the scientific notation is another way of writing a number with a lot of digits in a more compact way. It is based on the fact that the decimal comma can be moved by multiplying or dividing by 10. When we multiply one number by 10, the comma is moved one space to the right. So if we multiply by 10 two times (that is, multiply by 100) the comma is moved two spaces to the right. The same concept applies to dividing. When dividing, the comma is moved to the left.
One shorter way to indicate how many times we are multiplying or dividing by ten, is by using an exponent notation. So in this example
it means that we are multiplying by 10 a total number of 5 times, which means that we are moving the comma 5 spaces to the right. Anothe example is
which means that we are diving by 10 a total number of 3 times (note the negative sign!), which means that we are moving the comma 3 spaces to the left.
Consider, for example, the number
In this case we are moving the comma two spaces to the right , so we get the following number
Note that we moved the comma/decimal point two spaces to the right.
Now, consider the following number
which means that we are dividing by 10 three times. So, we need to move the decimal point 3 spaces to the left. So we get the following number
The main purpose of scientific notation is to write numbers with a lot of zeros in a more compact way. The first step to convert one number to scientifc notation is identify the numbers that are not zero. In our case the number 9380000 has 4 zeros and the non zero numbers are 9,3 and 8.
So we first write the numbers in the same order they appear. If we do so, we get the following
Now, we should put a comma somewhere in between this numbers. You can choose where. I will place the comma between the 9 and the 3, so I get
Now, I'm going to define how many spaces I should move the comma in this number so I get the original number. The original number is 9 380000. Note that I left a space inbetween the 9 and the the rest, to identify that we need to move the comma 6 spaces to the right so we get the initial number. So, we write the number we wrote and multiply it by the corresponding power of ten. We will get
Remember that this is because we want to move the comma 6 spaces to the right, which means that we must multiply by 10 a total number of 6 times.
As a way to practice. Consider the number 0.0000002. We would like to write this in scientific notation. So the non zero numbers are
We will place the comma on the right of the number, so we will have
now, we have the number 0.0000002 with the comma on the right of the 2. IN this case we should move the comma 7 spaces to the left so we get the initial number. Moving the comma to the left is equivalent to dividing by 10 seven times. This is summarized by multiplying by 10 raised to a negative power of 7. So the scientific notation for this number would be
Remember that in this case the exponent is negative because we are moving the comma 7 spaces to the left.