# The use of dots in mathematics

#### Prove It

##### Well-known member
MHB Math Helper
Hi everyone, just something for us to discuss. I have always been taught to use a full stop for a decimal point, and a centred dot to represent multiplication. This appears consistent with the output of CAS calculators and software. However, it appears that it people in older generations (I'm 27) do the exact opposite, having a full stop represent the multiplicative dot and the centred dot for a decimal point.

I was wondering which way you each prefer? It also got me thinking, what led to this "change". I would expect it would be the introduction of computers to aid calculations. Since calculators and computers usually have to perform floating point operations (i.e. deal with decimals), it makes sense to use a button on the keyboard for a decimal point, hence the switchover to the full stop. Does my reasoning seem correct? Or were there other reasons that you can think of?

#### MarkFL

Staff member
I was taught to use the centered dot for multiplication and and "full stop" or period for a decimal point. I'm 50 years old by the way, just for reference.

What threw me off the first time I saw it on a math help forum was the use of a comma for a decimal point.

#### Jameson

Staff member
Never heard of having a period indicate multiplication. That would not work today anywhere from what I can guess. Often when writing out equations I find I don't like any of the notations for multiplication. Sometimes the \cdot $\cdot$ seems too small and I like to use $\times$ but for some reason this was strongly discouraged when I was in school.

In Europe that comma and period are the opposite of what we use here in my experience, at least in many of the countries there. It threw me off the first time as well but you get used to it.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Coming from the Netherlands, I grew up using a comma as a decimal mark and a $\times$ for multiplication. On high school, the $\cdot$ became preferred for multiplication, presumably to distinguish it from the letter $x$. And I learned that in English speaking countries the point was used as a decimal mark. This was all before even calculators were introduced.

In a typical engineering expression, like the following, you have so many decimal points that a centered dot is easily mistaken for one.
$$w_{max} = \frac{5 Q l^3}{384 E I} = \frac{5 \cdot 2.1 \cdot 10^{6} \cdot (20,000 \cdot 10^{-3})^3}{384 \cdot 3.12 \cdot 10^9 \cdot 1.2 \cdot 10^{-6}}$$
(The maximum bending $w_{max}$ of a beam of length $l$ supported at its 2 end points under a equally divided load $Q$ of a material with elastic modulus $E$ and with a profile that has a second moment of inertia $I$.)
When writing by hand I myself tend to use $\times$ again next to $\cdot$, taking care there will be no confusion between $x$, $\times$, $\cdot$, and $.$. Typically I consistently use one version of multiplication for the multiplication with $10^n$ and the other for the other multiplications.

I have seen the $\cdot$ as a decimal mark only a couple of times by some engineers. According to Decimal Mark on wiki it is used as an alternative to the decimal point in: Ireland, Japan, Korea, Malaysia, New Zealand, Philippines, Singapore, Taiwan, Thailand, United Kingdom, United States (older, typically hand written).

#### Ackbach

##### Indicium Physicus
Staff member
I know Mathematica can use the full stop for the vector dot product. Otherwise, any thoughts I'd have had have already been written by others.

#### mathbalarka

##### Well-known member
MHB Math Helper
I have seen some abstract algebra textbook to use dot as an arbitrary group operation, although the usual is to use juxtaposition.

#### Deveno

##### Well-known member
MHB Math Scholar
Here was what I was able to glean from various internet sources:

Our numeral notation is, of course, Arabic in origin, and many of its conventions can be traced back to the usage of Arabic mathematicians.

The rational number $99\frac{9}{10}$ would often be notated like so:

99 9

or:

99|9.

Over time, the vertical stroke became shortened, like so:

99|9, and it was typical for the fledgling type-setting industry to substitute a comma or period instead, there being no comparable typographical character in most European alphabets.

Apparently, France and Italy were already using the "full-stop" (period) to delimit Roman numerals in such things as pagination, chapter headings and the like, so adopted the comma as the numerical decimal delimiter. This eventually became standard across most of Europe, except for the British Empire, which was already using commas to group large integers in "places of 3" such as: 123,500.

To avoid confusion with actual periods in written (and type-set) work, in the British Empire, the interpunct ("mid-dot") became common, and this was especially useful in hand-written ledger work, where a decimal point written as a period could be obscured by the rulings of the ledger.

Unfortunately for us, the interpunct (for multiplication) seems to have originated largely with Liebniz, who wrote in 1698 to John Bernoulli: "I do not like $\times$ as a symbol for multiplication, as it is easily confounded with $x$..." and later championed by his friend Christian Wolff, who was regarded by many as the leading philosopher of his day (and now regarded somewhat uncharitably as a Liebniz "follower"), and became wide-spread as an "abstract" representation of multiplication (now seen in vestigal form as the "dot-product" of vector spaces, and in place of the asterisk in some older abstract algebra books).

To see examples of the interpunct being used in American texts, one has to go back to the 1950's at least...by the time I first encountered "decimals" (around 1969 or so), that usage was already considered "archaic" (and not used in any of the textbooks I ever had, although I did see in in some engineering books my grandfather had).

So, anyway, "proper" usage and custom nowadays is not entirely "convergent": many countries use periods where Americans would use commas, and vice versa. Thus the rational number used as an example above, would be in official SI terms:

99,9

(Apparently France and England, in particular, have made it a point of honor to "do things differently", especially in mathematics...the contention over "the founder of Calculus" no doubt being one of the many reasons for this centuries-old feud. This can be seen most strikingly in Canada where the English-speaking provinces use the period as a decimal point, and Quebec uses the comma).

So...the entire situation is a bit of a muddle, and no doubt confusing to Americans who study abroad, or read foreign mathematical books, and likewise for Europeans who encounter the same difficulty with American or British oeuvres. What is "correct", therefore, depends on who and where you are.

Americans, of course, having invented the fastest computers, and best rocket-ships, like to think their word is final on the matter, and would no doubt be surprised to learn how much of the world disagrees.

Interestingly enough, much of the Arab world, having kicked off this glorious mess, now uses entirely different typography to denote numerical values, in particular, they use in some countries a (stylized) dot for "0". The tower of Babel, the sequel....coming soon to a global theater near you!

#### Klaas van Aarsen

##### MHB Seeker
Staff member
in official SI terms:

99,9
I believe that the French and the English stroke a deal in SI.
Probably because otherwise we wouldn't have a standard now.
I imagine they have had heated discussions on the subject.

Anyway, we now have SI (French style) that writes it as $99,9$ and SI (English style) that writes it as $99.9$.
In other words, SI allows both forms as long as they are consistently applied.