Mathematics for science and technology
Mathematics for science and technology

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Mathematics for science and technology

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’.

Activity 2 Orders of magnitude

Timing: Allow about 5 minutes

1.   What is the order of magnitude of the mass of the Earth, which is 6.0 × 1024 kg?

Answer

Mass of the Earth ~1025kg (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 × 1027 kg?

Answer

Mass of Jupiter ~1027kg (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

multiline equation row 1 Particle postfix times prefix times of lifetime equals 0.7 times prefix multiplication of 10 super negative 10 times s row 2 equals 0.7 times prefix multiplication of 10 super negative 11 times s row 3 tilde operator 10 super open negative 11 plus one close times s postfix times open since postfix times times seven postfix times prefix times of is postfix times greater postfix times than postfix times five comma the postfix times power postfix times of postfix times ten postfix times must postfix times be postfix times rounded postfix times up close row 4 tilde operator 10 super negative 10 times s

Worked example 2

Timing: Allow about 5 minutes

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

Answer

You had:

mass of Jupiter ~1027 kg

and mass of Earth ~1025 kg

So mass of Jupiter divided by mass of Earth ~10 super 27 divided by 10 super 25 ~10 super open 27 minus 25 close ~10 squared

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.

MST_1

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