Exponents and Scientific Notation

Exponents, Scientific Notation

In Science sometimes we have do deal with very large numbers, or very small numbers. Scientific notation solves the problem of writing out these numbers.

Exponents

Large multiples of 10 can be represented as:

number multiples of 10 exponent form
100 10 x 10 102
1,000 10 x 10 x 10 103
10,000 10 x 10 x 10 x 10 104
100,000 10 x 10 x 10 x 10 x 10 105

And so a large number like 100,000,000,000,000,000,000,000 can be written as 1023 because it is a 1 with 23 zeros following it.

Conversely for small powers of 10:

number fraction exponent form
0.1 1/10 10-1
0.01 1/100 10-2
0.001 1/1,000 10-3
0.000,1 1/10,000 10-4

Therefore a small number like 0.000,000,000,000,000,034 could be written as 3.4 x 10-17 because you'd have to move the decimal 17 times to the right in order to get 3.4 .

Notice that the negative exponent does not mean a negative number. It means a small number less than 1.

The combined table, with 1 and 10 included:

number exponent form
1,000 103
100 102
10 101
1 100
0.1 10-1
0.01 10-2
0.001 10-3
etc..  

Scientific notation

There are 602,000,000,000,000,000,000,000 atoms in 12 grams of carbon. 

602,000,000,000,000,000,000,000 could be written as 6.02 x 100,000,000,000,000,000,000,000

Using what we learned above, this can be written as 6.02 x 1023.

Alternatively, one atom of carbon weighs 0.000,000,000,000,000,000,000,0199 grams or 1.99 x 10-23 grams.

Multiplication and Division

Notice that if you multiply exponents they add, and if you divide they subtract:

100,000 x 100 = 10,000,000     or    105 x 102 = 107      (notice that 5 + 2 = 7)

100,000 / 100 = 100   or    105 / 102 = 103    (notice that 5 - 2 = 3)

100 / 100,000 = 0.001   or    102 / 105 = 10-3    (notice that 2 - 5 = -3 and also that 10-3 is not a negative number)