Writing Numbers In Exponential Notation

Written by: Ms. Bette Kreuz

Edited by: The Science Learning Center Staff

Writing in Exponential Notation consists of a __written module__ and a __post test__.

The objectives are to learn to:

1) write large and small numbers in exponential notation and

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 vise versa, as well as converting numbers already in exponential form into standard exponential notation. Some simple and easy to remember rules will be presented to aid in the conversion of a number from one form to another.

Directions

To complete this module you need only this written material. Within the body of this module, you will be given examples to work. Complete these examples on scratch paper, then proceed to the answers at the end of the module. Make sure that you understand all the examples before proceeding. When you have completed the module and feel that you understand the material, obtain and complete a post test from the Science Learning Center personnel. If you successfully complete the post test, make sure your name is recorded in the SLC database. If you do not pass the post test, you may review the module and retake the post test as many times as needed.

Introduction

In science as well as other disciplines, it is often necessary to work with extremely large or small numbers such as the following.

Speed of Light: 30,000,000,000 cm/sec

Number of Atoms in 1 Gram of Carbon: 50,150,000,000,000,000,000,000 atoms

Mass of One Hydrogen Atom: 0.000000000000000000000001674 gram

Solubility of Cadmium Sulfide Gas in Water: 0.0000000000000883 mole/liter

A simple way to avoid the problems involved with writing and working with large and small numbers is to always express them in standard exponential notation or standard scientific notation as shown below.

Speed of Light: 3.00 x 10^{10}

Number of Atoms in 1 Gram of Carbon: 5.015 x 10^{22} atoms

Mass of One Hydrogen Atom: 1.674 x 10^{-24} gram

Solubility of Cadmium Sulfide Gas in Water: 8.83 x 10^{-14}

From just observing the space occupied by these numbers now, you can see that it is more efficient to work with these numbers.

In order to make use of the exponential notation, you must be able to write both large and small numbers as exponential numbers. To understand precisely what these numbers mean, consider the example shown below.

2173.0 is also equivalent to writing 2.1730 x 1000 or 2.1730 x (10)(10)(10)

and since (10)(10)(10)=10^{3} , we may also write 2.173 x 10^{3}

The number is now in standard exponential notation!

Another example, this time a small number, is given below. The same process has been carried out here as in the previous example except that instead of multiplying by a large number (1000 or 10^{3} ) as in the first example, we are now multiplying by a small number, 0.0001 or 10^{-4}.

0.000716 is also equivalent to writing 7.16 x 0.0001 or 7.16 x (0.1)(0.1)(0.1)(0.1) and since (0.1)(0.1)(0.1)(0.1)=10^{-4} we may also write 7.16 x 10^{-4}

The number is now in standard exponential notation!

This method of conversion of a number to standard exponential notation is not the easiest method to use but it does give you an idea of where an exponential comes from.

From the previous two examples, a general rule can be seen. In standard exponential notation, the number is always written with only one digit to the left of the decimal. When we took the number 2173.0 we had to move the decimal 3 places to the left. As it turns out this is the same as the exponent on 10 when the number in standard exponential form. As shown below along with another example.

2173.0

DECIMAL MOVES 3 PLACES

2.1730 x 10^{3 }

EXPONENT ON 10 IS ALSO 3

ANOTHER EXAMPLE: 3,170,000,000

3,170,000,000.

DECIMAL MOVES 9 PLACES

3.17 x 10^{9}

EXPONENT ON 10 IS ALSO 9

In short, the exponent of 10 on a large number is equal to the number of places the decimal point has to be moved.

When working with small numbers, the same general pattern develops. Below the example 0.000716 is worked out.

0.000716

DECIMAL MOVES 4 PLACES

7.16 x 10^{-4 }

**EXPONENT ON 10 IS ALSO 4 BUT WITH A NEGATIVE SIGN**.

Again as a general rule, the exponent of a small number is a negative number equal to the number of places the decimal point moves.

Another example is given below:

0.0000575

DECIMAL PLACE MOVES 5 PLACES

5.75 x 10^{-5}

EXPONENT ON 10 IS ALSO 5 BUT WITH A __NEGATIVE__ SIGN.

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

1) Move the decimal point so that there is only one digit tot he left of the decimal.

2) Count the number of places the decimal moves.

3) If the number is large (or 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 number is small (or 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 in a number of the following general form:** C x 10 ^{n}**

PRACTICE PROBLEM SET #1

Before proceeding convert the following numbers to standard exponential form on a piece of scratch paper.

1) 3,760,000,000

2) 0.0000567

3) 0.0000000091

4) 476,100

The answers to the above exercise are given at the end of this document. **Check your answers. Make sure that you understand this process before proceeding.**

CONVERTING EXPONENTIAL NUMBERS TO STANDARD FORM

Often numbers that result from calculation are not in standard exponential notation or often to add and subtract numbers you need to change the values of the exponents. In the example below the number 81.7 x 10^{3 }is not in standard exponential notation (with one digit to the left of the decimal). To convert this number to standard exponential notation we need to move the decimal point to the ** left** and

8.17 x 10^{(3+1)}

1 IS ADDED TO THE CURRENT EXPONENT OF 10

8.17 x 10^{4}

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

2040.x 10^{-5}

DECIMAL MOVES 3 PLACES TO LEFT

2.040 x 10^{(-5+3)}

3 IS ADDED TO THE CURRENT EXPONENT OF 10

2.040 X 10^{-2}

If the number is small and the decimal must move to the ** right **, we

0.0414 x 10^{-3}

DECIMAL MOVES 2 PLACES TO RIGHT

4.14 x 10^{(-3-2)}

2 IS SUBTRACTED FROM THE CURRENT EXPONENT OF 10

4.14 x 10^{-5}

In another example

0.00051 x 10^{9}

DECIMAL MOVES 4 PLACES TO RIGHT

5.1 x 10^{(9-4)}

4 IS SUBTRACTED FROM THE CURRENT EXPONENT OF 10

5.1 x 10^{5}

We can formulate two general rules:

1) If the decimal point moves to the left or the number is large, add the number of places the decimal move to the current exponent of ten.

2) If the decimal point moves to the right or the number is small, subtract the number of places the decimal moves from the current exponent of ten.

PRACTICE PROBLEM SET #2

Before proceeding, convert the following numbers to standard exponential notation on a piece of scratch paper.

1) 0.00416 x 10^{6}

2) 24.8 x 10^{-3}

3) 0.716 x 10^{-4}

4) 3410 x 10^{2}

Check your answers with those at the end of this document.

ALTERNATE METHODS

If you feel that you might get the rules confused or prefer not to remember the rules, there are alternate methods for converting numbers into and out of standard exponential notation.

The following example shows how you can reason through the conversion process.

CONVERT 0.0052 X 10^{4} TO STANDARD EXPONENTIAL NOTATION

Since standard exponential notation requires one digit to the left of the decimal, you must first change

0.0052 to 5.2 -- by moving the decimal point three places to the right: 0005.2.

The number has now been made larger by three decimal places: (5.2 > 0.0052)

Therefore, to compensate, the exponent of 10 must be made smaller by three numbers. So, subtract three from the exponent:

10^{(4-3)} = 10^{1}

Thus,

0.0052 x 10^{4} = 0.0053 x 10^{(4-3)} = 5.2 x 10^{1}

In the next example, conversion of

714.24 x 10^{7}

to standard exponential notation requires that the decimal point be moved two places to the left

7.1424

which makes the number smaller by two decimal places

(7.1424 < 714.24).

Therefore, the exponent of 10 must be made larger by two to compensate

10^{7+2 }= 10^{9}

Thus,

714.24 x 10^{7} = 7.1424 x 10^{7+2 }= 7.1424 x 10^{9}

The same reasoning also holds when the exponent is negative. For example, convert the following to standard exponential notation:

0.0026 x 10^{-4}

First move the decimal three places to the right

0002.6

which makes the number larger

(2.6 > 0.0026)

therefore the exponent of 10 must be made smaller by subtracting three:

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

Thus, 0.0026 x 10^{-4} = 0002.6 x 10^{-4-3} = 2.6 x 10^{-7}

As another example, convert

826.44 x 10^{-7}

to standard exponential form. Move the decimal point two places to the left

8.2644

which makes the number smaller by two decimal places (8.2644 < 826.44).

Therefore, the exponent of 10 must be made larger by two to compensate:

10^{-7+2} = 10^{-5}

Thus, **826.44 x 10 ^{-7} = 8.2644 x 10^{-7+2} = 8.2644 x 10^{-5}.**

NOTE: When you work with negative numbers like -0.023 x 10^{6 }or -52.7 x 10^{-4}, you may disregard the negative sign while you convert to standard exponential notation. Put some thought into the following examples and you should convince yourself why this is so.

1) -0.023 x 10^{6}

(-) 002.3 x 10^{6-2}

-2.3 x 10^{4}

2) -52.7 x 10^{-4}

(-) 5.27 x 10 ^{-4+1}

-5.27 x 10^{-3}

ALWAYS CHECK YOUR WORK FOR REASONABLENESS!!!

Of course you should always check calculations or conversions of any kind to be certain they appear reasonable. You can check these conversions by writing them out as regular numbers. In the previous examples, you converted 0.0052 x 10^{4}to 5.2 x 10^{1}. Write out both of these numbers. They should be equal if the conversion was done correctly.

0.0052 x 10^{4} = 52

CHECK!

5.2 X 10^{1} = 52.

Another example:

714.24 x 10^{7} = 7, 142, 400, 000

CHECK!

7.1424 X 10^{9} = 7, 142,400,000

As you can see this process becomes tedious if the exponent is a large positive or negative number. However, it often provides a worthwhile check on your work, and you will not always need to write the numbers out entirely to see if your notation appears reasonable or not.

When you feel you understand the material presented in this module, obtain and complete a Post test. Have the test checked before you leave. If you complete the test correctly, make sure your name is recorded in the Science Learning Center data base. If you make any mistakes in the Post test you may review this module and retake the test as many times as necessary.

ADDITIONAL COMMENT

Many present day calculators have the capability of handling exponential notation. If you have such a calculator make sure that you know how to enter numbers with both positive and negative exponents 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 book 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.

ANSWERS TO PRACTICE PROBLEM SET #1

1) 3.76 x 10^{9}

2) 5.67 x 10^{-5}

3) 9.1 x 10^{-9}

4) 4.761 x 10^{5}

ANSWERS TO PRACTICE PROBLEM SET #2

1) 4.16 X 10^{3}

2) 2.48 X 10^{-2}

3) 7.16 X 10^{-5}

4) 3.41 X 10^{5}