Division algorithm for reals

Standard

You must have seen long-division method to compute decimal representation for fractions. Astonishingly, I never pondered about how one would divide an irrational number to get decimal representation. Firstly, this representation will be approximate. Secondly, we have been doing this in name of “rationalizing the denominator” stating the reason that division by irrationals is not allowed. But, in fact, this is the same problem as faced while analysing division algorithm for Gaussian integers.

Bottom line: Numbers are just symbols. We tend to assign meaning to them as we grow up. Since the set of real numbers, rational numbers and integers  form an Euclidean domain, we can write a division algorithm for them. For example, we don’t have special set of symbols for 3 divided by π, but 3 divided by 2 is denoted by 1.5 in decimals.

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