# 1.4.1 Scientific Notation

As chemistry is the study of atoms and molecules, which have extremely low masses and are present in extremely large numbers, a chemist has to deal with numbers as large as 602, 200,000,000,000,000,000,000 for the molecules of 2 g of hydrogen gas or as small as 0.00000000000000000000000166 g mass of a H atom.

Similarly, other constants such as Planck’s constant, speed of light, charges on particles, etc., involve numbers of the above magnitude. It may look funny for a moment to write or count numbers involving so many zeros but it offers a real challenge to do simple mathematical operations of addition, subtraction, multiplication or division with such numbers.

You can write any two numbers of the above type and try any one of the operations you like to accept as a challenge, and then, you will really appreciate the difficulty in handling such numbers.

This problem is solved by using scientific notation for such numbers, i.e., exponential notation in which any number can be represented in the form N × 10n , where n is an exponent having positive or negative values and N is a number (called digit term) which varies between 1.000... and 9.999....

Thus, we can write 232.508 as 2.32508 ×102 in scientific notation. Note that while writing it, the decimal had to be moved to the left by two places and same is the exponent (2) of 10 in the scientific notation.

Similarly, 0.00016 can be written as 1.6 × 10–4. Here, the decimal has to be moved four places to the right and (–4) is the exponent in the scientific notation. While performing mathematical operations on numbers expressed in scientific notations, the following points are to be kept in mind.

Multiplication and Division

These two operations follow the same rules which are there for exponential numbers, i.e.

(5.6 X 105) X (6.9 X 108) = (5.6 X 6.8) X 105+8 =38.64 X 1013 = 3.864 X 1013

(9.8 X 10-2) X (2.5 X 10-6) = (9.8 X 2.5) X 10-2+(-6) =24.50 X 10-8 = 2.450 X 10-7

(2.7 X 10-3) / (5.5 X 104) = (2.7 / 5.5) X 10-3-4 = 0.4909 X 10-7  =4.909 X 10-8

For these two operations, first the numbers are written in such a way that they have the same exponent. After that, the coefficients (digit terms) are added or subtracted as the case may be.

Thus, for adding 6.65 × 104 and 8.95 × 103 , exponent is made same for both the numbers. Thus, we get (6.65 × 104 ) + (0.895 × 104 )

Then, these numbers can be added as follows (6.65 + 0.895) × 104 = 7.545 × 104

Similarly, the subtraction of two numbers can be done as shown below:

(2.5 × 10-2) – (4.8 × 10-3)  = (2.5 × 10-2) – (0.48 × 10-2)

= (2.5 – 0.48) × 10-2= 2.02 × 10-2