decimals and fractions

the decimal number system is a very good way of representing whole numbers. it does however have its drawbacks when we use decimals as a way of representing the fractional parts of numbers. (here 'fractional' means a real number less than 1 and not less than 0, rather than a fraction a/b.)

for the purpose of clarity, we write a fractional number in a decimal as
0.a_1 a_2 a_3 a_4 a_5... where a_n is a digit 0 to 9. the representations are infinite, even in cases where numbers end in 0000000..., even though we usually ignore these trailing zeroes when we write out decimal fractions.

for example, we write 31/57 as 0.5438791...

some of these drawbacks are as follows

-some very simple rational numbers such as 1/3 have non-terminating decimal representations and so cannot be expressed exactly by decimals.

-approximating decimals by fractions is not straightforward. eg we approximate 0.143 by 143/1000, and have no obvious way of getting the simpler 1/7

-representations are not unique: 1/2 can be written as 0.500000... or as 0.49999...
if you're not convinced that these two are equal, try subtracting the latter from the former, and stop when you get convinced.

-working out the decimal representations of rational numbers, ie long division, is a pain in the arse.

the egyptians used a counting system in which the natural numbers (1, 2, 3..) and their reciprocals (1, 1/2, 1/3..) were considered as equally essential, while nowadays in mathematics the natural numbers are often seen as being more basic. the egyptians wrote fractional numbers as the sums of reciprocals of natural numbers.

for example 31/57 = 1/2 + 1/23 + 1/2622

the way i worked this out was simply to take the largest reciprocal smaller than 31/57, (= 1/2) subtract that from 31/57 to get 5/114, and repeat the process on that.

this isn't a great system. although we can write any rational number finitely, these are not unique, and it's not easy to find the best one for a given number. the 'best one' here might mean the one with the fewest terms, or the one with the smallest denominators. when we add or multiply this type of number, we do obtain a sum of reciprocals as the answer, (except when we add 1/n to 1/n, but 2/n is in general not hard to simplify) but it is unlikely to be a very simple form.

it is probably redundant to say that the egyptians adopted this way of writing numbers for aesthetic and ideological reasons. most of the concepts we evolve in mathematics are based on an implicit view of the world as it is, however much mathematicians pretend otherwise.

however, i don't think the egyptians were completely off the right track. perhaps if they had refined their system a little, they might have arrived at a method of writing fractional numbers that i believe would have been too useful ever to be properly superseded by decimal fractions. they could have done this by inventing continued fractions.

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