WRITING NUMBERS IN EXPONENTIAL NOTATION

    This module consists of a written script and a post test.  The objectives of this module are to:

1. write large and small numbers in exponential notation

2. convert numbers to standard exponential notation

PURPOSE:

This learning module consists of a review of standard exponential notation.  This review includes converting numbers as we usually write them into standard exponential notation and vice versa, as well as converting numbers all ready in exponential form into standard exponential notation.  Some simple and easy to remember rules will be presented to aid in the conversion of one form to another.

DIRECTIONS

To complete this module, you need only this written script.  Within this script are sample problems to work.  Do these problems on scratch paper, then compare the answers you get, with the correct answers on the following page.  Make sure that you understand all the examples before you proceed.  When you have completed this module and feel that you understand the material, obtain and complete a post-test from the Science Learning Center staff.  If you do not pass the post-test, you may review the module and retake the post-test as many times as needed.  If you have any questions, please ask!  You may contact the Science Learning Center at slc@mail.uccs.edu

PART I: INTRODUCTION

Why do we need to know how to express numbers in exponential notation?  In science as well as in other disciplines, it is often necessary to work with extremely large or extremely small numbers.  For example:

29,980,000,000 cm/sec is the speed of light in a vacuum

602,000,000,000,000,000,000,000 is the number of atoms in 1 mol Carbon

0.000000000000000000000001674 g is the mass of one hydrogen atom

It is extremely cumbersome to do calculations with such numbers.  A simple way to avoid the problems involved in working with very large and very small numbers is to express the numbers in standard exponential notation (or standard scientific notation.)  In exponential notation, numbers are expressed as a coefficient times an integral power of ten (the exponent):

c x 10ⁿ

where c is the coefficient and n is the exponent.

The above numbers, expressed in exponential notation, are:

2.998 x 10¹⁰ cm/sec (the speed of light in a vacuum)              

6.02 x 10²³ atoms (the number of atoms in 1 mol of carbon)

1.674 x 10^-24 g (the mass of one hydrogen atom)

Notice that in standard exponential notation, the number is always written with only one digit to the left of the decimal.

PART II: CONVERTING NUMBERS TO STANDARD EXPONENTIAL NOTATION

To convert numbers to standard exponential notation, we need to write the number in the form c x 10ⁿ; that is, as a coefficient with only one digit to the left of the decimal place times ten raised to an exponent.

Consider the number 2173.0.  This is equivalent to writing

2173.0 = 2.1730 x 1000
2173.0 = 2.1730 x (10)(10)(10)

Since(10)(10)(10) = 10³, we  may also write 2173.0 = 2.1730 x 10³ . This number is now in standard exponential notation!

Let's look at another example, this time a very small number. The number is 0.000716.  This is equivalent to writing:

0.000716 = 7.16 x 0.0001

0.000716 = 7.16 x (0.1)(0.1)(0.1)(0.1). 

Since (0.1)(0.1)(0.1)(0.1) = 10^-4, we can also write: 0.000716 = 7.16 x 10^-4  This number is now in standard exponential notation!

An easier method of converting a number to exponential notation is to note the direction and number of places you had to move the decimal point to leave just one digit to the left of the decimal point.  For example, in the number 2173.0 the decimal was moved three places to the left in order to have just one digit to the left of the decimal. Therefore, the exponent on the ten is 3 and the number should be written 2.1730 x 10³

Another example:Consider the number 6,200,000,000,000.  the decimal moves 12 places, so the exponent is 12.  The number 6,200,000,000,000 should be written as 6.2 X 10¹² ]

Practice Problem 1:  Write the number 3,170,000,000 in standard exponential notation. Solution: Even though the decimal is not written in, it is assumed that it is at the end of the number.  To write this number in exponential notation, it is necessary to move the decimal 9 places to the left.  The answer is:  3.17 x 109

When working with small numbers, the same general pattern develops.  Again, consider the number 0.000716.  In order to write this small number with only one digit to the left of the decimal, the decimal must be moved 4 places to the right. In standard exponential notation, this number becomes 7.16 x 10^-4  where the negative sign on the exponent means that the decimal had to be moved to the right and the 4 means that the decimal was moved 4 places to the right.

As another example, consider the number 0.0000000123.   To write the number in standard exponential notation, the decimal must move 8 places to the right, so the exponent is a - 8.

Practice Problem 2:  Write the number 0.0000575 in standard exponential notation. Solution: The decimal must be moved 5 places to give the number 5.75 x 10-5

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    In summary, to convert a number to standard exponential notation:

1. Move the decimal point so that there is only one digit to the left of the decimal.

2.Count the number of places the decimal moves.

3. If the decimal moves to the left, the exponent on 10 is equal to the number of places the decimal was moved and has a positive sign.

4. If the decimal moves to the right, the exponent on 10 is equal to the number of places the decimal was moved and  has a negative sign.

This process results on a number of the general form: c x 10ⁿ where c is a number between 1 and 9.999 and n is a positive or negative integer.

Practice Problems: 

Before proceeding to the next section, convert the following numbers to standard exponential form.   (Use a piece of scratch paper.  If you don't have any, check at the desk.)   When you are finished with the problems, check your answers on the last page of this module.  If your answers were correct, move on to the next section; if you missed some of the problems, review the section.   Make sure you understand this process before proceeding to the next section.

	Problem:	Answer:
	3,760,000,000	3.76 x 109
	0.000005678	5.678 x 10-6
	0.000000000000000090	9.0 x 10-17
	476,100	4.761 x 105

PART III:  CONVERTING TO STANDARD EXPONENTIAL NOTATION

Often numbers that result from a calculation are not in standard exponential notation.  To add or subtract numbers you need to change the values of the exponents.  You will need to convert from exponential notation to standard exponential notation. In the example below, the number 81.7 x 10³ is not in standard exponential notation.  (It has more than one digit to the left of the decimal point.)  To convert this number to standard exponential notation, we need to move the decimal point to the left by 1 place and increase the exponent by 1 as shown.

81.7 x 103 DECIMAL MOVES 1 PLACE 8.17 X 10(3+1) 1 IS ADDED TO THE CURRENT EXPONENT OF 10 8.17 x 104

Below is an example with a negative exponent of 10.  We again added a number to the exponent equal to the number of places the decimal was moved.

2040. x 10-5 DECIMAL MOVES 3 PLACES 2.040 x 10(-5+3) 3 IS ADDED TO THE CURRENT EXPONENT OF 10 2.040 X 10-2

If the decimal must be moved to the right we subtract the number of places the decimal moves from the current exponent.  This is shown in the following examples.

0.0414 x 10-3 DECIMAL MOVES 2 PLACES TO THE RIGHT 4.14 x 10(-3-2) 2 IS SUBTRACTED FROM THE CURRENT EXPONENT OF 10 4.14 x 10-5

 

Practice Problem 3: Write 0.00051 x 10-2 in standard exponential notation. Solution: The decimal must be moved 4 places to the right.  It must be subtracted from the current exponent: 5.1 x 10(-2-4) = 5.1 x 10-6

We can formulate two general rules for converting numbers to standard exponential notation:

1. If the decimal point moves to the left, add the number of places the decimal moves to the current exponent of ten.

2. If the decimal point moves to the right, subtract the number of places the decimal moves from the current exponent often.

Alternative Method: 

Another method exists for converting numbers in non-standard exponential notation to standard exponential notation.  As an example, consider the number 0.0052 x 10⁴ and convert it to standard exponential notation:

Solution:  First change the 0.0052 to 5.2 since standard exponential notation requires that there be only one digit to the left of the decimal. By changing 0.0052 to 5.2, you have made it larger by 3 decimal places. Therefore, to compensate, the exponent of 10 must be made smaller by 3 numbers.  So, subtract 3 from the exponent:

10^(4-3) = 10¹

0.0052 x 10^(4-3) = 5.2 x 10¹

Thus, 0.0052 x 10⁴ = 0.0052 x 10^(4-3) = 5.2 x 10¹

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Here's another example: Convert 714.24 x 10⁷ to standard exponential notation:

Solution:  The decimal point must be moved 2 places to the left which makes the number smaller by 2 decimal places. Therefore, the exponent of 10 must be made larger by 2 to compensate:

10⁽⁷⁺²⁾ = 10⁹

714.24 x 10⁷ = 7.1424 x 10⁹

Thus 714.24 x 10⁷ = 714.24 x 10⁽⁷⁺²⁾ = 7.1424 x 10⁹

The same reasoning applies when the exponent is negative.  For example, convert the following to standard notation: 0.0026 x 10^-4. First move the decimal 3 places to the right. This makes the number larger. Therefore, the exponent of 10 must be made smaller by subtracting 3:

10^(-4-3) = 10^-7

Therefore,   0.0026 x 10^(-4-3) = 2.6 x 10^-7

Note:   When working with negative numbers like -0.023 x 10⁶ or -52.7 x 10^-4 , you may disregard the negative sign while you convert to standard exponential notation.  Convince yourself why this is so.  Many calculators have the capability of handling exponential notation.   If you have such a calculator make sure you know how to enter numbers with positive and negative integers and how to read the display when in exponential notation.  If you do not know how to do this it would be to your advantage to get out the instruction booklet that came with your calculator and learn how to do it. Also, if you plan to continue in more science courses and are thinking about purchasing a calculator, you might think about one that can handle exponential notation.

Here are some practice problems.

You can practice for the post test in the Practice Questions. When you feel confident that you understand the material, make your second attempt at the Math Skills Assessment.