Scientists and engineers often work with very large or very small numbers, which are more easily expressed in exponential form or *scientific notation*. A classic chemistry example of a number written in scientific notation is Avogadro's number (6.022 x 10^{23}). Scientists commonly perform calculations using the speed of light (3.0 x 10^{8} m/s). An example of a very small number is the electrical charge of an electron (1.602 x 10^{-19} Coulombs). You write a very large number in scientific notation by moving the decimal point to the left until only one digit remains to the left. The number of moves of the decimal point gives you the exponent, which is always positive for a big number. For example:

3,454,000 = 3.454 x 10^{6}

For very small numbers, you move the decimal point to the right until only one digit remains to the left of the decimal point. The number of moves to the right gives you a negative exponent:

0.0000005234 = 5.234 x 10^{-7}

### Addition Example Using Scientific Notation

Addition and subtraction problems are handled the same way.

- Write the numbers to be added or subtracted in scientific notation.
- Add or subtract the first part of the numbers, leaving the exponent portion unchanged.
- Make sure your final answer is written in scientific notation.

(1.1 x 10^{3}) + (2.1 x 10^{3}) = 3.2 x 10^{3}

### Subtraction Example Using Scientific Notation

(5.3 x 10^{-4}) - (2.2 x 10^{-4}) = (5.3 - 1.2) x 10^{-4} = 3.1 x 10^{-4}

### Multiplication Example Using Scientific Notation

You do not have to write numbers to be multiplied and divided so that they have the same exponents. You can multiply the first numbers in each expression and add the exponents of 10 for multiplication problems.

(2.3 x 10^{5})(5.0 x 10^{-12}) =

When you multiply 2.3 and 5.3 you get 11.5. When you add the exponents you get 10^{-7}. At this point, your answer is:

11.5 x 10^{-7}

You want to express your answer in scientific notation, which has only one digit to the left of the decimal point, so the answer should be rewritten as:

1.15 x 10^{-6}

### Division Example Using Scientific Notation

In division, you subtract the exponents of 10.

(2.1 x 10^{-2}) / (7.0 x 10^{-3}) = 0.3 x 10^{1} = 3

### Using Scientific Notation on Your Calculator

Not all calculators can handle scientific notation, but you can perform scientific notation calculations easily on a scientific calculator. To enter in the numbers, look for a ^ button, which means "raised to the power of" or else y^{x} or x^{y}, which means y raised to the power x or x raised to the y, respectively. Another common button is 10^{x}, which makes scientific notation easy. The way these button function depends on the brand of calculator, so you'll need to either read the instructions or else test the function. You will either press 10^{x} and then enter your value for x or else you enter the x value and then press the 10^{x} button. Test this with a number you know, to get the hang of it.

Also remember not all calculators follow the order of operations, where multiplication and division are performed before addition and subtraction. If your calculator has parentheses, it's a good idea to use them to make certain the calculation is carried out correctly.