Scientific Notation (a.k.a. Exponential Notation)
We routinely deal with very large or very small numbers in the sciences. For example, the number of atoms in a sample of carbon that has a mass of 12 grams (a little less than half an ounce) is around 602,000,000,000,000,000,000,000 atoms. Clearly it is inconvenient to right all of those zeros. The solution to this problem is called scientific notation. In this notation, the number is written as a product of a number between 1 and 10 and a power of 10.
I am going to begin with a simple example of scientific notation. Let's express the number 300 in scientific notation. First recognize that 300 is just 3 x 100. Note that 100 is 10 x 10, which is 102. So 300 may be written as 3 x 102. I have just expressed the number 300 as a product of a number between 1 and 10 (3 in this case) and a power of 10 (2 in this case), which is scientific notation. So it follows that 3000 can be written as 3 x 103 and 30 can be written as 3 x 101. Can the number 3 be expressed in scientific notation? Yes, any number can be written in scientific notation. 3 is just 3 x 100. A number raised to the power of zero, such as 100 in our case, is defined as 1.
The previous examples of scientific notation are not very impressive because it seems like more work to write 3 x 102 rather than simply writing 300. However, consider the number 602,000,000,000,000,000,000,000. This number expressed in scientific notation is 6.02 x 1023, which is much more compact and efficient. Note that 60.2 x 1022 is the same as 6.02 x 1023, but it is not considered proper scientific notation because 60.2 is not a number between 1 and 10. The same goes for the number 0.602 x 1024.
Now you may be asking yourself how I knew that the exponent on 6.02 x 1023 should be 23. All you have to do is count the number of places you move the decimal, because each movement of the decimal to the left increases the exponent by one (referred to as an order of magnitude). The example below illustrates how the number 93,000,000 can be easily converted to 9.3 x 107 by counting over decimal places. To convert a number in scientific notation to standard notation you just reverse the process.
Very small numbers are also conveniently written in scientific notation. I'll use a simple example: 0.003 may be written as 3 x 1/1000 which can be written as 3 x 1/103. But, 1/103 is the same as 10-3. So 0.003 expressed in scientific notation is 3 x 10-3. That's the long way of doing it. The short way is to count the number of decimal places you move to the right, which decreases the exponent by one for each movement of the decimal place. Numbers between zero and one, when expressed in scientific notation, will always contain negative exponents. One last example, the number 0.000573 in scientific notation is 5.73 x 10-4. Again, note that 5.73 is a number between 1 and 10. 57.3 x 10-5 is also equivalent to 0.000573, but it is not written in proper scientific notation because 57.3 is not a number beteen 1 and 10.