8 More about powers and units

8.1 Using positive and negative powers with units

Some important general results were introduced in Section 7.2, namely that

and that

Note that and are rarely used in scientific writing; it is usual to write simply 10, 1 or 0.1 instead. However, the use of positive and negative powers provides a useful notation that can also be used with symbols and units.

can be expressed as and (which could also be written as ) can also be expressed as

This way of converting between positive and negative powers is often used when expressing units concisely. Let’s take an example that you have already met, the unit of speed, which is metres per second, abbreviated in Section 5.1 to or m/s.

The conventional scientific way of expressing the unit of speed is , and a variety of units of measurement can be expressed in a similar way using positive and negative powers.

Notice that we have left a space between m and in the unit of speed, and we do this whenever we write a unit that is a combination of two or more other units. This is different from the way that prefixes for multiples of units are written; they are always written without a space between the prefix and the basic unit. Thus, ‘ms’ means ‘millisecond’, but ‘m s’ means ‘metre second’. This separation of the different components of a unit, but not for multiples of units, avoids confusion.