2.5 Small numbers and scientific notation

By using negative powers of ten it is possible to write numbers less than one in scientific notation. Let’s take the example of 0.03.

0.03 is three one hundredths, so this can be shown as a fraction, with 100 as the denominator. This is important, as 100 is a multiple of 10, and can therefore be shown using powers of 10 as follows:

From the previous section you know that .

Hence,

Now 0.03 is shown in scientific notation, since there is a number between 1 and 10, that is then multiplied by a power 10.

This shows you why the process works, but it may not necessarily be the most straightforward way of writing numbers that are less than one in scientific notation. Another way of thinking about this is in two steps, as with large numbers:

• Write the number as one between 1 and 10.
• From this, decide on the power of 10 required.

To establish the power of 10 required, work out how many times you would need to divide the number from step 1 by ten to reach the original number. Each time you divide by 10, the negative power of 10 reduces by 1, starting from -1.

Following these steps for 0.03 again:

1. Write the number as one between 1 and 10.

This gives us 3.

2. From this, decide on the power of 10 required.

To do this divide by 10 and then by 10 again to return to the original number. Hence, the negative power will be .

To convert a number in scientific notation back into decimal form, write down the negative power of 10 as a fraction and then divide the numerator by the denominator. For example:

Now use what you have learned in this section, as well as your previous knowledge from the week, to complete the following activity and hone your skills.