If addition and multiplication are fundamentally connected, then why is $x * y$ operable explicitly but not $x+y$? [closed]

This sounds baffling simplistic and I apologize for how naive I sound asking this, but I can't really provide a good explanation for myself as to why this is the case intuitively, other than it just being the way we do math.

But, the way I see it, multiplication is a glorified, repeated form of addition, and that sounds more niche to me; naturally sounding less fundamental and specialized. Yet, when doing math I'm much more happy to deal with multiplication than I am to deal with the stubbornness of addition unless I'm doing calculus. I know why I can simplify equations because of properties of multiplication, but I don't know why they are the way they are and why they tend to be much more lenient than addition despite being a more advanced form of addition, for lack of a better word. It's much easier to put a multiplicative element in a formula in physics than addition, for example, since addition requires both variables being added to have the same units.

Where did this all come to be? If anything about my question needs clarifying, please do comment and ask.

closed as unclear what you're asking by Sahiba Arora, Ben, user91500, José Carlos Santos, J. M. is a poor mathematicianAug 6 '17 at 7:41

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• This is more-so meant for whomever answers OP: I'd love to see a group theoretic approach to this – Andrew Tawfeek Aug 1 '17 at 17:47
• I'm much more happy to deal with multiplication than I am to deal with the stubbornness of addition. What do you mean by that? Also maa.org/external_archive/devlin/devlin_06_08.html – Paracosmiste Aug 1 '17 at 17:49
• @whatever I suppose that statement is over exaggerated and a bit unfair, since the example in my head would be something like $\frac{3x}{6x^2}$ being operable while $\frac{x+4}{x+3}$ is not. So it's more me trying to say I find multiplication tends to have less demanding properties in arithmetic. – sangstar Aug 1 '17 at 17:53
• Only in the nonnegative integers is multiplication repeated addition. For arbitrary integers, real numbers, etc. it is a distinct but closely related operation (it distributes over addition). – Matt Samuel Aug 1 '17 at 17:55
• @pjs36 "is it motivated by the fact that we can multiply quantities of different dimensions (e.g. "Newton meter" is a fine unit, but "Newton + meter" isn't)?" That's pretty much what I'm trying to get at in my question. – sangstar Aug 1 '17 at 18:08

Firstly, the operations of addition and multiplication are just binary operators, that is an operator say, $\#$, that works on any two elements out of a mathematical set, $S$.

We can say that those two elements $a,b \in S$, are in the set and can be operated on $a\; \# \;b$ to get another $c \in S$. This property, of taking a binary operation on two elements of a set to get another element of a set is the fundamental axiom of what we call a group. There are infact other properties that $\#$ must have to qualify for a group, notably there must be some $e \in S$ that $a\; \# \;e = e \; \# \; a = a$, and inverses that exist. However, you can look into that more on your own.

Back to your original question, addition and multiplication as you describe them, are simple two binary operations that have come to mean a whole lot to the world as a whole as they generally do a good job at combining numbers we observe in every day life.

The property you describe as "ease" of multiplication I assume is in large part due to its distributivity, as both multiplication and addition are communities (the order of the elements being operated on does not matter) and associate (the order of successive calls of the operator does not matter). Which I should note, these properties, especially commutativity are rare and special to come by!

As multiplication is in fact a shorthand to express addition, distributivity can be "proved" as follows:

$$(a+b)*(c) = \sum^{c}_{1}{a+b} = \sum^{c}_{1}{a} + \sum^{c}_{1}{c} = a*c + b*c.$$

I also quite like this picture

This property is in large part why it is so much easier to deal with values as you describe in physics, where simplification in large part is due to the grouping and solving of sub expressions that can be most readily extracted out using distributivity.

I also suggest you ponder over further opertators that are extensions atop multiplication themselves! Such as exponent, $2^3 = 2*2*2$, or even the awesome arrow notation, $2 \uparrow \uparrow 3 = 2^{2^2}$.

It is a simple question, but one at the heart of abstract algebra and number theory, two topics that would appear to interest you. Rings, groups, fields, and on! I hope this helps.

• This helped a lot. I'm going to start trying to learn those disciplines as soon as I can. Thanks very much. – sangstar Aug 1 '17 at 18:09