Demonstration of the Distributive Law

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The applet helps make sense of the distributive law for positive integers, a.k.a. counting numbers. Multiplication over the set of counting numbers is little more than counting by groups of objects of the same size. 4×5 objects combine into 4 groups of 5 objects each. Therefore, an instance of the distributive law, like

is an expression of the fact that 5 = 2 + 3. Of course, in mathematics, such reasoning does not prove anything, altough it may be helpful in grasping the concept. But when it comes to a proof, the latter may depend on a chosen system of axioms. One such axiomatic approach could be found in the classic Foundations of Analysis by Edmund Landau. The book also derives the law for other number systems - whole, rational, irrational and complex numbers.

References

1. E. Landau, Foundations of Analysis, Chelsea Pub Co, 2001

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