Thursday, February 26, 2015


Power to the Bases

Often without being aware of it, we express quantities using polynomials. I vaguely remember learning about a variety of other ways of representing numbers in writing, several decades ago in a history of mathematics course. My notes are not at hand, though, so take for example the Roman numerals as an example of non-polynomial representation.

The symbols used in Roman numerals are I, V, X, L, C, D, and M. Let us recall that these correspond, in our decimal (powers of ten) representation, to 1 (=I), 5 (=V), 10 (=X), 50 (=L), 100 (=C), 500 (=D), and 1000 (=M). Our 4 is written "IV", one before five; and 6 is written "VI", one after five. Similarly, 40 is written "XL", ten before (or less than) fifty; however, 49 is not written "IL" but "XLIX", nor is 45 written "VL" but "XLV": the decrementing prefix is only the allowed to be the next smaller symbol. Or is it? "LC" makes no sense (it equals "L") nor does "DM" (equal to "D"), but I believe "CM" is permitted, as is "IX". The rule might be "the next smaller symbol unless that is half the decremented symbol, in which case it is the next smaller than the half." Then "M" could be decremented by "C" but not by "X", "V" or "I". In any case, numbers are sums of symbolised quantities, with a little subtraction thrown in to shorten the symbol string, analogous to measuring with weights on a balance.

In a polynomial (or whole powers of the base) representation, the number of symbols used is the same as the base, and there is one for each unit increment. In other words, they use counting symbols: 1, 2, 3, … and a "0". The symbols are used to express numbers as sums of whole multiples of the base, and the base is generally understood and not explicitly written. There is no subtraction involved.  A small example may help clarify how this works.

Let's take base 3: its symbols may be 0, 1, 2 (for their familiarity to us). The following table shows the way some numbers in our familiar base-10 (decimal) system are represented in that base. Numbers less than the base are written with their symbol; a number as big as the base is written "10" meaning "1 time the base and no more", followed by "11". "20" means "two times base + 0" and "21" means "two times base plus one". Then "100" means "one time base times base + 0 time base + 0". "1000" means "base times base times base + 0 times base times base + 0 times base + 0". In the last column, 42 is 27+9+(2x3)+0, since 3x3x3=27 and 3x3=9.


Since there is a symbol for each whole number less than the base, the position of the symbol can be understood to correspond to the multiple of the base it is counting, with a "0" used to fill a position not needed for the sum. Consequently, if the base is known it does not have to be written with each term. We can write "11203" instead of "(1x3x3x3)+(1x3x3)+(2x3)+0" (here the subscript 3 indicates the base in use since it might not otherwise be known).

Is the nature of the following sequence now apparent?

101010, 1120, 1012, 132, 110, 60, 52, 46, 42, 39, 36, 33, 30

And now?

1010102, 11203, 10124, 1325, 1106, 607, 528, 469, 4210, 3911, 3612, 3313, 3014

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