Place value, superposition, and all that

Consider the number `535.49`.  It can be written as a series of multiples of powers of ten, e.g.

`\qquad 535.49 = 5 (10 xx 10)  +  3 (10)  +  5 (1)  +  4 (1/10)  +  9 (1/10 xx 1/10),`
`\qquad\qquad = 5 (100)  +  3 (10)  +  5 (1)  +  4 (0.1)  +  9 (0.01),`

We have already seen a some correspondences between how decimals work and how wiggle works.  Consider then

`\qquad\qquad\qquad 5_{7}2_{24}10;_{60}45_{10}7  =  898\frac{457}{600}`

We can write

`5_{7}2_{24}10;_{60}45_{10}7 `
`=  5  (7 xx 24) +  2 (24)  +  10 (1)  +   45 (1/60)  +  7 (1/60 xx 1/10)`
`=  5  (1_{7}0_{24}0) + 2 (1_{24}0) + 10 (1) +  45 (0;_{60}1) + 7 (0;_{60}0_{10}1).`

This decomposition tells what each superdigit contributes to the total.  The contribution is the product of the size of the superdigit itself (e.g. `5`) and the place value associated with where the superdigit is placed (e.g. `7 xx 24`).  The correspondence with decimals should be obvious.

Another thing to notice is that superposition holds just as well for wiggle expressions as it does for mixed units and for decimal notation.  Superposition is the idea that each superdigit contributes a known amount to the sum, in fixed proportion to its size, and independent of what the other superdigits are contributing.