Scientific notation is a method of expressing numbers commonly used by scientific notation calculators and also by humans. The scientific notation style includes exponents and provides a way to write numbers that are too big or too small to write in traditional styles. Engineering fields commonly use this notation in order to simplify equations and formulas.

For example, 1 trillion dollars would be written in a traditional style like this:

1000000000000

That is '1' followed by 12 zeroes. The number poses problems because it's altogether too easy for human beings to miscount or miswrite the string of zeroes, thereby introducing error into the expression.

Take a look at these two measurements. Do you know the difference between the two numbers?

1.5 X 10² meters

1.50 X 10² meters

Are they equivalent? Perhaps they are, perhaps not. The first number is written a little less precisely than the second. The first number, 1.5 X 10², has 2 significant figures. The number can also be written in traditional notation as 150, with no trailing decimal point. This implies that it is accurate to the nearest 10 meters. The second number, 1.50 X 10², has three significant figures. The trailing zero implies that is is accurate to the nearest meter. This number can also be written as 150., with a trailing decimal point. In science and engineering, these two similar-looking numbers are indeed very different.

Therefore, when we write 150 meters (with no trailing decimal point) we are implying that the measurement can be anywhere from 145 to 154 meters. When we write 150. meters (with a trailing decimal point) we are implying that our measurement is greater than or equal to 149.5 meters and less than 150.499 (repeating) meters.

What if we write "150.0 meters"? Well, in this case we are implying that our measurement is greater than or equal to 150.0 meters and less than 150.05 meters.

We can add numbers that are written in scientific notation:

    1.0 X 1012

 +  2.0 X 1012

 _____________

    3.0 X 1012

It's easy: line up the numbers on the decimal point and make sure the exponents are identical. Then, add the numbers to the left of the multiplication symbol and simply bring down the exponent. If the exponents don't match, you can reformat one or both numbers in order to get them to match. For example:

    1.0 X 1011

 +  2.0 X 1012

 _____________

We can't perform this addition without massaging the numbers a little. The decimal points appear to line up, but exponents don't match. To fix this problem all we have to do is adjust the top number by incrementing the exponent and at the same time decreasing the number by an order of magnitude:

    0.1 X 1012

 +  2.0 X 1012

 _____________

    2.1 X 1012

Very easy! The trick is to notice that

0.1 X 1012

is exactly the same as

1.0 X 1011

Instead, we can use scientific notation to write the number as:

1 X 10¹²

which is much easier to read and write. There are many ways to write the same number in scientific notation. We could also write the above number as:

10 X 10¹¹

which is an equivalent value, but a slightly different representation. It's acceptable, but it's not normalized. A special form of scientific notation, referred to as normalized scientific notation, requires that the absolute value to the left of the multiplication symbol must be greater than or equal to 1 and less than 10. The first example, above, is in normalized scientific notation form.