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April 06, 2008

How much accuracy is needed?

Mausi sometimes wonder how civilisations could survive prior to the invention of pocket calculators and computers. It is amazing how quickly people seem to loose the feeling for figures and numbers once they have enslaved themselves to electronic calculators. All of us are familiar with the blank stare we sometimes get from the cashier in the supermarket when the cash machine fails for some reason and he or she has to calculate the change in her head.

Mausi is a member of the generation who finished school before pocket calculators were for sale to ordinary people. She and her classmates had to use logarithm tables or slide rules instead. Quite a good exercise because when dealing with very large or very small numbers one usually did a rough calculation first to determine the magnitude of the result. When the exact number was finally calculated one would automatically do a plausibility check against the rough estimate. A few years later this approach had completely gone out of fashion. As a student Mausi used to coach pupils through their school exams in mathematics and noticed that they only knew the order in which to push certain buttons on their calculators. Any feeling for orders of magnitudes or the need to check the result for plausibility, as it is quite easy to get a comma in the wrong place on a pocket calculator, had completely vanished, at least among those who needed extra coaching.

Today there is rather too much emphasis put on numbers themselves. Very often integers would suffice instead of adding decimals to it. If I knew I had to drive 50 km to some destination that would be an accurate enough number to let one calculate my travel time, one would not have to know that the "exact" distance is 50.367 km. Mausi rather likes the approach the Aztecs took to for example calculate the area of their plots. That employed a few rules of thumb and some other very simple means.

The Acolhua tribe tried to keep their calculations as simple as possible. They had a basic unit which corresponds to 2.5 m, called "T" by modern scientists. These have also deciphered the signs for 1/2 T, 3/5 T and 1/5 T. Fractions of T had obviously been used when the length of a plot wasn't an exact multiple of T. Astonishingly, however, the plot areas in old documents are given only in integer multiples of T*T. How did the Acolhuas achieve this?

They started their calculations with integers in the first place. And they used some simple rules. A rectangular plot would be easy: Area = a*b, same as today. If the sides opposite each other weren't of the same length they would multiply mean values: Area = (a+c)/2 * (b+d)/2. Interestingsly, this last rule was also employed by European land surveyors. Plots were sometimes also divided into to triangles whose areas would then be added up: Area = a*b/2 + c*d/2.

In a geometrical sense the calculations performed by the Acolhuas were not as accurate as we like to do them nowadays. Still, they obviously served the purpose. Mausi wishes, we modern people would again develop our feeling for figures and numbers again and be able to do rought estimates instead of slavishly believing the results of our pocket calculators which can be rather wide off the mark. Mausi also thinks it would make maths at school a lot more fun than it is nowadays.

Posted by Mausi at April 6, 2008 02:51 PM

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