Scientific notation is a method for writing very large or very small numbers in compact form. It is also useful for entering numbers into a calculator. Let's look at scientific notation by considering some examples. We'll begin with large numbers.
The numbers in bold are scientific notation. Notice that the decimal point is always after the first digit.
Look at the number below and we'll show you how it works:
132500 in scientific notation must have the decimal point after the 1, so we'll change the number to 1.32500
However, 132500 does not equal 1.32500, so to make them the same we have to multiply 1.32500 by 100000, or 105 in order to move the decimal place back where it belongs:
132500 = 1.32500 x 100000 = 1.32500 x 105
The process may be clearer if we use a big number where the decimal point is actually visible. Let's use 4156.13
Scientific notation requires the decimal point in 4156.13 to be after the 4. But if we put it there to get 4.15613, this will only equal the original number if we also multiply by 1000, or 103
4156.13 = 4.15613 x 1000 = 4.15613 x 103
Changing a number to scientific notation is just a matter of moving the decimal point to a spot right after the first digit, and then multiplying the result by a power of 10 that will put it back where it was.
Another example: 823100000
823100000 = 8.231 x 108 The 8 is determined by counting places from 8.231 to 823100000.
You can also use scientific notation to represent very small numbers. For example, consider 0.0000852
Using the same technique, we want to place the decimal after the first (non-zero) digit, the 8.
In order to make 8.52 the same as 0.0000852, we will have to divide it by 100,000 to move the decimal back (left) to where it was. This is equivalent to multiplying by 1 over 100,000, or 10-5
0.0000852 = 8.52 x 1/100000 = 8.52 x 10-5
You can review how negative exponents of 10 work here.
Here are some more examples:
Notice that the negative exponent means we're moving the decimal point to the left. Again, we get the exponent by counting places back to where the decimal point originally was.
Entering a number as scientific notation in a calculator makes your work easier because you can sometimes omit brackets. More about that here.