Scientific Notation
There are many very large and very small numbers in scientific studies. How would you like have to calculate with:
1 mol = 602,200,000,000,000,000,000,000 atoms
or
1 Dalton = 0.000,000,000,000,000,000,000,00165 g
You can streamline large or small numbers with scientific notation. The standard is that you place the decimal point after the first significant digit and adjust the exponent of ten so that there is no change in the value of the number. Think of the change as creating a new number with two parts, a digit part and an exponent part, from the old number. To put the decimal point behind the first digit, you must divide or multiply the original number by some integer power of ten. Then you must do the opposite (inverse) to the exponent part of the new expression so that there is no change in the value of the number.
(0.000,000,000,000,000,000,000,00165 x 1024) x 1/1024 = 1.65 x 10-24 or 1.65 E-24
(602,200,000,000,000,000,000,000/1023) x 1023 = 6.022 x 1023 or 6.022 E23
The original numbers have the same value as the exponential forms, but the exponential forms have the decimal point in the right place. The 'right place' is to the right of the first digit. The "E" in the number stands for exponent. Your scientific calculator will use the numbers in the shortened form, usually best represented by the "E" form. Don't get caught making too much of this. You have seen it before. The number "five point two million" is the same as 5.2 E6. The number of the power of ten only indicates how many places you need to move the decimal to get the long form of the number back. The only question you might have trouble with is WHICH WAY to move the decimal. The easy way to remember that is: numbers that are less than one have negative exponent numbers in the scientific notation form, and numbers that are larger than one have positive exponent numbers. Very often chemistry professors will tell you they want answers in scientific notation if the number is larger than one thousand or smaller than one thousandth. Keep your professor happy. Find out exactly what is required in your course and follow the instructions to the letter.
The table below shows the fully written out number, the way we say the number in English, and the way to write the number in scientific notation. The numbers in scientific notation are to two significant digits.
Full number | Words for number | Sci. notation styles |
5,000,000,000 | five billion (USA) | 5.0 x 10^{9 } or 5.0 E9 |
500,000,000 | five hundred million | 5.0 x 10^{8} or 5.0 E8 |
50,000,000 | fifty million | 5.0 x 10^{7} or 5.0 E7 |
5,000,000 | five million | 5.0 x 10^{6} or 5.0 E6 |
500,000 | five hundred thousand | 5.0 x 10^{5} or 5.0 E5 |
50,000 | fifty thousand | 5.0 x 10^{4} or 5.0 E4 |
5,000 | five thousand | 5.0 x 10^{3} or 5.0 E3 |
500 | five hundred | 5.0 x 10^{2} or 5.0 E2 |
50 | fifty | 5.0 x 10^{1} or 5.0 E1 |
5 | five | 5.0 x 10^{0} or 5.0 E0 |
0.5 | five tenths | 5.0 x 10^{-1} or 5.0 E-1 |
0.05 | five hundredths | 5.0 x 10^{-2 } or 5.0 E-2 |
0.005 | five thousandths | 5.0 x 10^{-3 } or 5.0 E-3 |
0.000,5 | five ten-thousandths | 5.0 x 10^{-4} or 5.0 E-4 |
0.000,05 | five hundred-thousandths | 5.0 x 10^{-5} or 5.0 E-5 |
0.000,005 | five millionths | 5.0 x 10^{-6 } or 5.0 E-6 |
0.000,000,5 | five ten-millionths | 5.0 x 10^{-7} or 5.0 E-7 |
0.000,000,05 | five hundred-millionths | 5.0 x 10^{-8} or 5.0 E-8 |
0.000,000,005 | five billionths | 5.0 x 10^{-9 } or 5.0 E-9 |