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I just would like to know whether or not there exists an historical reason to prefer the expression $a b$ to $a \times b$. Why does the sign $\times$ vanish (whereas $+$ stays)?

I thought that $\times$ was replaced with $\cdot$ not to be confounded with the variable $x$, and only after, $\cdot$ vanished. However, I do not know whether this explanation could be plausible.

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    $\begingroup$ $2$ dozens means $2\times 12$ $\endgroup$
    – user5402
    Aug 2, 2014 at 22:02
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    $\begingroup$ In the Sciences, many basic formulas are multiplicative. One might as well make life easy there by using compact notation. Nobody wants to see $E=m\times c\times c$. $\endgroup$ Aug 2, 2014 at 23:48
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    $\begingroup$ Edsger Dijkstra, a famous mathematician, considered the invisible multiplication operator a mistaken, and complained bitterly about it. $\endgroup$
    – MJD
    Aug 4, 2014 at 17:42
  • $\begingroup$ If we have $3+\frac 12=\frac 7 2$ and $3 \cdot \frac 12=\frac 32$, it must follow that $3\frac 12 = \frac 32$ doesn't it? Just kidding. More to the point, as metacompactness already mentioned, the juxtaposition is quite natural if you consider that the intuitive $3\text{ m}$ actually means $3 \cdot \text{meter}$. $\endgroup$ Aug 4, 2014 at 18:35

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I think the $\times$ symbol for multiplication wasn't the first one used to denote multiplication, since Greeks used to denote multiplication side-by-side.

The dot $\cdot$ notation was introduced as a symbol for multiplication by Leibniz. On July 29, 1698, he wrote in a letter to Johann Bernoulli: "I do not like $\times$ as a symbol for multiplication, as it is easily confounded with $x$..."
Quoted in F Cajori, A History of Mathematical Notations. [1]

Leibniz also used the cap symbol $\cap$ symbol for multiplication. Thomas Harriot (1560-1621) used the dot $\cdot$ for multiplication much before Leibniz. The asterisk $*$ was used by Johann Rahn (1622-1676) in 1659 in Teutsche Algebra.

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  • $\begingroup$ Re-quoted in "The Earliest Uses of Mathematical Notation" ... specifically, "Symbols of Operation". The page notes that juxtaposition of multiplied numbers dates back to 8/9/10th-century India, and the 15th century further west. $\endgroup$
    – Blue
    Aug 2, 2014 at 22:02
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    $\begingroup$ @Blue Thanks for the additional references. :) $\endgroup$
    – Hakim
    Aug 2, 2014 at 22:05
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The following examples may offer an explanation.

Consider the expression $2a+3b$ in the commonly understood sense. In that alternate universe, we would have to write this as $(2\cdot a)(3\cdot b)$. Note that the parantheses would be necessary to avoid ambiguity. In the classical setting, we spent $5$ characters; in the alternate setting we spent $10$ characters.

One can say "but $(2+a)(3+b)$ would be easier," but then we would have to write this as $(2a)\cdot (3b)$. In the classical setting, we spent $10$ characters; in the alternate setting we spent $9$ characters.

On average (unless I cheated above by adding unnecessary extra parantheses), it appears that especially when multiplication and addition are mixed, the setting we use right now is the more economic setting. Of course, this is far from a conclusive result due to the fact that the sample space is so small.

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  • $\begingroup$ Except of course in transfinite arithmetic, where $\omega3$ means three omegas ($\omega+\omega+\omega$) and $3\omega$ means omega threes ($3+3+3+\cdots$). $\endgroup$
    – bof
    Sep 9, 2014 at 9:11
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One aspect that I think hasn't been explicitly mentioned yet is that in many languages if you count things you just put the number in front of what you've counted - as in "six oranges". So, in a way, you can "read" something like $3a$ as "three $a$'s". It's only an analogy of course, but note that for example you rarely see people writing $a3$ instead or even $a\cdot 3$.

(And maybe also worth mentioning is that of course a CAS like Mathematica has to be told how to interpret something like $ab$ and thus we'll have to write "$\mathtt{a}\;\mathtt{b}$" with space inbetween which makes it look and feel more like "$a$ $b$'s".)

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