Using Scientific Notation
Dr. MJ Patterson
When scientists work with very large or very small numbers, a shorthand called scientific notation is often employed. The shorthand allows you to skip writing a bunch of zeroes just to show where the decimal place should go.
A number we will encounter very soon in our chemical studies is Avogadro's number, NA.
NA=602000000000000000000000
If you had to write this out every time you wanted to use it, you would soon quit your chemistry class. To save our enrollment numbers, we use a shortcut.
First of all, write down the significant or important part of the number (We'll define just what numbers are significant in a later lesson. For now, just take it on faith that any zero that merely specifies where the decimal goes is not important. In this example, all of the zeroes following the 2 are not important). We'll write it so that the important part of the number has exactly one digit to the left of the decimal. This is the base of the number.
base = 6.02
To specify how many places we just moved the decimal, we'll multiply by a power of ten. The exponent on the ten is simply the number of places the decimal moved.
exponent = 23
NA in scientific notation = 6.02 x 1023
Examples:
1. Write the following numbers in scientific notation:
a. 1999 b. 1405
2. Write the following numbers in conventional notation:
a. 4.6 x 102 b. 7.8 x 105
Solutions:
1a. Move the decimal so that there is exactly one digit to the left of
the decimal. The base is 1.999. You must move the decimal 3 places
to do so. The exponent is 3. The
number in scientific notation is 1.999 x 103.
b. Again, you must move the decimal three
places. The number is 1.405 x 103.
2a. The 2 exponent means that you must move the decimal 2 places. 4.6 x 102 = 460
b. The 5 exponent means that you must move the decimal 5 places. 7.8 x
105 = 780,000
If you'll look carefully, you'll see that the three examples we've used so far have all been larger than 1. Anytime the number is smaller than 1, we use the same ideas, except that the exponent becomes negative.
For example, in SI units, the charge on a single electron is 0.00000000000000000016 Coulombs. The base is 1.6, and we had to move the decimal 19 places to get there. Since the number is less than 1, the exponent is -19. In scientific notation then, the number is 1.6 x 10-19.
Examples:
1. Write the following numbers in scientific notation:
a. 0.0257 b. 0.00089
2. Write the following numbers in conventional notation:
a. 1.1 x 10-3 b. 9.87 x 10-5
Solutions:
1.a. The number is less than 1, so it will have a negative exponent. If
you move the decimal 2 places, you have a base of 2.57. The number is
2.57 x 10-2.
b. Again, the number is less than 1, so it will have a negative exponent.
If you move the decimal 4 places, you have a base of 8.9. The number is
8.9 x 10-4.
2.a. Move the decimal three places to give a number
less than one since the exponent is negative. Thus, 1.1 x 10-3 = 0.0011
b. Again, move the decimal 5 places to give a number less than one since the
exponent is negative. Thus, 9.87 x 10-5 = 0.0000987
To summarize writing a number in scientific notation: