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?

Advertisements

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.
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s