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

Question

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?

Answer 1

I have no idea the answers to your historical linguistic questions, which are better asked in other forums, but will provide a mathematical answer since you posted here. :)

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

The first interpretation is sometimes called the measurement interpretation, while the second is called the partitive interpretation.

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 $$\omega\cdot 2=\underbrace{\omega+\omega}_{2\text{ terms}}>\omega$$