Exponents, Scientific Notation and Calculators
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)
Calculators
It depends on your calculator, but most scientific calculators have an exponent button and a to the power of button. Using these you should be able to use scientific notation. Consider the various ways of typing in 6.02 x 1023:
| Button used | Buttons pressed | Display shows | Which reads as |
|---|---|---|---|
| EE | [6] [.] [0] [2] [EE] [2] [3] | 6.02 E23 | 6.02 exponent 23 |
| EXP | [6] [.] [0] [2] [EE] [2] [3] | 6.02 E23 | 6.02 exponent 23 |
| ^ | [6] [.] [0] [2] [x] [1] [0] [^] [2] [3] | 6.02 x 10^23 | 6.02 times 10 to the power of 23 |
| 10x | [6] [.] [0] [2] [x] [10x] [2] [3] | 6.02 E23 | 6.02 exponent 23 |
| yx | [6] [.] [0] [2] [x] [1] [0] [yx] [2] [3] | 6.02 E23 | 6.02 exponent 23 |
Bottom line: consult your calculator manual for details.
An example using Microsoft's Calculator Program:
1.99 x 10-23:
Type: 
To see:
Or alternatively type:
to get
Then press 
to get it back into
scientific notation
