In both of your examples, there is a **convention** to write the numerical coefficient, or the scalar multiplier, on the left. This does support the suggestion that we most naturally think of a multiplication as “multiplier times multiplicand”.

On the other hand, it is not **illegal** to write “x^2 3” or “v3”; for evidence of the latter, see Wikipedia, which talks about both left and right scalar multiplication (which are equivalent when the scalar comes from a commutative ring). So it is only a convention, not a requirement, that the “multiplier” is written first.

As for your grammatical point, I have pointed out that “times” in “a times b” serves not as a verb, but as an adjective or a preposition (my dictionary says the latter); and “multiplied by” is a participle, not a finite verb that would be followed by an object but a phrase followed by an agent that does the action. The latter reading definitely puts the multiplier on the right. I think the main grammatical point to make is ambiguity: we have different ways to read it in English.

Considering only symbolic forms and ignoring language, consistency with addition, subtraction, and division supports thinking of the multiplier as the second number as you suggest.

Ultimately, at best your thoughts suggest a tendency; but this relates only to the specific applications you are using, which was my main point anyway. It is not the symbols written, but the **application** for which they are used, that determines which is the multiplier.

8 x 9 is 8 times 9, and 9 multiplied by 8, and 9 over 8 (when written as a multiplication vertically).

Example 1. Polynomials are the central object of study in algebra (non-abstract). We’ve arranged our notation to be able to write them easily (multiplication has higher precedence than addition). Further, if p(x) = 3x^2 + 2x + 1, most would agree that 3x^2 is more naturally thought of as 3 groups of x^2, or x^2 three times, rather than x^2 groups of 3, or 3, x^2 times. In this example, the multiplier is on the left, not the right.

Example 2. If you’re familiar with vector spaces, you’d agree that we do the same with vector notation. For a vector v in V over R, we write the scalar multiplier on the left, e.g. 3v rather than v3. Since there is generally no concept of adding 3, v times, 3v can only be interpreted as v 3 times, meaning the multiplier is on the left.

I think this concept is complex because English is a subject-verb-object language, it would be natural to see the first number as the one being multiplied, and in fact with addition or division, the first number is the main object to which something is “done” (added, divided).

]]>