7.2 Going down: powers of ten for small numbers

Let’s see how the powers of ten notation can be extended to cover small numbers, such as 0.000 000 000 25 m (the diameter of a water molecule).

Write down the next two numbers in each of the following two sequences.

In the first sequence, each successive number is divided by 10 (i.e. had one zero taken off the end) so the number that follows 100 is The next number in that sequence must result from another division by 10. That is, we must divide 10 by 10 and Therefore, the second answer is 1. In the second sequence of numbers, each successive number has 1 subtracted from its power, so the first answer is because 2 – 1 = 1. For the second answer, we must subtract 1 from the power 1. Because 1 – 1 = 0, the next answer is

In fact, both sequences are the same because 10 000 is , 1 000 is , 100 is , and 10 is . The implication is that and hence This makes perfectly good sense if you recall that, in the second sequence given above, the power is the number of times that 1 is multiplied by ten (e.g. ). For , 1 is multiplied by 10 no times at all, leaving it as 1.

Why stop at 1 or ? Using the same rules, write down the next number in each of these sequences.

In the first sequence, dividing 1 by 10 gives or 0.1 as the next number. In this box, we’re keeping to decimals, so the answer we want is 0.1. But what about the second sequence? The answer is more straightforward than it may seem. We continue to subtract 1 from the powers of ten so that the next number in the sequence has a negative power of ten, , because 0 – 1 = –1. Remembering that the two sequences are equivalent, it seems that This is exactly right! We could equally write

Just as a positive power of ten denotes how many times a number is multiplied by 10, so a negative power of ten denotes how many times a number is divided by 10. For , we must divide 1 by 10 just once and we end up with 0.1.

Another way to think about powers of ten for very small numbers involves shifting the decimal point. A negative power of ten denotes the number of places that the decimal point moves to the left. For example, think of , which we will write as to remind us of the position of the decimal point. Starting with the number 1.0, the power of –2 requires us to move the decimal point 2 places to the left. One place to the left gives 0.1 and two places 0.01.

We therefore have

Let’s try an example. Suppose a raindrop has a breadth of about 0.002 m. This distance could be given in scientific notation as m. This is clear from the following series.

• Start with:
• Divide by ten:
• Divide by ten again:
• And again:

Alternatively, in considering the meaning of ‘two times ten to the power minus three,’ you may wish to start with the number 2.0 and move the decimal point three places to the left to give 0.002.

You know from Section 7.1 that when expressing large numbers in scientific notation, the power of ten (which is positive) denotes the number of places that the decimal point moves to the right. Similarly, when expressing small numbers in scientific notation, a negative power of ten denotes the number of places that the decimal point moves to the left.

You have seen that a negative power of ten tells you how many times you need to divide by ten, so that

But, of course, , and so

This relationship between positive and negative powers of ten is quite general, so

Convention requires that, when writing large numbers in scientific notation, the power of ten should be accompanied by a number that is equal to or greater than 1 but less than 10. The same convention is used when dealing with small numbers and hence negative powers of ten. This is why 0.002 m, the breadth of the raindrop, is given in scientific notation as m, and not as m or m.