New bases for free, by hundreds and by thousands

Consider the number `535.49165552`.  This is a number written in base ten.  This fact can be used immediately to write an equivalent wiggle expression : `5_{10}3_{10}5;_{10}4_{10}9_{10}1_{10}6_{10}5_{10}5_{10}5_{10}2`.  It turns out, however, that there are other equivalent wiggle expressions that can be written at sight.  There is of course the trivial possibility---simply consider `535.49165552` to be a single superdigit.

Be we can also write `535.49165552` immediately in base a hundred, `5_{100}35;_{100}49_{100}16_{100}55_{100}52,` or in base a thousand, `535;_{1000}491_{1000}655_{1000}520,` or in base a million `535;_{1000000}491655_{1000000}520000.`  Clearly, other positive whole powers of ten will work easily too.

Nor is it necessary to have a constant base.  Yet another equivalent of `535.49165552` is `5_{100}35;_{10}4_{100}91_{10}6_{1000}555_{10}2.`

So far we have clumped together digits of `535.49165552` into superdigits (making sure that there is clump boundary immediately after the unit digit), changed the decimal point to a semicolon, and then in the spaces between superdigits, written for an interbase a `1` with as many `0`s after it as there are digits in the superdigit to its immediate right.

Clumping is not the only possibility.  With a little more effort, we may also dissect.  We could, for instance, first rewrite each decimal digit as a number in base 5, `(1_{5}0)_{10}(0_{5}3)_{10}(1_{5}0);_{10}(0_{5}4)_{10}(1_{5}4)_{10}(0_{5}1)_{10}(1_{5}1)_{10}(1_{5}0)_{10}(1_{5}0)_{10}(1_{5}0)_{10}(0_{5}2),` and then remove brackets, compensating for the newly exposed `5`s by removing a factor of 5 from each 10 interbase, `1_{5}0_{2}0_{5}3_{2}1_{5}0;_{2}0_{5}4_{2}1_{5}4_{2}0_{5}1_{2}1_{5}1_{2}1_{5}0_{2}1_{5}0_{2}1_{5}0_{2}0_{5}2,`

This is a composite base, which can be called `2 xx 5`. The arithmetic of base `2 xx 5` is exactly the arithmetic of the Japanese abacus, or soroban.  The order here is important.  It is given by the order of the interbases in the repeating group, where the semicolon marks a break between one repeating group and the next.  On the soroban, the lone heaven bead on a rod is worth `5`, and each of the four earth beads is worth one `1`.

An alternative ten-friendly composite base is `5 xx 2`.  An abacus built according to that scheme would have four heaven beads, each worth `2`, and one earth bead worth `1`.  A regular soroban turned upside down could be used in that way.

BTW, Happy Australia Day.