Understanding the Definition and Application of Multiplication

[roger]

What is the rigorous definition of multiplication and how can it be shown to be like repeated addition ?

 

[matt grime]

The rigorous definition of multiplying positive integers is that it is repeatedly adding them.

It is extended to rationals algebraically, and to the reals by continuity, and thence the complexes by algebraic means.

See the VSI (A Very Short Introduction to) book on Mathematics.

 

[roger]

The rigorous definition of multiplying positive integers is that it is repeatedly adding them.

It is extended to rationals algebraically, and to the reals by continuity, and thence the complexes by algebraic means.

See the VSI (A Very Short Introduction to) book on Mathematics.

but why isn't the set Z3,+ a subgroup of Z7,+ ?

 

[HallsofIvy]

but why isn't the set Z3,+ a subgroup of Z7,+ ?

? It is a subgroup! Or, rather, is isomorphic to a subgroup. Mapping {0, 1, 2} to {0, 9, 18}, in that order, is an isomorphism.

 

[matt grime]

Not sure I either understand what roger is getting at or that i agree with halls, though exactly what Z3 and Z7 are is ambigous, but reading them as Z mod 3 and 7 together then the former is not a subgroup of the latter (additvely). But who says that they must be? (apart from roger)

 

[HallsofIvy]

? It is a subgroup! Or, rather, is isomorphic to a subgroup. Mapping {0, 1, 2} to {0, 9, 18}, in that order, is an isomorphism.

Sorry. For some reason my eyes bollixed on me and I read Z₇ as Z₂₇!

 

[mathwonk]

i tend to think of multiplication as any operation which distributes over adition.for positive integers I like to think of multiplication as counting the elements of a cartesian product.

I.e. if a set S has n elements and a set T has m elements then their cartesian product, i.e. the rectangle you build with base S and height T, has nm elements.

this also works for infinite sets. it also illustrates why commutativity is true, by turning the rectangle on its side.

of course from a certain point of view, considering them separately, addition on the real and multiplication on the positive reals, they are prety much the same, under the exponential mapping.