now solve similar exercises in the worksheet!
References:
1. Experiencing Algebra, Thomason and Pessut, Prentice Hall, 1999.
2.
Mathematical Ideas, Miller, Heeren and Hornsby, Addison Wesley, 2001.
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Worksheet
A.
B.
Write the following in scientific notation:
1.
Michael Jackson’s album Thriller is one of the world’s best selling albums
of all time. It has sold over 40 000 000 copies.
2.
The lightest atom, hydrogen, weighs 0.0000000000000000000000017
grams.
Answer the problems given below:
1.
The pyramids at Giza are the most colossal funeral monuments ever built.
Each weighs over 6 000 000 tons. What is this weight in pounds, given
that 1 ton equals 2 000 pounds?
2.
The GNP of Switzerland was 2.49 × 1011 US dollars
in 1992. During the same time the GNP of Romania
was 2.49 × 1010 US dollars. By how much did the
GNP of Switzerland exceed that of Romania?
3.
On October 28, 1998, IBM announced that it was
going to offer a computer that is capable of 3.9 ×
108 operations per second. This was 15 000 times
faster than the normal desktop computer at that
time. What was the number of operations that
the normal desktop could do?
4.
A parsec, a unit of length used in astronomy, is 19 × 1012 miles. The mean
distance of Uranus from the sun is 1.8 × 107 miles. How many parsecs is
Uranus from the sun?
5.
A light-year is the distance that light travels in one year. Find the number
of miles in a light-year if light travels 1.86 × 105 miles per second.
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ANSWERS
A.
1.
2.
4 × 107
1.7 × 10-24 grams
1.
2.
3.
4.
5.
1.2 × 1010 pounds
2.241 × 1011 dollars
26,000 or 2.6 × 104 operations
approximately 9.474 × 10-7 parsec
approximately 5.87 × 1012 miles
B.
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