Why does two terms immediately adjacent "mean" multiply?

Question

I am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication?

I understand the rule that two terms next to each mean multiply those two terms together, regardless as to what those terms look like. But did this notation that we now all understand come from shorthand? Or was it easier to read? Or is there some other reason entirely?

Any insight that you can provide will be appreciated. I would get very angry with teachers in my life who explained things by saying, because that is the way it is. If this is the case, then I at least want to be able to tell them that honestly.

Answer 1

I would argue a case for how we describe multiples using natural language. When I say, "I have three boxes", I don't need any other words between 'three' and 'boxes'. Similarly, "I have three $x$", in shorthand, becomes simply $3x$.

Contrast this with addition: "I have an apple and two oranges". The 'and' separates objects that are together but are not necessarily of the same form. "I have an $x$ and two $y$s" becomes $x+2y$.

Answer 2

Another thing to consider and not mentioned is that the multiplication symbol $\times$ also happens to look a lot like the variable $x$, especially when writing them out by hand. It is one reason $\cdot$ is sometimes used for multiplication as well.

Answer 3

Theophile's answer is greatly intuitive, but what's really behind all this?

I believe in the end it comes down to the precedence of the operations. Say you have expression

$$2 \cdot x \cdot y + 3 \cdot x$$

The meaning of the expression actually is:

It can be clearly seen that the $\cdot$ operation is the "tightest" bond between those operands - so if you want to cluster the operands, i.e. if you want to save writing by omiting some of the operators, you must clearly start at the very bottom level of the syntactic tree. So you write

$$2 x y + 3 x$$

Just imagine what would happen if you try to omit the $+$ operator instead:

$$2 \cdot x \cdot y 3 \cdot x$$

That would mess up the syntax completely.

PS: the syntax tree was generated using http://mshang.ca/syntree/

Answer 4

Teophile's answer is great.

I would like to add that multiplication is the "natural" mathematical operation, not addition, as some people might think. A group has multiplication, but no addition. Addition is only introduced in rings, once we already have a multiplication.

Also, function concatenation is a multiplication, and mathematicians like to write this without an operator (look e.g. into the Lambda calculus), unless we talk about function application. But even then, mathematicians do not need an additional symbol - note how there is no extra symbol on the right hand side.

$$ (fg)(x) = f(g(x)) $$

(you might more be used to write $(fg)(x)$ as $f\circ g(x)$ ($\circ$ here being function concatenation))

Answer 5

While I happen to think this notation is bad¹, there's a rather compelling way to think about it: a numeral can be interpreted as a linear operator. This is natural as a "base case" of the way we write linear mappings $\mathbb{R}^n \to \mathbb{R}^m$ as $m\times n$ matrices.

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¹What I really dislike about juxtaposition multiplication is how it clashes with general (i.e. non-linear) function application. It's very natural to shorten $\sin(x)$ to $\sin x$ – after all, the parens don't really group anything! Yet people will get confused if you write $f\: x$ to notate $f(x)$ for some general function $f$, and possibly mistake it for something like $f(x)\cdot x$; especially in physics, where it's commonplace to omit the function argument and assume you'll put on the "usual" symbol.