Scientific Notation - The Idea

The idea behind scientific notation is simply that a number can be thought of as several factors multiplied together. In other words 6 can be thought of as 3*2 or for that matter 2*3.

Big Numbers

So to work with very large numbers we break them into two factors. The first factor is the "interesting" digits and the other is a factor of 10. For instance:
2300 = 2.3 * 1000
of course, 2300 is also 23 * 100 or 230 * 10, but for consistency sake we always use a decimal number between 1 and 10.
Using that logic, 45,000,000 is 4.5 * 10,000,000 and 367.9 = 3.679 * 100
The next step is simply to re-write the factor of 10 as a power. So:
2300 = 2.3 * 1000 = 2.3 * 10³
45,000,000 = 4.5 * 10,000,000 = 4.5 * 10⁷
A hint: the exponent on the 10 is the number of zeros that follow the 1 in the factor.

Little Numbers

When the number to be written is very small, we use the same process (breaking it into to two factors and writing the second as a power of 10) but in this case the power of 10 will be negative.
So, 0.0023 = 2.3 * 0.001 = 2.3 * 10^-3 and 0.000 000 000 774 = 7.74 * 10^-10
A hint: the exponent on the 10 is negative and the number is the number of zeros after the decimal point + 1.

What it all means

In the end, if a number is written in scientific notation, we can get a quick feel for the size of the number by looking at the exponent on the 10. If the exponent is positive, the number is larger than 1 and the bigger the exponent, the bigger the number. If the exponent is negative, then the number is less than 1, and the more negative the exponent is, the smaller the number is.
Of course understanding the theory is only half the battle. You also need to be able to do math using scientific notation.