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I am not a mathematician. I did additional maths O’level back in the stone age but did not pursue maths further (much to my regret).

I am reading David Acheson’s fascinating book ‘The Story of Calculus’ and have just about kept up till I got a use of ‘$\cdot$' (dot) that I do not understand. It is in his Chapter $14$ ‘an Enigma’ and first occurs here in the context of chain rule:-

Suppose, for instance, that $y$ is some function of $x$, and that $x$ itself is a function of some other variable - say $t$. Then we can, if we wish, consider $y$ as a function of $t$, and then $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$

What is the dot doing? I looked at the suggested previous questions about the dot without success. Does it mean $\&$ (as it does in propositional logic, where $P.Q$ stands for $P \& Q$?

The (or a) mysterious dot corps up again in Chapter $23$, about $e$ numbers, on the topic of the Taylor series. Here we find the series

$$e^x=1+x+\frac{x^2}{1.2}+\frac{x^3}{1.2.3}+...$$

What is the '$.$' doing here, please? Is it in some way a concatenation? Or what is it?

mrtaurho
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Tuffy
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    Sometimes a dot is used for multiplication – J. W. Tanner Mar 13 '19 at 13:37
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    Such a use of a dot when used for multiplication however usually occurs centered vertically as such: $a\cdot b$ typed as a \cdot b as opposed to lower like a decimal point as such: $a.b$. – JMoravitz Mar 13 '19 at 13:39
  • @J.W.Tanner Thank you. Yes, that makes sense. – Tuffy Mar 13 '19 at 13:39
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    If you just type a\cdot b without initiating mathmode, it doesn't do anything special of course... you need to initiate mathmode first using dollar signs like $a\cdot b$. See more about how to type with MathJax and $\LaTeX$ here by visiting this tutorial – JMoravitz Mar 13 '19 at 13:45
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    One final comment from me before leaving this thread, I would personally avoid using the lower dots to mean multiplication and would only use center dots as it is more common on an international site to interpret $5.3$ as the number $5 + \frac{3}{10}$ rather than the number $15$. Yes, some countries use commas rather than periods to denote decimal points so it might not have been ambiguous to them, but it will appear strange and frustrating to those from countries where that isn't the case. It is like how $\sin^{-1}$ means different things based on your location ($\csc$ vs $\arcsin$). – JMoravitz Mar 13 '19 at 13:50
  • Historical note : "dot" is sometimes used as "and" in propositional logic because prop logic originated as boolean algebra and Boole used "product" (juxtaposition : $xy$) as logical multiplication to mean both intersection and conjunction. – Mauro ALLEGRANZA Mar 13 '19 at 14:02
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    In my (almost entirely English-language) experience the lowered dot for multiplication is used in British sources. Acheson is British. – Michael Lugo Mar 13 '19 at 14:11
  • Even if one accepts the use of . for multiplication, I would still object that x^2/1.2 should mean (x^2/1) . 2 and not x^2 / (1.2). – Federico Poloni Mar 13 '19 at 17:30
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    @FedericoPoloni I would say that it is simply the missing parathesis which leads to this case of weird formatting. Since it is given that the series should equal the Taylor Series Expansion of $e^x$ it is clear that it refers to your latter option. – mrtaurho Mar 13 '19 at 17:38
  • It seems to me that the notation $1.2.3$ for $1\cdot 2\cdot 3$ is more archaic British use as well, but I'm not British so I don't know for sure. – JMJ Jun 18 '19 at 22:52
  • @SZN Well, after all that's the whole point of this question, isn't it? That's why I decided to keep the eye catching title. Additionally it's still used today in some fields. – mrtaurho Jun 18 '19 at 22:59

3 Answers3

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It is a quite common notation, if used, for multiplication, i.e.

$$5.3=5\cdot3=5\times3=15$$

In your case

$$dy/dx.dx/dt=\frac{dy}{dx}\times\frac{dx}{dt}$$ and $$e^x=1+x+\frac{x^2}{1.2}+\frac{x^3}{1.2.3}+\cdots=1+x+\frac{x^2}{1\times2}+\frac{x^3}{1\times2\times3}+\cdots$$

mrtaurho
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    I think that’s it, thank you. But does that mean in the second example that “x^3/1.2.3” means “x^3/1x2x3”? – Tuffy Mar 13 '19 at 13:50
  • @Tuffy Precisely! $1.2=1\times2=2$, $1.2.3=1\times2\times3=6$, etc. – mrtaurho Mar 13 '19 at 13:52
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    Is the low dot actually really "commonly" used for multiplication? Where? I can understand $x.y$, and $5 \cdot 3$ is obviously multiplication, but wouldn't $5.3$ get confused with the number $5 + 3/10$ really fast?! – ilkkachu Mar 13 '19 at 17:07
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    It is completely nuts for "5.3" to be the same as 5×3. 5.3 is 5 + 3/10, and you can't overload the same symbol to mean something totally different. (letting dx.dx be dx × dx is tolerable, because dx.dx does not already mean dx + dx/10). – Monty Harder Mar 13 '19 at 17:14
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    @ilkkachu Especially in the field of algebra, specifically in Linear Algebra and Abstract Algeba, I have encountered this notation quite often denoting multiplication, e.g. an inner product, in various ways. For myself, as German native speaker, I am used to $5\color{red}{,}3$ as equivalent to $5+\frac3{10}$ from where it cannot be mistaken with $5.3$. I have to admit that I have seen this notation rarely in connection with actual multplication of numbers. – mrtaurho Mar 13 '19 at 17:31
  • @mrtaurho, Right, I meant with numbers. With letters and symbols it can't really get confused with anything. We actually use the comma as the decimal separator, here, too, but what with English being so influential, the low dot between numbers would be really weird. Something like $5,3.2,3$ would look a list of three numbers to me... (even without Mathjax formatting it like that, with small spaces after the commas) – ilkkachu Mar 13 '19 at 17:38
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    @ilkkachu In fact, I have solely encountered this notation in connection with either inner products of vectors spaces or as product of two groups. Take a look a this video:https://matterhorn.dce.harvard.edu/engage/player/watch.html?id=4c666bab-7cdd-4cac-af67-eb3ce8eb2c14 , right in the begin defining the group $G$ as product of two other groups, namely $G=\mathbb R^2\color{red}{.}O(2)$. – mrtaurho Mar 13 '19 at 17:43
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    @mrtaurho, ok, good, that does alleviate my initial shock. :) – ilkkachu Mar 13 '19 at 17:45
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    @Monty Harder: It is completely nuts for "5.3" to be the same as 5×3. 5.3 is 5 + 3/10 --- Given that Acheson's book appears to take a heavily historical approach (based on what little I can see via google sample previews), it seems pretty obvious to me that he's doing this so as to be using the notation originally used in the 1700s and 1800s. For example, see p. 109 here and p. 49 here. – Dave L. Renfro Mar 13 '19 at 17:58
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    When I was in high school, the local university organised a study group for university prep stuff and used this notation for multiplying integers on a worksheet, I believe using just a regular period. It confused and bewildered me enough that I never went back, so I would definitely not recommend using it for integers but I have seen it. – llama Mar 13 '19 at 17:59
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    To complete the confusion, I have seen some old book use a raised dot as a decimal point! – hmakholm left over Monica Mar 13 '19 at 23:16
  • @MontyHarder: Books that use a baseline dot for multiplication also tend to use a raised, center dot for the decimal point, so $1{\cdot}3$ means $\frac{13}{10}$ and $1.3$ means 3. – Nick Matteo Mar 14 '19 at 03:36
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    @MontyHarder "you can't overload the same symbol to mean something totally different" Sure we can; we're humans, not computers. There are too many mathematical concepts for no symbol (or word) to ever do two things. The important thing is for authors to help readers to code switch, especially if the authors can't help but use multiple meanings within one work. – J.G. Mar 14 '19 at 05:57
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    @Henning Makholm: Now that you mention it, I recall often seeing a raised dot as a decimal point in 1800s English literature. In fact, almost all 1800s English mathematical literature I've seen used this, while using the lower dot for multiplication. See, for example, p. 98 of Todhunter's 1859 Plane Trigonometry for raised dots and p. 95 for lowered dots. I checked some of the items I cited here, and it seems the French used commas for decimal points and raised dots for mult. – Dave L. Renfro Mar 14 '19 at 07:13
  • @DaveL.Renfro . I had an early cheap pocket calculator and saw that the French part of the instructions used commas for decimal points, and where the English would use commas between blocks of digits, the French used empty spaces. 19th and early 20th century (English) books didn't use $!$ for factorial. And in geometry you would see $A:B::C:D,$ which, if none of $A ,B,C,D$ is $0$, means $A/B=C/D$ and in general is equivalent to $A\times D=B\times C .$ – DanielWainfleet Mar 14 '19 at 10:25
  • @DanlWainfleet: Regarding the factorial notation, about 3 months ago I posted in mathematics stack exchange some history about the factorial symbol. See my answer to History of notation: “!”. – Dave L. Renfro Mar 14 '19 at 13:13
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Sometimes a dot is used for multiplication. Cf. this Wikipedia article.

J. W. Tanner
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As @J.W.Tanner said though we usually write $a$ times $b$ as $$ab$$ or $$a \times b$$ the urge of denoting it by $$a \cdot b$$ is also common.

MATHS MOD
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    I'm not sure who "we" is in this answer. Among mathematicians, multiplication of numbers is almost universally denoted by $ab$ or $a\cdot b$. $\times$ is used to denote other kinds of products, like the cross product of vectors. – Wojowu Mar 13 '19 at 14:21
  • @Wojowu: note that $\times$ is "times" in MathJax – J. W. Tanner Mar 13 '19 at 14:38
  • @J.W.Tanner I know, I have used that in my comment. – Wojowu Mar 13 '19 at 14:45
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    @Wojowu I think I'd be a little more precise: multiplication of numerical variables is almost always $ab$ or $a\cdot b$, with $\times$ used for other kinds of products. But for multiplying literal numbers, a lot of people will write, e.g., $3\times 5$ because $3\cdot5$ looks a lot like $3.5$ (and, obviously, $35$ is thirty-five, not fifteen). – David Richerby Mar 13 '19 at 16:15
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    @DavidRicherby I admit I have meant numerical variables there; of course concatenation would be a terrible choice of a notation. I would still think that, for concrete numbers, $3\cdot 5$ would be more common than $3\times 5$ (though I admit I'm having hard time finding evidence for that - most math papers nowadays don't multiply numbers!) – Wojowu Mar 13 '19 at 16:22