# In its earliest usage, did the notation "5 x 3" mean "five groups of three" or "five, placed into each of three groups"?

In today's time, if I take a look at what language we use to read out "5 x 3", it could be read out as "five-times three" (that is: three, five-times, as in five groups of three), or "five multiplied by three" (that is, five, placed into each of three groups").

My main question is:

• In its earliest usage, did the notation "5 x 3" mean "five groups of three", or instead "five, placed into each of three groups"?

My more minor questions are:

• Historically, when did the vocabulary of "times" (as in "five times three") first start to be used, to describe multiplication (as in "5 x 3"), and in what contexts (eg educational contexts? financial contexts? personal correspondence that mathematicians would write to each other? etc) ? (Implicit in this question, is an understanding that the equivalent of "times" might have first been used in a language different than English).

• Similarly, historically, when did the language of "multiplied by" (as in "five multiplied by three") appear, and in what contexts?

• I suspect this would be better asked on the English StackExchange, along with the origin of the usage: "Take 5, and times it by 3". The only thing mathematics has to contribute to this discussion is, "Well, multiplication is commutative, so it's fine?" Oct 12, 2022 at 20:58
• You say "but very rarely '3x' or '2x' to communicate the same idea" I have the opposite impression. I see both. "the 'five times' seems to be the most interesting thing" That is your personal opinion and not rooted in much fact. That is a matter of language more than mathematics... words appearing first in some languages do impart a somewhat greater weight when it comes to importance, but when it comes to math such things are usually done devoid of emotion. Oct 12, 2022 at 20:58
• You seem to be getting at the nuanced difference between the multiplicand and the multiplier... see wikipedia. In the end, the order in which multiplicands and multipliers are written is irrelevant, situational, and ultimately up to personal preference. Even the whole notion of "multiplication is repeated addition" is merely a naive teaching tool which does not tell the whole story of what multiplication really is and it fails to work as a metaphor for things like $\pi\times \sqrt{2}$. It is best to grow out of using that interpretation. Oct 12, 2022 at 21:01
• Funny, I hear 2x or 3x multipliers and have never heard of an x2 multiplier. Oct 12, 2022 at 21:43
• @Blue thanks very much for the idea Oct 13, 2022 at 4:34

As indicated in the comments, because ordinary whole number multiplication is commutative, the number $$m\times n$$ can be interpreted equivalently as $$m\times n=\underbrace{m+\cdots+m}_{n\text{ terms}}$$ or as $$m\times n=\underbrace{n+\cdots+n}_{m\text{ terms}}$$
There is a similar dual interpretation of whole number division. If $$n$$ divides $$m$$, then the quotient $$m\div n$$ can be interpreted equivalently as either
• $$m\div n$$ = the number of groups if $$m$$ objects are partitioned into equal groups of $$n$$ objects
• $$m\div n$$ = the number of objects in each group if $$m$$ objects are partitioned into $$n$$ equal groups
In contexts where multiplication is not commutative, there is no such equivalence. For example with ordinal multiplication, $$\alpha\cdot\beta$$ means (intuitively) $$\alpha$$ repeated $$\beta$$ times'', so $$2\cdot\omega=\underbrace{2+2+\cdots}_{\omega\text{ terms}}=\omega$$ while $$\omega\cdot 2=\underbrace{\omega+\omega}_{2\text{ terms}}>\omega$$