Lesson 5: Scientific Notation

In the technical world, when extremely large or small numbers need to be expressed, a technique called scientific notation is often used to simplify writing these numbers.

In scientific notation, a number is simplified by moving the decimal place around until only one nonzero digit is to the left of it. Then, the number of digits the decimal was moved is compensated by multiplying the new number by some power of ten. This is a confusing concept to understand at first so if you are confused, just stick with it because the examples will help clarify what is going on.

Let's think of the number 2 000 000 000. In scientific notation, this can be expressed as 2.0 * 10^9 (two times ten to the nineth power). So what is happening here, is that the number 2 is being multiplied by 10 to the ninth power which equals 2 000 000 000. Lets take this process step by step so we can see what is happening.

1. We take 2 000 000 000 and move the decimal place over to the left until we have only 1 nonzero digit to the left of the decimal place. So now we have 2.000 000 000 which can be simplified to 2.0
2. We take note that we move the decimal place over 9 digits so then we multiply 2.0 by 10^9. The original number was made smaller so we must make the exponent on the ten larger.
3. Now, we have our final product which is 2.0*10^9 and is exactly equal to 2 000 000 000.

This process can work the other way. We can use this notation to make extremely small numbers easier to write. Lets go through an example to illustrate this. We will use .0000007

1. We will shift the decimal to the right seven places which will give us 7.0
2. We next multiply 7.0 by 10^-7 because we moved the decimal place 7 places to the right. We made the original number bigger so we must make the exponent smaller.
3. This then gives us 7.0*10^-7.

It is important to remember a few things when using this notation. The modified number should always have a decimal in it in order to indicate the amount of precision of the measurement (We will cover this in more detail later), and when we move the decimal to the left, or make the number smaller, the exponent on the ten must get bigger. In contrast, when we move the decimal place to the left, or make the number bigger, the decimal must get smaller.