Basic Math and Pre-Algebra For Dummies (70 page)

When you get used to writing numbers in scientific notation, you can do it all in one step. Here are a few examples:

When you understand how scientific notation works, you're in a better position to understand why it works. Suppose you're working with the number 4,500. First of all, you can multiply any number by 1 without changing it, so here's a valid equation:

Because 4,500 ends in a 0, it's divisible by 10 (see Chapter
7
for info on divisibility). So you can factor out a 10 as follows:

Also, because 4,500 ends in two 0s, it's divisible by 100, so you can factor out 100:

In each case, you drop another 0 after the 45 and place it after the 1. At this point, you have no more 0s to drop, but you can continue the pattern by moving the decimal point one place to the left:

What you've been doing from the beginning is moving the decimal point one place to the left and multiplying by 10. But you can just as easily move the decimal point one place to the right and multiply by 0.1, two places right by multiplying by 0.01, and three places right by multiplying by 0.001:

As you can see, you have total flexibility to express 4,500 as a decimal multiplied by a power of ten. As it happens, in scientific notation, the decimal must be between 1 and 10, so the following form is the equation of choice:

The final step is to change 1,000 to exponential form. Just count the 0s in 1,000 and write that number as the exponent on the 10:

The net effect is that you moved the decimal point three places to the left and raised 10 to an exponent of 3. You can see how this idea can work for any number, no matter how large or small.

A good question to ask is why scientific notation always uses a decimal between 1 and 10. The answer has to do with order of magnitude.
Order of magnitude
is a simple way to keep track of roughly how large a number is so you can compare numbers more easily. The order of magnitude of a number is its exponent in scientific notation. For example,

Every number starting with 10 but less than 100 has an order of magnitude of 1. Every number starting with 100 but less than 1,000 has an order of magnitude of 2.

Multiplying numbers that are in scientific notation is fairly simple because multiplying powers of ten is easy, as you see earlier in this chapter in “Adding exponents to multiply.” Here's how to multiply two numbers that are in scientific notation:

1. Multiply the two decimal parts of the numbers.
   

   Suppose you want to multiply the following:
   

   

   Multiplication is commutative (see Chapter
   4
   ), so you can change the order of the numbers without changing the result. And because of the associative property, you can also change how you group the numbers. Therefore, you can rewrite this problem as
   

   

   Multiply what's in the first set of parentheses —
   — to find the decimal part of the solution:
   

   

2. Multiply the two exponential parts by adding their exponents.
   

   Now multiply
   :
   

   

3. Write the answer as the product of the numbers you found in Steps 1 and 2.
   

   

4. If the decimal part of the solution is 10 or greater, move the decimal point one place to the left and add 1 to the exponent.
   

   Because 8.6 is less than 10, you don't have to move the decimal point again, so the answer is 8.6 × 10
   ¹²
   .
   

   Note:
   This number equals 8,600,000,000,000.