# 2.6 Negative powers

Now look at what happens when the power is negative. What does 10^{−3} mean? What is the result of the following calculation?

100 ÷ 100 000

What you are actually being asked to find is:

But look at the calculation again. Using the rule for the division of powers of numbers gives:

10^{2} ÷ 10^{5} = 10^{2−5} = 10^{−3}

So 10^{−3} = 0.001. But you can also write this result as:

This means that 10^{−3} can be thought of as 1 divided by 10^{3}.

This result is true for all negative powers, not just powers of ten. For example:

One over any number is called the **reciprocal** of the number.

For example, the reciprocal of 10 is = 10^{−1} and the reciprocal of 100 is = = 10^{−2}

A number raised to a negative power is the reciprocal of the number raised to the corresponding positive power.

So 10^{−5} = = 0.00001.

## Example 8

The decimal place value table columns are headed tenths, hundredths, thousandths etc. Write these as powers of ten.

### Answer

a tenth = =10^{−1}

a hundredth = = 10^{−2}

a thousandth = = 10^{−3}

So the headings of the place value table are all powers of ten. To the left of the units they are positive powers, the units column is 10^{0}, and to the right the column headings are negative powers of ten.

Notice that

The negative power indicates the position of the decimal point — how many times it has moved to the left from 10^{0} = 1.

10^{−1} = 0.1 (move one to the left); 10^{−2} = 0.01 (move 2 to the left) etc.