Very often want to be able to guess an answer before we do a detailed calculation.
e.g.
How many lightbulbs are sold in Canada in one year?
How many do we each buy ?
1 < n < 20
Hence N~ (no. of people in Canada) x n
N ~20x106x10~108
Note that in order of mag. calcs, want nearest factor of 10 only
e.g. My calculation says that the volume of an atom is 1020 m3.
It's very small, so it must be correct?
O.K. , I did it again and got 10-20 m : that really is very small, so it must be right???
Turn Off Your Taps!
If a tap is left dripping just once a minute, it can waste enough water in a month to fill 20 baths.
Source: Ottawa Citizen (Ontario Ministry of the Environment)
........Wait a moment.......!!!!!!!!!!!!
Dimensional Analysis
e.g. Wallpaper
My room is 8' x 10' x 15'. How much wallpaper do I need?
8x10x15 = 1200.
1200 what?
1200 square feet, obviously. (????????)
This works out to cu. ft, not sq. ft.
Always check that units are consistent: we cannot have an equation with kg = m . Never substitute numbers into formula until the last step
Can also use this to guess formulae sometimes
Theoretical Physics
To make any further progress in understanding, we must develop some basic theory. Will look at one of the oldest areas of physics: the understanding of motion
A digression before we start: why do we need to emphasise the maths? why can't we just describe what goes on?
The Sapir Whorf hypothesis.
(Very loosely).
The strong form: you cannot discuss a concept unless you have the language to describe it.
The weak form: your views of the universe are modified and restricted by language.
A famous wrong example: Eskimos have 56 words for snow
Most of these distinctions are meaningless to non-skiers, but they allow a concepts to be transferred in compact form.
The Pirahã:
an extraordinary example: the Pirahã: language has
No words for colour
No words for number
No creation myths
No words for ancestors
Even taught the Portugese words for numbers, "not one of them had learned to count to 10 or even to add 1 and 1" New Scientist, March 18th 2006
In physics, we need to describe concepts which don't map well on to the English language, and don't necessarily have translations. e.g. "mass" and "weight" are almost synonymous in English and many languages have the same word. They also carry cultural baggage. Further, we have concepts which have no correct mapping e.g. "electron".
Maths represents a (more-or-less) culture-free way of describing concepts. Also it allows predictions in a quantitive way.
No human investigation can be called real science if it cannot be demonstrated mathematically Leonardo da Vinci
