Download Scientific Notation

What Is Scientific Notation And How Is It Used?
Today we will discuss the use of scientific notation also referred to as exponential notation. The
notation is based on powers of base number 10. The general format looks something like this:
N X 10x where N= number greater than 1 but less than 10 and x=exponent of 10.
Placing numbers in exponential notation has several advantages.
1.
2.
For very large numbers and extrememly small ones, these numbers can be placed in
scientific notation in order to express them in a more concise form.
In addition, numbers placed in this notation can be used in a computation with far greater
ease. This last advantage was more practical before the advent of calculators and their
abundance.
In scientific fields, scientific notation is still used. Let's first discuss how we will express a number
greater than 10 in such notational form.
Numbers Greater Than 10
1.
2.
3.
4.
We first want to locate the decimal and move it either right or left so that there are only one
non-zero digit to its left.
The resulting placement of the decimal will produce the N part of the standard scientific
notational expression.
Count the number of places that you had to move the decimal to satisfy step 1 above.
If it is to the left as it will be for numbers greater than 10, that number of positions will equal
x in the general expression.
As an example, how do we place the number 23419 in standard scientific notation?
1.
Position the decimal so that there is only one non-zero digit to its left. In this case we end up
with 2.3419
2.
3.
Count the number of positions we had to move the decimal to the left and that will be x.
Multiply the results of step 1 and 2 above for the standard form:
So we have: 2.3419 X 104
How about numbers less than one?
We generally follow the same steps except in order to position the decimal with only one non-zero
decimal to its left, we will have to move it to the RIGHT. The number of positions that we had to
move it to the right will be equal to -x. In other words we will end up with a negative exponent.
1
Negative exponents can be rewritten as values with positive exponents by taking the inversion of
the number.
For example:
10-5 can be rewritten as 1/ 105.
Here is an example to consider:
Express the following number in scientific notation:
0.000436
1.
First, we will have to move the decimal to the right in order to satisfy the condition of
having one non-zero digit to the left of the decimal. That will give us: 4.36
2.
Then we count the number of positions that we had to move it which was 4. That will equal
-X or x = -4
And the expression will be 4.36 X 10-4
What about numbers that are between 1 and 10?
In those numbers we do not need to move the decimal so the exponent will be zero. For example:
7.92 can be rewritten in notational form as: 7.92 X 100
Now it is your turn. Express the following numbers in their equivalent standard notational form:
1.
2.
3.
4.
5.
6.
123,876.3
1,236,840.
4.22
0.000000000000211
0.000238
9.10
One of the advantages of this notation that was mentioned earlier in the lesson was the ability to
compute with them in an easier fashion than with actual numerical equivalents.
Let's discuss how one would multiply with such notations. The general format for multiplying using
scientific notation is as follows:
(N X 10x) (M X 10y) = (N) (M) X 10x+y
1.
2.
3.
First multiply the N and M numbers together and express an answer.
Secondly multiply the exponential parts together by ADDING the exponents together.
Finally multiply the two results for your final answer.
2
For example:
(3 X 104) (1 X 102)
1.
2.
First 3 X 1=3
Second (104) (102) = 104+2 = 106
3.
Finally 3 X 106 for the answer
Another example:
(4 X 103) (2 X 10-4)
1.
2.
First 4 X 2 = 8
Secondly (103) (10-4) = 103+(-4) = 103-4 = 10-1
3.
Finally 8 X 10-1 would be the answer
Express the product of the following:
1.
(3 X 105) (3 X 106) = ?
2.
(2 X 107) (3 X 10-9) = ?
3.
(4 X 10-6) (4 X 10-4) = ?
As a footnote I need to discuss the need to express all answers in STANDARD notational form.
That is where the N or M number has a value between 1 and 10. If you should get a computed value
where the number that appears before the power of ten is greater than 10 or less than 1.
For example:
165 X 108
would have to have its decimal position adjusted to make it a standard notational form. You would
have to move the decimal to the left two positions. In the case where you move the decimal to the
left, you would add a positive 1 to the exponent for EACH position that you moved the deimal. So
here we would have to move the decimal two positions to the left which means adding a +2 to the
exponent.
That would make the answer 1.65 X 108+2 = 1.65 X 1010
On the other hand if you were required to move the decimal in the answer to the RIGHT in order to
express the standard form, you would add a -1 for each position to the right you repositioned the
decimal. So for example in the exponential equivalent
3
.0078 X 105
it is required that the decimal be moved 3 positions to the right. So the answer would be:
7.8 X 105+(-3) = 7.8 X 105-3 = 7.8 X 102
Remember to move the decimal to the left ADD a +1 to the exponent for each position that it is
moved.
To move the decimal to the RIGHT add a -1 to the decimal for each position the decimal is moved
to the right.
Division of Exponentially Notated Numbers
Now let's discuss Division using scientific notation. The general format is as follows:
N X 10x / M X 10y = N/M X 10x-y
Before we get into some examples of this type of mathematical operation, lets review the
conventional rules for combining signs that have been multiplied together. First there are two types
of SIGNS in math. There are operational signs which indicate a specific math operation to be
performed. A plus sign might indicate summation where a negative sign might indicate a
subtraction. Then there are value signs that ascribe a certain value to a number such as a -5
contrasted with a +5 in value. One would have a value five units to the left of the zero position on a
number line and the other would have a value of 5 positions to the right of the zero position of a
number line.
According to a set of conventional rules if an operational sign is multiplied by a value sign and both
are in the same mode (both plus or both minus) then they are replaced by a single sign which is
positive.
So for example: -(-) = + or +(+) = +
On the other hand if the operational and value signs are of opposite mode (one plus while the other
is negative) then the rule states that the two signs are replaced by a negative sign.
So for example: -(+) = - or +(-) = Keeping these sign rules in mind let us approach division with an example.
6 X 105 / 2 X 102 =
1.
2.
Perform the division on the N and M numbers , 6/2 = 3
Perform the division on the exponential parts by subtracting the exponent in the lower
number from the exponent in the upper number.
4
So 105 / 102 = 105-2 = 103
3.
Multiply the two results together, 3 x 103
Let's take another example,
8 X 10-3 / 2 X 10-2
1.
2.
First 8 / 2 = 4
Subtracting (operational sign is negative) a -2 (where the value sign is also negative) from a
-3. According to the sign rules the opperational sign that is negative and the value sign
which is also of the SAME mode (negative) will be replaced by a positive sign so -(-2) = +2.
3.
So: 10-3+2 = 10-1
4.
Multiply the result of step 1 above with the result of step 3 above to get the answer: 4 X 10-1
Express the quotients of the following:
1.
3.45 X 108 / 6.74 X 10-2 = ?
2.
6.7 X 107 / 8.6 X 103 = ?
(Be sure to express the answer in STANDARD form)
3.
4.7 X 10-2 / 5.7 X 10-6 = ?
Addition and Subtraction Using Exponential Notation
We can discuss these together since the important thing to remember about adding or subtracting is
that the exponents must be the same before the math operation can be performed. The general
format would be:
(N X 10x) + (M X 10x) = (N + M) X 10x
or
(N X 10y) - (M X 10y) = (N-M) X 10y
If the exponential equivalents do not have the same exponent then the decimal of one has to be
repositioned so that it's exponent is the same as all the rest of the numbers being added or
subtracted. The reason for that is that when we add or subtract numbers we must line all the
decimals up in the same position before we add or subtract columns of numbers. So for example:
(2.3 X 10-2) + (3.1 X 10-3)
5
We recognize that the two exponents are not the same so either the exponent of the first number has
to be changed to a -3 or the exponent of the second number has to be changed to a -2. It is really
arbitrary which one is changed. Let's change the first one.
2.3 X 10-2
Remember for each position to the right we add a -1 to the exponent and for each position to the left
we add a +1 to the exponent. In this case we must reposition the decimal one position to the right so
it becomes:
23. X 10-2 +(-1) = 23. X 10-3
Now both numbers will have the SAME exponent value.
(23. X 10-3) + (3.1 X 10-3) = (23. + 3.1) X 10-3 = 26.1 X 10-3
Here is another example:
(4.2 X 104) - (2.7 X 102) =
Lets adjust the exponent of the second number this time. Remember it is arbitrary which one we
adjust.
2.7 X 102
must be repositioned two places to the left so that we will add a +2 to the exponent to make it the
same value as the exponent of the first number.
2.7 X 102 becomes .027 X 102+2 = .027 X 104
Now the problem reads
(4.2 X 104) - (.027 X 104) = (4.2 - .027) X 104 = 4.173 X 104
Identify the sums or differences of the following:
1.
(8.41 X 103) + (9.71 X 104) = ?
2.
(5.11 X 102) - (4.2 X 102) = ?
3.
(8.2 X 102) + (4.0 X 103) = ?
4.
(6.3 X 10-2) - (2.1 X 10-1) = ?
R. H. Logan, Instructor of Chemistry, Dallas County Community College District, North Lake
College.
http://members.aol.com/profchm/sci_not.html
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Section A: Convert to Scientific Notation.
1.
1001
=
1.001 x 103
2.
53
=
5.3 x 101
3.
6,926,300,000
=
6.9263 x 109
4.
-392
=
-3.92 x 102
5.
0.00361
=
3.61 x 10-3
6.
0.13592
=
1.3592 x 10-1
7.
-0.0038
=
-3.8 x 10-3
8.
0.00000013
=
1.3 x 10-7
9.
0.567
=
5.67 x 10-1
Section B: Converting Back (Scientific => Decimal)
10. 1.92 x 103
=
1,920
11. 3.051x101
=
30.51
12. 4.29 x 102
=
429
13. 6.251 x 109
=
6,251,000,000
14. 8.317 x 106
=
8,317,000
15. 1.03 x 10-2
=
.0103
16. 8.862 x 10-1
=
.8862
17. 9.512 x 10-8
=
.00000009512
18. -6.5 x 10-3
=
.0065
19. 3.159 x 102
=
315.9
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Section C: Multiplication, Division and ... with Scientific Notation
20. 4.1357 x 10-15 X 5.4 x 102
=
2.2333 x 10-12
21. 1.695 x 104 ÷ 1.395 x 1015
=
1.2151 x 10-11
22. 4.367 x 105 X 1.96 x 1011
=
8.5593 x 1016
23. 6.97 x 103 X 2.34 x 10-6
=
1.631 x 10-2
24. 5.16 x 10-4 ÷ 8.65 x 10-8
=
5.9653 x 103
25. 9.68 x 104 - 3.2 x 10-2
=
9.68 x 104
26. 1.23 x 102 - 1 x 101
=
1.13 x 102
27. 3.5 x 106 + 1.25 x 106
=
4.75 x 106
6.75 x 108 + 9.9 x 101
=
6.75 x 108
28.
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