In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + c) = ab + ac$].

What's bothered me for a long time is that while combining like terms [$ax + bx = (a + b)x$] is equivalent to distribution, it seems more basic and fundamental. It's used as an addition process (instead of a multiplicative one); it seems commonsensical in terms of unit addition (3 feet + 5 feet = 8 feet), which was referenced by some as the "great principle of similitude"; and so forth.

So what is the rationale for taking distribution as axiomatic, and proving combination afterward? Why is it not better pedagogy to take combining terms as fundamental, and then prove distribution from it?

(Edit) I've cross-posted this question on the Mathematics site: https://math.stackexchange.com/questions/1417856/why-is-distribution-prioritized-over-combining

(Edit) I put a 50 point bounty for this on the Mathematics site post above.

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + c) = ab + ac$].

What's bothered me for a long time is that while combining like terms [$ax + bx = (a + b)x$] is equivalent to distribution, it seems more basic and fundamental. It's used as an addition process (instead of a multiplicative one); it seems commonsensical in terms of unit addition (3 feet + 5 feet = 8 feet), which was referenced by some as the "great principle of similitude"; and so forth.

So what is the rationale for taking distribution as axiomatic, and proving combination afterward? Why is it not better pedagogy to take combining terms as fundamental, and then prove distribution from it?

I've cross-posted this question on the Mathematics site: https://math.stackexchange.com/questions/1417856/why-is-distribution-prioritized-over-combining

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + c) = ab + ac$].

What's bothered me for a long time is that while combining like terms [$ax + bx = (a + b)x$] is equivalent to distribution, it seems more basic and fundamental. It's used as an addition process (instead of a multiplicative one); it seems commonsensical in terms of unit addition (3 feet + 5 feet = 8 feet), which was is referenced by some as the "great principle of similitude"; and so forth.

So what is the rationale for taking distribution as axiomatic, and proving combination afterward? Why is it not better pedagogy to take combining terms as fundamental, and then prove distribution from it?

(Edit) I've cross-posted this question on the Mathematics site: https://math.stackexchange.com/questions/1417856/why-is-distribution-prioritized-over-combining

(Edit) I put a 50 point bounty for this on the Mathematics site post above.

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + c) = ab + ac$].

What's bothered me for a long time is that while combining like terms [$ax + bx = (a + b)x$] is equivalent to distribution, it seems more basic and fundamental. It's used as an addition process (instead of a multiplicative one); it seems commonsensical in terms of unit addition (3 feet + 5 feet = 8 feet), which was referenced by some as the "great principle of similitude"; and so forth.

So what is the rationale for taking distribution as axiomatic, and proving combination afterward? Why is it not better pedagogy to take combining terms as fundamental, and then prove distribution from it?

(Edit) I've cross-posted this question on the Mathematics site: https://math.stackexchange.com/questions/1417856/why-is-distribution-prioritized-over-combining

(Edit) I put a 50 point bounty for this on the Mathematics site post above.