Exponential and Scientific Notation

 

A. Exponential Notation

Frequently in science we must write very large or very small numbers. A convenient method of expressing these types of numbers is by using exponential notation. This notation expresses the numbers as powers of ten or tenths.The number 1,000 is the result of the following multiplication: 10 x 10 x 10. Written in exponential terms, this is:

 

 

This would normally be referred to as "ten to the third power." In science, the base is normally 10.

 

How can you easily convert a decimal number to an exponential number? Consider the following relationships.

 

Decimal

 

 

 

Exponential

10

=

10

=

101

100

=

10 x 10

=

102

1,000

=

10 x 10 x 10

=

103

10,000

=

10 x 10 x 10 x 10

=

104

 

Now, what about numbers that are smaller than one? The number 0.01 is the result of multiplying 0.1 x 0.1. Written in exponential terms, this is:

10-2

Once again, consider the relationship between decimal and exponential versions of numbers that are smaller than one.

 

Decimal

 

 

 

Exponential

0.1

=

0.1

=

10-1

0.01

=

0.1 x 0.1

=

10-2

0.001

=

0.1 x 0.1 x 0.1

=

10-3

0.0001

=

0.1 x 0.1 x 0.1 x 0.1

=

10-4

 

Notice that again, the number of zeros in the decimal number, including the one to the left of the decimal point, determines the exponent (except it is always a negative value). The number 1 written in exponential terms is 100.

 

B. Scientific Notation
Frequently numbers consist of digits other than ones and zeros. The decimal form of numbers like these are frequently expressed in a form of exponential notation, called scientific notation, in which a decimal number between 1 and 10 is followed by a power of ten. For example, the distance from the sun to the earth is 93,000,000 miles. Written in scientific notation, this becomes:

 

 

Notice that in this case you cannot simply count zeros. In order to determine the exponent, count the number of places needed to move the decimal point (real or imaginary) until you obtain a number between 1 and 10.

 

Numbers Larger than 10

Write 670,620 in scientific notation.

Step 1 - Move the decimal point of 670,620 to the left to get a number between 1 and 10 times a power of ten.

670,620 = 6.7062 x 10exponent

Step 2 - Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.

exponent = 5

Step 3 - Write the number in scientific notation.

670,620 = 6.7062 x 105

 

Numbers smaller than 1

Write 0.00234 in scientific notation.

Step 1 - Move the decimal point of 0.00234 to the right to get a number between 1 and 10 times a power of ten.

0.00234 = 2.34 x 10exponent

Step 2 - Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.

exponent = -3

Step 3 - Write the number in scientific notation.

0.00234 = 2.34 x 10-3

 

In some cases you may want to convert a number from scientific notation to decimal notation. If the exponent is positive, move the decimal point rightward the number of places indicated by the exponent. If the exponent is negative, move the decimal point leftward the number of places indicated by the exponent. In both cases it may be necessary to add zeros.

Convert the following scientific notation to decimal notation.

 

7.33 x 10-4 = 0.000733

This is the result of moving the decimal point four places.

 

 

C. Multiplication
When exponential numbers are multiplied, the exponents are added.


10a x 10b = 10a + b

Thus, if 100 is multiplied by 10,000, in exponential terms this would be:


102 x 104 = 102 + 4 = 106

 

Be careful when working with negative exponents. Consider the following two examples.


105 x 10-7 = 105 + (-7) = 105 - 7 = 10-2

 

10-6 x 10-8 = 10-6 + (-8) = 10-6 -8 = 10-14

 

If there is a coefficient in front of the exponentials, multiply them, and then multiply the exponential component. Finally, put the answer in scientific notation if not already so, and then round off to the appropriate number of significant figures. Letís try the following calculations.

 

(3.8930 x 104) (2.600 x 10-8)

††††† =

10.1218 x 10-4

 

 

†† ††††=

1.01218 x 10-3

Since the coefficient has become smaller by a factor of 10, the exponential portion becomes bigger by a factor of 10.

 

†††††† =

1.012 x 10-3

 

 

 

 

 

(4.658 x 10-2) (4.2 x 10-5)††

†††

††† =

 

19.5636 x 10-7

 

 

††† =

 

1.95636 x 10-6

 

 

 

††† =

2.0 x 10-6

 

 

D. Division

When exponential numbers are divided, the exponents are subtracted.

 

10a

 

 

-----

=

10a - b

10b

 

 

 

Letís try dividing 100,000 by 100 using exponential notation.

 

 

 

 

 

100,000

/ 100=

105 / 102

= 105 - 2 = 103

 

 

 

 

As with the multiplication of exponentials, be careful when working with negative exponents. Consider the following two examples.

 

 

 

 

 

 

108/ 103††† =

108 - (-3) = 108 + 3 = 1011

 

 

10-5 /10-11†† =

10-5 - (-11) = 105 +11 = 106

If there is a coefficient in front of the exponentials, divide them, and then divide the exponential component. Finally, put the answer in scientific notation if not already so, and then round off to the appropriate number of significant figures. Letís try the following calculations.

 

 

 

(6.750 x 105)

/(4.20 x 103)=

1.607143 x 102 = 1.61 x 102

 

 

 

 

 

 

(5.99 x 10-6) /(30.01 x 10-3)=

0.1996001 x 10-3 = 1.996001 x 10-4=2.00 x 10-4

 

 

 

E.Addition and Subtraction

The key to adding or subtracting numbers in Scientific Notation is to make sure the exponents are the same. For example,

(2.0 x 102) + (3.0 x 103)

can be rewritten as:

(0.2 x 103) + (3.0 x 103)

Now you just add 0.2 + 3 and keep the 103 intact. Your answer is 3.2 x 103, or 3,200. We can check this by converting the numbers first to the more familiar form.

So:

2 x 102 + 3.0 x 103 =200 + 3,000 =
3,200=
3.2 x 103

Let's try a subtraction example.

(2.0 x 107) - (6.3 x 105)

The problem needs to be rewritten so that the exponents are the same. So we can write

(200 x 105) - (6.3 x 105) =
193.7 x 105

which in Scientific Notation would be written 1.937 x 107.

Let's check by working it another way:

2 x 107 - 6.3 x 105 = 20,000,000 - 630,000 =
19,370,000 =
1.937 x 107