How do we prove the distributive property of multiplication?

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    Multiplication Property
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Discussion Overview

The discussion revolves around the proof of the distributive property of multiplication, specifically contrasting it with the commutative property. Participants explore various mathematical frameworks and axioms that underpin these properties across different number systems, including natural numbers, integers, rational numbers, and real numbers.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants clarify that the equation 3 x 2 = 2 x 3 exemplifies the commutative law, not the distributive law.
  • Others note that proofs of the distributive and commutative laws vary depending on the number system and the axioms used.
  • A participant suggests that the proof of the commutative law for real numbers can be found in Rudin's "Principles of Mathematical Analysis," indicating it is complex and layered.
  • One participant describes a personal theory on distribution in equations, including an example involving e=mc², which is met with skepticism regarding the terminology used.
  • Another participant discusses the construction of real numbers and the axiomatic approach, mentioning that axioms like the distributive law may not require proof if accepted axiomatically.
  • Several participants express curiosity about the definitions and constructions of real numbers, including Dedekind cuts and equivalence relations on sequences of rational numbers.

Areas of Agreement / Disagreement

Participants generally agree on the distinction between the distributive and commutative laws, but there is no consensus on the best approach to proving these properties or the nature of real numbers. Multiple competing views on definitions and proofs remain unresolved.

Contextual Notes

Participants mention various mathematical frameworks and constructions, indicating that the proofs of the distributive law depend on the definitions and axioms chosen, which may not be universally accepted.

Who May Find This Useful

This discussion may be of interest to students and educators in mathematics, particularly those exploring foundational concepts in number theory and mathematical proofs.

  • #31
greswd said:
What should I read as prep before Goldrei?
Not sure you will need anything to prep for that, but you may find some of the stuff recommended in this thread useful, in particular the book linked to in post #2 and the 10-page pdf linked to in #5.

I've been discussing similar things with a guy in this thread, and he seems to find both of those useful.
 
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  • #32
Fredrik said:
Not sure you will need anything to prep for that, but you may find some of the stuff recommended in this thread useful, in particular the book linked to in post #2 and the 10-page pdf linked to in #5.

I've been discussing similar things with a guy in this thread, and he seems to find both of those useful.

That's great, mate. :smile:

Damn, that thread stretched 8 pages.
 
  • #33
I'm struggling with No.5 in Book of Proof.
 
  • #34
greswd said:
I'm struggling with No.5 in Book of Proof.
Chapter 5? Problem 5? You may need to be more specific. :smile:

If it's an exercise that you're stuck on, you can start a thread about it in the homework forum. If it's a concept, you can start a thread in the forum that seems the most appropriate, probably "general math" or "set theory, logic, probability, statistics". Make sure to include the link to the online version of the book and a statement about what specifically you're having difficulties with.
 

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