2.2 Power notation
The notation in Example 6 is called power notation, or index notation. In a number such as 2^{5}, the 5 is called the power, or index, of the number.
The squares are particular examples of powers: 9^{2}, for example, can be thought of as ‘9 to the power 2’.
For most numbers, calculating powers by hand soon becomes tedious: while you might be quite happy to find 2^{5} or 9^{2}, it would take a long and fairly dull time to find 2^{50} or 9^{20} by hand. So you will be using your calculator for most calculations involving powers, even when the numbers themselves are quite simple. However, there is one number whose powers are quite easy to find, namely 10. For example:
one hundred = 10 × 10 = 10^{2} = 100;
one thousand = 10 × 10 × 10 = 10^{3} = 1000;
one million = 10 × 10 × 10 × 10 × 10 × 10 = 10^{6} = 1000 000.
In the same way that 2 can be written as 2^{1} so 10 can be written as 10^{1}.
It is also easy to find powers of 1 and 0. 1 × 1 × 1 × … = 1 and 0 × 0 × 0 = 0.
Example 7
The headings on the place value tables in the OpenLearn course Numbers, units and arithmetic [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] are tens, hundreds, thousands etc. … Write these as powers of ten. What do you think the units columns heading would be as a power of ten?
Answer
ten | = 10 = 10^{1} |
hundred | = 100 = 10^{2} |
thousand | = 1000 = 10^{3} |
ten thousand | = 10 000 = 10^{4} |
hundred thousands | = 100 000 = 10^{5} |
million | = 1000 000 = 10^{6} |
The power of ten increases for each column. So to be consistent, the units column should be 10^{0}. Putting 10^{0} into a calculator gives 10^{0} = 1. So the units column is 1 = 10^{0}.