2 Scales of measurement

In thinking about the sizes of things, it is sometimes useful to do so in quite rough terms, just to the nearest power of ten. For example, 200 is nearer to 100 than it is to 1000, but 850 is nearer to 1000 than it is to 100. So, if you were approximating to the nearest power of ten you could say 200 was roughly 10², but 850 was roughly 10³. This process is called reducing the numbers to the nearest order of magnitude.

The approximate value of a quantity expressed as the nearest power of ten to that value is called the order of magnitude of the quantity.

The easiest way to work out the order of magnitude of a quantity is to express it first in scientific notation in the form a × 10ⁿ. Then a if is less than 5, the order of magnitude is 10ⁿ. But if a is equal to or greater than 5, the power of ten is rounded up by one, so the order of magnitude is 10ⁿ⁺¹. For example, the diameter of Mars is 6762 km. This can be written as 6.762 × 10³ km, and because 6.762 is greater than 5, the diameter of Mars is said to be ‘of order 10⁴ km’.

This is normally written as:

• diameter of Mars ~ 10⁴ km

where the symbol ~ denotes ‘is of order’.

An idea related to order of magnitude is the logarithmic scale, and that is the subject of the next section.