Basic Math and Pre-Algebra For Dummies (69 page)

Similarly, to find the value of 10
^–7
, start with 1 and make it smaller by moving the decimal point seven spaces to the left:

 Negative powers of 10 always have one fewer 0 between the 1 and the decimal point than the power indicates. In this example, notice that 10
^–7
has six 0s between them.

As with very large numbers, using exponents to represent very small decimals makes practical sense. For example,

As you can see, this decimal is easy to work with in its exponential form but almost impossible to read otherwise.

 An advantage of using the exponential form to represent powers of ten is that this form is a cinch to multiply. To multiply two powers of ten in exponential form, add their exponents. Here are a few examples:

• 

  Here, I simply multiply these numbers:
  

• 

  Here's what I'm multiplying: 100,000,000,000,000 × 1,000,000,000,000,000 = 100,000,000,000,000,000,000,000,000,000
  

  You can verify that this multiplication is correct by counting the 0s.
  

• 

  Here I'm multiplying a googol by 1 (any number raised to an exponent of 0 equals 1), so the result is a googol.
  

In each of these cases, you can think of multiplying powers of ten as adding extra 0s to the number.

The rules for multiplying powers of ten by adding exponents also apply to negative exponents. For example,

Scientific notation
is a system for writing very large and very small numbers that makes them easier to work with. Every number can be written in scientific notation as the product of two numbers (two numbers multiplied together):

• A decimal greater than or equal to 1 and less than 10 (see Chapter
  11
  for more on decimals)
  
• A power of ten written as an exponent (see the preceding section)
  

 Here's how to write any number in scientific notation:

1. Write the number as a decimal (if it isn't one already).
   

   Suppose you want to change the number 360,000,000 to scientific notation. First, write it as a decimal:
   

   • 360,000,000.0
     
2. Move the decimal point just enough places to change this number to a new number that's between 1 and 10.
   

   Move the decimal point to the right or left so that only one nonzero digit comes before the decimal point. Drop any leading or trailing zeros as necessary.
   

   Using 360,000,000.0, only the 3 should come before the decimal point. So move the decimal point eight places to the left, drop the trailing zeros, and get 3.6:
   

   • 360,000,000.0 becomes 3.6.
     
3. Multiply the new number by 10 raised to the number of places you moved the decimal point in Step 2.
   

   You moved the decimal point eight places, so multiply the new number by 10
   ⁸
   :
   

   • 3.6 × 10
     ⁸
     
4. If you moved the decimal point to the right in Step 2, put a minus sign on the exponent.
   

   You moved the decimal point to the left, so you don't have to take any action here. Thus, 360,000,000 in scientific notation is 3.6 × 10
   ⁸
   .
   

Changing a decimal to scientific notation basically follows the same process. For example, suppose you want to change the number 0.00006113 to scientific notation:

1. Write 0.00006113 as a decimal (this step's easy because it's already a decimal):
   
   • 0.00006113
     
2. To change 0.00006113 to a new number between 1 and 10, move the decimal point five places to the right and drop the leading zeros:
   
   • 6.113
     
3. Because you moved the decimal point five places to the right, multiply the new number by 10
   ^-5
   :
   

   

   So 0.00006113 in scientific notation is
   .