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Skill 6 of 13
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Units

Mathematical skill ●●○○○

The chair is 2 away’ and ‘The popcorn needs 10 more’ make no sense to say. But ‘The chair is 2 metres away’ and ‘The popcorn needs 10 seconds more’ do make sense. A unit is clearly necessary. We must invent units for our properties.

• We could invent the unit metre1 for position (the ‘where’) and
• the unit second2 for time (the ‘when’).

What about quantity? While it doesn’t make sense to say ‘How far is it? It is three away’ without a unit, it does make sense to say ‘How many are there? There are three’ or ‘You two go up the stairs.’ We could say ‘three pieces’ and ‘two guys’ if we wanted to. But clearly the unit can be implied and is thus not necessary, so let us keep quantity unitless.

The sizes need units as well. Let us choose:

• the unit metre again for length,
• the unit square metre for area (since ‘square’ is a word for a 2D shape) and
• the unit cubic metre for volume (since ‘cube’ is a word for a 3D shape).

Since sizes are just ways to talk about chunks of space, it makes sense that their units overlap with the unit for position, the property of space. But while ‘metre’ makes sense for length (it is after all just a difference between positions), how should we understand ‘square metre’ as area and ‘cubic metre’ as volume? Let us invent a definition for these units:

• a square metre can be the space covered by 1 metre lengthwise and crosswise, while
• a cubic metre can be the space covered by 1 metre lengthwise, crosswise and vertically.

Now you may be asking ‘Why metre and second?

For no special reason. We could have chosen anything: ‘The cable is two coffee tables long,’ ‘one sofa-length long,’ ‘as long as the garden apple tree is tall’ etc. Any would be fine as a length unit if we just pick one and stick to it. Someone picked something back in the past and called it the metre, and we have stuck to it ever since. How long that metre was chosen to be was not important as long as something was chosen (it just had to be measurable).

References:

1. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com