SELF STUDY SHEET - SCIENTIFIC NOTATION
The student should be able to:
1. Write any decimal fraction in corresponding scientific, or exponential notation.
2. Express a number originally in exponential notation in corresponding decimal form.
3. Add and subtract numbers in exponential notation and express the answer in proper form.
The numerical values encountered in most science courses tend to be extremely large or extremely small-for example, the distance from the earth to the sun 93,000,000 miles or the radius of an atom 0.000,000,01
of a centimeter. In order to make it easier to handle such numbers, they are expressed in exponential, or as
it is called, scientific notation.
1.
Definition of Exponents
If a number is multiplied with itself, we call the product a power of the original number. For
example:
1. 2  2  2 = 8 or 23 =8
In expression number 2, the number is called the base and the number 3 is called the exponent.
Expression number 2 is read "two to the third power equals eight". The statement 2 3 means
multiply two with itself three times.
Since our computation system is based upon the number ten, the remainder of this discussion will be
primarily concerned with expressions involving powers of ten.
2.
Power of Ten
The number ten may be expressed as 101; that is, ten to the first power. We may express other
numbers as exact powers of ten also.
For example:
100 = 10  10 or we can write 102
1000 = 10  10  10 or 103
QUESTION III - Write the number ten thousand (10,000) as multiples of 10
QUESTION IV - Write 10,000 as a power of ten; that is in exponential notation .
QUESTION V - What is the base for the number you gave as the answer to question IV?
QUESTION VI - What is the exponent of the number in the answer to question IV?
.
If you answered all of the above questions correctly, proceed to part 3.
3.
Negative Exponents of Ten
Fractions are generally denoted in two ways: First, as "common fractions" or ratios; for example,
1/2 or 3/4. Fractions may also be expressed in decimal form--thus 0.5 or 0.75. The fraction 1/10 may be
written as 0.1 and 1/100 as 0.01, etc.
QUESTION VII - Write 100 in exponential form _________________.
We can write the fraction 1/100 as 10-2. One of the basic laws of operations with exponents says that we
can divide two numbers with the same base by subtracting their exponents. We therefore write
1/100 = 1/102 = 100 /102 = 100-2 = 10-2
Therefore, 0.01 is written as 10-2 in exponential notation.
QUESTION VIII - Write 0.1 in exponential form _____________________.
QUESTION IX - Write 0.001 in exponential form ___________________.
4.
Numbers Other Than Exact Multiples of Ten
Suppose we wish to write the number 200 in exponential notation. We cannot express this number
exactly as a power of ten. Instead, we can write it as 2 x 100, or 2 x 10 2. This last expression is
the correct one.
We shall say that a number is written in "standard notation" when it is written with one non-zero digit to
the left of the decimal point and expressed as a multiple of the appropriate power of ten. For example:
125 in standard notation is 1.25  102.
Expressing 0.0045 in standard notation gives 1.5  10-3 .
QUESTION X - Tell whether each of the following numbers is standard notation or not. It is not, then
express it in that form:
3  10-2 _____________________________________
7.061  1010 _________________________________
-1.2  10-3 __________________________________
0.1  10-2 ___________________________________
101  10-2 ___________________________________
In order to write a number in standard scientific notation, you may imagine performing the following
operations:
1.
Move the decimal point to the left or right as many places as necessary in order to have only onezero digit to the left of the decimal.
2.
Count the number of places that the decimal was moved. This number is the exponent of ten that
is to be used in writing the number.
3.
The sign of the exponent is positive if the original number is greater than one and negative if the
original number is a fraction.
Examples: Each of the following numbers can be written in standard notation as follows:
1.
730...The decimal is moved two places to the left to give 7.30  102. Note that the ten is
raised to the second power.
Question XI - Why is two the exponent of 10 in the above example?
2.
10,400...Here the decimal is shifted four places to the left to give 1.04  104. (It is not
necessary to keep the last two zeroes.)
3.
-5,250...The decimal is shifted three places to the left this time. The negative sign of the
number be kept, thus -5.25  103.
4.
0.00333...In this case the decimal must be shifted to the left. It must be shifted three
places before a non-zero digit accrues to the left of the decimal. Note also that the
number is less than one; therefore, the exponent has a negative value. The result is 3.33 
10-3.
It should be clearly understood at this point that the value of a number is not changed by changing the
form in which it is written.
QUESTION XII - Express each of the following numbers in standard notation:
236 _______________________
0.124 _____________________
-67,800 ___________________
0.0134 ____________________
If you successfully answered all of the above questions, continue on to the next page.
5.
Summary of Results Thus Far
A number written in standard exponential notation or "scientific notation" is said to be made up of
two factors. The first factor is always a number between 1.00 . . . and 9.99 . . . ; that is, a number
with one non-zero digit to the left of the decimal place. The second factor is an appropriate
power of ten. Thus, the distance from the earth to the sun is written as 9.3  107 miles. In this
number, the first factor is 9.3 and the second is 10 7. The exponent of ten is seven in this example.
We now wish to give formal statements of some definitions from algebra.
a. Let “a” represent any number; then the product of “a” with itself some number of times is a 
a  a  . . . n times = an.
3.1  3.1  3.1  3.1 = 3.14. In this example the number corresponding to “a” is 3.1 and “n”
had the value 4.
b. 1/an = a-n. This statement simple means that a negative exponent implies that the number is a
fraction.
c. Any number raised to the zero power is defined to be exactly 1.
Example: 1,0000 = 1
1/20 = 1
6.
Addition and Subtraction of Numbers Written in Exponential Form
We may carry out the normal arithmetic operations of addition and subtraction with numbers in
standard form, but it must be remembered that the exponential factor will not be involved directly.
Example: Suppose we wish to form the sum 20 + 30. The answer, of course, is
50. Written 2  101 + 3  101 = 5  101
We may state the following rule: We add the non-exponential factors. The exponent of ten is not
changed.
Some special situations need to be considered:
a. It may be that in finding a sum, the result is greater than ten. In this case, the answer must be
rewritten so that it will be standard form.
Example:
3.1  104 + 7.5  104 + 1.1  104 = 11.7  104
The answer is not in standard form, so we rewrite it as:
1.17  105
b. We may wish to add or subtract numbers that do not all have the same exponential factor. In
this case, it is necessary to rewrite the numbers so that they all have the same exponential
term. It may be necessary to express the number in other than standard form to do this, but the
final answer should be put in standard form.
QUESTION XIII - Find the indicated sum or difference:
1.
7.6  102 + 1.8  102 =
2.
4.15  10-3 + 2.10  10-3 =
3.
8.4  109 - 6.2  109 =
4.
7.8  103 + 1.3  103 - 2.5  103 =
5.
4.4  103 + 1.1  104 =
6.
3.30  10-2 - 1.1  10-3 =
7.
1.2  104 + 7.3  102 + 1.1  102 =