How do we prove the distributive property of multiplication?

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    Multiplication Property
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SUMMARY

The discussion clarifies the distinction between the commutative law and the distributive law of multiplication, specifically addressing the equation 3 x 2 = 2 x 3 as an example of the commutative law. The distributive law is defined as a x (b + c) = a x b + a x c. Various mathematical systems, including natural numbers, integers, rational numbers, and real numbers, require different approaches for proving these laws, with references to foundational texts such as Rudin's "Principles of Mathematical Analysis" and Hrbacek and Jech for set theory. The complexity of these proofs varies, with the construction of real numbers being notably intricate.

PREREQUISITES
  • Understanding of the commutative law and distributive law in mathematics.
  • Familiarity with number systems: natural numbers, integers, rational numbers, and real numbers.
  • Basic knowledge of set theory, particularly regarding definitions and constructions of real numbers.
  • Access to mathematical texts such as Rudin's "Principles of Mathematical Analysis" and Hrbacek and Jech.
NEXT STEPS
  • Study the proofs of the commutative and distributive laws in natural numbers and integers.
  • Learn about the construction of real numbers using Dedekind cuts and Cauchy sequences.
  • Explore the field axioms that govern real numbers and their implications for mathematical proofs.
  • Review set theory fundamentals to better understand the definitions of real numbers.
USEFUL FOR

Mathematicians, educators, and students seeking a deeper understanding of multiplication laws and their proofs across various number systems.

  • #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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