Why does commutivity of real numbers exist for multiplication

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Discussion Overview

The discussion centers on the commutativity of multiplication for real numbers, specifically exploring why the equation a*b = b*a holds true. The scope includes theoretical considerations and historical context regarding the definition and construction of real numbers from natural numbers.

Discussion Character

  • Exploratory, Conceptual clarification, Historical

Main Points Raised

  • Some participants propose that the commutativity of real numbers stems from the commutative property of natural numbers and the construction of real numbers from them.
  • Others question the foundational reason for the commutativity of multiplication within natural numbers, seeking deeper understanding.
  • A participant notes that the definition of multiplication itself leads to commutativity, suggesting that it is a matter of how the operations were defined.
  • Historical context is provided, indicating that the motivation for defining real numbers was related to geometric measurements, where commutativity is desirable for concepts like area.

Areas of Agreement / Disagreement

Participants express differing views on the foundational reasons for commutativity, with some focusing on definitions and others on historical motivations. The discussion remains unresolved regarding the deeper implications of these perspectives.

Contextual Notes

Limitations include the lack of exploration into the axiomatic foundations of multiplication and the dependence on definitions that may not be universally accepted.

roger
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Why does commutivity of real numbers exist for multiplication ie why a*b=b*a ?




Roger
 
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It boils down to the natural numbers being commutative plus the way the set of real numbers is constructed (in a few steps) from N.
 
The entirely unenlightening answer is because that's how we defined them.


Historically speaking, the motivation behind the real numbers was measurement in geometry. For example, the product of two sides of a rectangle is the area of that rectangle, and the "rectangle with sides of length a and b" is congruent to the "rectangle with sides of length b and a", so commutativity was desired.
 

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