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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. 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.

  2. 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. 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.

  2. The resulting placement of the decimal will produce the N part of the standard scientific notational expression.

  3. Count the number of places that you had to move the decimal to satisfy step 1 above.

  4. 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. Count the number of positions we had to move the decimal to the left and that will be x.

  3. Multiply the results of step 1 and 2 above for the standard form:

    So we have: 2.3419 X 10 4

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.

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. 123,876.3

  2. 1,236,840.

  3. 4.22

  4. 0.000000000000211

  5. 0.000238

  6. 9.10

To determine how you did with the above problems, click here

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. First multiply the N and M numbers together and express an answer.

  2. Secondly multiply the exponential parts together by ADDING the exponents together.

  3. Finally multiply the two results for your final answer.

For example:

(3 X 104) (1 X 102)

  1. First 3 X 1=3

  2. Second (104) (102) = 104+2 = 106

  3. Finally 3 X 106 for the answer

Another example:

(4 X 103) (2 X 10-4)

  1. First 4 X 2 = 8

  2. Secondly (103) (10-4) = 103+(-4) = 103-4 = 10-1

  3. Finally 8 X 10-1 would be the answer

Now you try some examples.

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) = ?

In order to check to see if you did well on the above examples, click here

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

.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. Perform the division on the N and M numbers , 6/2 = 3

  2. Perform the division on the exponential parts by subtracting the exponent in the lower number from the exponent in the upper number.

    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. First 8 / 2 = 4

  2. 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

Now it is your turn:

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 = ?

In order to check your results on the above problems click here

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)

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

Now it is your turn,

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) = ?

In order to check your results of the above examples click here

That concludes our discussion on exponential notation and computation with such forms. Another particularly useful Web site on this topic is ChemTeam's Scientific Notation. If you have any questions on the reading, click on the mailbox icon below and leave e-mail to me.

R. H. Logan, Instructor of Chemistry, Dallas County Community College District, North Lake College.

Acknowledgements

Send Comments to R.H. Logan: [Email] rhlogan@ix.netcom.com 

All contents copyrighted (c) 1995
R.H. Logan, Instructor of Chemistry,DCCCD
All Rights reserved


Revised: 7/11/2001



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Express the following numbers in their
equivalent standard notational form:

    

  1. 123,876.3 = 1.238763 X 105


    

  2. 1,236,840. = 1.23684 X 106

    
 
  3. 4.22 = 4.22 X 100

    
 
  4. 0.000000000000211 = 2.11 X 10-13

    

  5. 0.000238 = 2.38 X 10-4


    
 
  6. 9.10 = 9.10 X 100


Return To the lesson





Express the product of the following:

    
 
  1. (3 X 105) (3 X 106) = 9 X 1011


    
 
  2. (2 X 107) (3 X 10-9) = 6 X 10-2

    
 
  3. (4 X 10-6) (4 X 10-4) = 1.6 X 10-9



Return To The Lesson





Express the quotients of the following:

    
 
  1. 3.45 X 108 / 6.74 X 10-2 = 5.12 X 109


    

  2. 6.7 X 107 / 8.6 X 103 = 7.80 X 103 (Be sure to express the answer in STANDARD form)

    

  3. 4.7 X 10-2 / 5.7 X 10-6 = 8.25 X 103



ReturnTo The Lesson





Identify the sums or differences of the following:

    

  1. (8.41 X 103) + (9.71 X 104) = 10.55 X 104 = 1.055 X 105


    

  2. (5.11 X 102) - (4.2 X 102) = .91 X 102 = 9.1 X 101

    

  3. (8.2 X 102) + (4.0 X 103) = 48.2 X 102 = 4.82 X 103


    

  4. (6.3 X 10-2) - (2.1 X 10-1)=-14.7 X 10-2 = - 1.47 X 10-1


Return To The Lesson

URL:http://edie.cprost.sfu.ca/~rhlogan/sci_not.html