Is Subtraction of Real Numbers Associative or Not?

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Discussion Overview

The discussion revolves around the associativity of subtraction in the context of real numbers. Participants explore whether subtraction can be considered associative by relating it to the operation of addition, as well as examining the implications of treating subtraction as a distinct operation.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that subtraction can be viewed as addition of the inverse, suggesting that in this context, it appears associative.
  • Others argue that if subtraction is treated as a separate operation, it does not exhibit associativity, providing examples to illustrate this point.
  • One participant notes that mathematicians generally do not consider subtraction a distinct operation due to its lack of associativity, emphasizing that it is shorthand for adding the additive inverse.
  • A later reply affirms that the initial remark about subtraction was valid and not silly, indicating a supportive response to the exploration of the topic.

Areas of Agreement / Disagreement

Participants express differing views on the nature of subtraction and its associativity, with no consensus reached on whether subtraction should be considered associative or not.

Contextual Notes

The discussion highlights the dependence on definitions of operations and the implications of viewing subtraction as distinct from addition, which remains unresolved.

Kocur
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Well, you might consider my remark to be really silly :smile:. I was thinking about subtraction of reals and the following ideas came to my mind:

From one side, we may consider subtraction to be addition of the inverse of an element, for example:

(5 - 3) - 2 = (5 + (-3)) + (-2).

This "version" of subtraction is associative and (5 + (-3)) + (-2) = 5 + ((-3) + (-2)).

On the other side, if we treat subtraction separately from the addition, it is not associative:

(5 - 3) - 2 is different from 5 - (3 - 2).

Kocur.
 
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Yes,it's actually the "+" operation (called addition) that comes into abstract fields,rings and algebras definition.

As u can check,"+" is an associative operation.

Daniel.
 
In general mathematicians do not consider "subtraction" to be a distinct operation- largely because it would not be associative and associative is a very nice property.
Subtraction is simply shorthand for "add the additive inverse" and, of course, addition is associative.
 
Hopefully, Kocur, you've understood from the previous replies that your remark wasn't silly at all.
 

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