This post was inspired by “Who Can Name the Bigger Number?“.
“oooooooooooo”, “1100”, “12” and “1.2e1” are different ways of representing the same number.
In coming up with a new representation, the first step is to invent a process by which all members of the old representation can be written in the new one. For example, to convert the “ooo” form to “11” (binary), one would do the following procedure:
1) “ooo” – “”
2) “oo” – “1”
3) “o” – “10”
4) “” – “11”
Addition and Multiplication are defined for the old rep, and to do the same on the new rep, one can convert to the old rep, do the operation, and then convert back. However, most useful reps also have direct ways of doing the same operation. In fact, the usefulness of a rep is often determined by how efficient the operations in it are. (Transforms are used to the same effect.)
Also, in principle the different reps describe the same objects, but this is assuming we have infinite memory and time. For example, to write “1e100” in the “o”-form is impossible, since that would require more “o”s than there are particles in the universe (~1e79).
Representations neglect information that we do not need. Scientific notation, for example, lists only the leading digits of a number and allows us to discard inaccurate digits early in a computation and thus reap the results of error propagation by compacting the expression.
The equivalence of an operation in a higher and lower rep must be proven using mathematics, and not by going through all possibilities, since the higher rep is capable of expressing values that the lower rep is not.
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In music, (more…)