SCIENTIFIC NOTATION

                                When solving problems in chemistry, we often come up with very large or very small numbers. For example, 12.9 grams of carbon contains

6.02214150000000000000000 carbon atoms

and each carbon atom has a mass of

0.0000000000000000000000199 g

Having these numbers when solving will make us prone to errors especially in writing those zeros. To avoid missing those zeros, we use scientific notation. 

Scientific notations are written like this:

a x 10^b ( a times ten to the power of b)

where:  a = coefficient  ( any real number between 1 to 10)

               b = exponent 9integer, whole number)

For every large numbers, the exponent is positive while for small numbers, the exponent is negative.

Let’s say you are asked to write 459 000 000 in scientific notation. The first thing to do is to find b. We do this by counting the number of places to decimal point must be moved to get the value of a (that is between 1 to 10).

• The exponent is POSITIVE if the decimal point was moved to the LEFT.
• The exponent is NEGATIVE if the decimal point was moved to the RIGHT.

The following shows how to use scientific notation.

1.                   Write 67940000 in scientific notation

              67940000      =        6.794 x 10⁹

               9 steps to the LEFT          exponent is positive

2.                Write 0.000000000000321 in scientific notation

               0.000000000000321     =    3.21 x 10^-13

               13 steps to the right           exponent is negative

In the example 0.000000000000321, it is not written as 32.1 x 10^-14 or 0.321 x 10^-12 because it will violate the rule the coefficient must be between 1 to 10.

Addition and Subtraction of Scientific Notation

                                When adding or subtracting numbers in scientific notation, we first write each quantity with the same exponent, add or subtract the digits, then copy the exponent.

                                           8.45 x 10⁴                 →             9.2      x 10⁴

                                        + 7.23 x 10³⁽⁺¹⁾       →            0.723 x 10⁴

                                                                                         9.923  x 10⁴

If the coefficient is less than 1 or has two whole number digits, move the decimal point so that the coefficient is between 1. to 10 and adjust the exponent.

                                            9.2 x 10⁷         →        9.2 x 10⁷

                                         + 8.5 x 10⁶         →       0.85 x10⁷

                                                                             10.05 x 10⁷⁽⁺¹⁾ →  1.005 x 10⁸

                                           4.25 x 10^-3         →        4.25 x 10^-3

                                         - 6.15 x 10^-4         →        0.615 x10^-3

                                                                                  3.635 x 10^-3

Multiplication and Division of Scientific Notation

                                When multiplying or dividing numbers in scientific notation, we multiply or divide the coefficient in the usual manner, but add or subtract the exponents.

                                                     2.3 x 10⁴

                                                  x 3.9 x 10⁵

                                              (2.3 x 3.9) (10⁴⁺⁵)   →   8.97 x 10⁹

                                                    5.6 x 10^-4

                                                ÷  4.2 x 10^-2

                                                   (5.6 ÷ 4.2) (10^(-4)-(-2))   →   1.33 x 10^-2