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I am aware that commutativity, associativity and distributivity of integer addition and multiplication follow from their standard set theoretic definitions but I am looking for something suitable for novices or younger students to understand. From what I understand you can alternatively argue with case by case analysis but there's a lot of cases and it doesn't look that satisfying.

I have came up with some geometric proofs of my own and would happily share them since I have found nothing like this online, but I get the feeling that some already exist and are as refined and streamlined as they will ever be, so if you happen to know any please do share :)

P.S. Just to make myself clear I am not talking about just natural numbers here, there's plenty of that stuff online for sure.

Edit: Thanks for your comments, I attached a picture and will explain my arguments for commutativity and associativity. I am aiming to be as elementary as possible and give a good reason to believe the theorems rather than a rigorous proof.

I am modelling integers as arrows pointing to equally spaced points on a line, addition defined in the obvious way.

The top of the image below demonstrates commutativity. Call the blue arrow x and the orange arrow y. Drawing x and y next to each other makes a combined arrow of length |x| + |y|, moving the arrows so that they start from the ends of x and y preserves the length, thus they meet at the same point. This shows that going along blue then orange gets you to the same place as orange then blue, in other words x+y = y+x. Similar argument should hold for any two integers with different signs.

integerthms

Next is associativity, equivalent to show (x+y)+z = (z+y)+x by commutativity. The first part is (x+y)+z, then (z+y)+x can be represented by travelling in reverse with the left direction increasing as shown next (read upwards). The arrow representing (z+y)+x is the reflection of (x+y)+z, so it has the same length, thus in this interpretation it is the same integer.

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  • $\begingroup$ You can get a geometric definition for operations on real numbers using parallels, and their well-definedness and basic properties, like distributivity, can be proved by Desargue and Pappos theorems. $\endgroup$
    – Berci
    May 29, 2020 at 11:17
  • $\begingroup$ I doubt that properties such as commutativity of natural numbers can be proved strictly geometrically. Synthetic(axiomatic) geometry at the start usually lacks notions of numbers themselves and is built e.g. on congruence. You can check whether segments are equal without referring to their lengths(real numbers). On the other hand analytic geometry has all such properties as a prerequisite i.e. you must prove them before you build geometry. $\endgroup$
    – Kulisty
    May 29, 2020 at 14:38
  • $\begingroup$ Maybe the proof of consistency of arithmetic under the assumption of consistency of geometry can be viewed as a geometrical proof of basic properties (definitely not for novice students). $\endgroup$
    – Kulisty
    May 29, 2020 at 14:59
  • $\begingroup$ "I have came up with some geometric proofs of my own and would happily share them" ... Please include them in your question. This will give a better sense of what you're looking for, and it'll help people avoid wasting time (theirs or yours) duplicating your effort and/or telling you things you already know. (It's no fun putting time into an answer only to have the asker respond, "Yeah, I did it that way myself.") $\endgroup$
    – Blue
    May 29, 2020 at 18:35

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In Elements II, 1, Euclid proves for rectangles the geometric analog to the distributive law of multiplication. And Elements VII, 16 gives a purely arithmetic proof of the commutative law of multiplication for natural numbers. These, or something along these lines, might be useful for novice or younger students.

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