# 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 10^{2}, but 850 was roughly 10^{3}. 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* × 10* ^{n}*. Then a if is less than 5, the order of magnitude is 10

*. But if*

^{n}*a*is equal to or greater than 5, the power of ten is rounded up by one, so the order of magnitude is 10

^{n}^{+1}. For example, the diameter of Mars is 6762 km. This can be written as 6.762 × 10

^{3}km, and because 6.762 is greater than 5, the diameter of Mars is said to be ‘of order 10

^{4}km’.

This is normally written as:

- diameter of Mars ~ 10
^{4}km

where the symbol ~ denotes ‘is of order’.

## Activity 2 Orders of magnitude

1. What is the order of magnitude of the mass of the Earth, which is 6.0 × 10^{24} kg?

### Answer

Mass of the Earth ~10^{25}kg (since 6.0 is greater than 5, the power of ten has been rounded up).

2. What is the order of magnitude of the mass of Jupiter, which is 1.9 × 10^{27} kg?

### Answer

Mass of Jupiter ~10^{27}kg (since 1.9 is less than 5, the power of ten remains unchanged).

3. What is the order of magnitude of the average lifetime of unstable ‘sigma plus’ particles, which is 0.7 × 10^{−10} s?

### Answer

## Worked example 2

To the nearest order of magnitude, how many times more massive is Jupiter than the Earth?

### Answer

You had:

mass of Jupiter ~10^{27} kg

and mass of Earth ~10^{25} kg

So ~ ~ ~

Jupiter is two orders of magnitude (i.e. roughly 100 times) more massive than the Earth.

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