# What is the dot in “$1.2.4$”?

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?

• Sometimes a dot is used for multiplication – J. W. Tanner Mar 13 at 13:37
• 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 at 13:39
• 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 at 13:45
• 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 at 13:50
• In my (almost entirely English-language) experience the lowered dot for multiplication is used in British sources. Acheson is British. – Michael Lugo Mar 13 at 14:11

## 3 Answers

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 $$e^x=1+x+\frac{x^2}{1.2}+\frac{x^3}{1.2.3}+\cdots=1+x+\frac{x^2}{1\times2}+\frac{x^3}{1\times2\times3}+\cdots$$

• 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 at 13:50
• 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 at 17:07
• 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 at 17:14
• @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 at 17:31
• @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 at 17:58

Sometimes a dot is used for multiplication. Cf. this Wikipedia article.

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.

• 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 at 14:21
• @Wojowu: note that $\times$ is "times" in MathJax – J. W. Tanner Mar 13 at 14:38
• @J.W.Tanner I know, I have used that in my comment. – Wojowu Mar 13 at 14:45
• @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 at 16:15
• @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 at 16:22