If we have 1/2 times 1/3 as an area, then we are defining area as this product. This is like measure theory in advanced calculus. We set up measure theory first. Then we apply measure theory to geometric point sets. The same is done with fractions. Area is a definition that we can make after we define the fraction concept.

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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.
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