Commutators always zero

P(n) = n-1. S(n) = n+1.

Operator commutator

[P,S]n = PSn – SPn = 0

Here we assume Pn is defined so this is true. For n=0 in natural numbers, this is a problem. Note this is one way to lead to negative numbers. However, negative numbers mean we generalize natural numbers to ordered pairs of a sign and a natural number, (+,n) and (-,n) and define new functions P and S on this data.

A handy name for such ordered pairs is a sign and a size. This applies to fractions and reals as well.

Here P and S are unary operators.

Project: Define a commutator for binary operators, e.g. the commutator for addition and multiplication. What type of test data is needed?


About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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