# School math notation - x for multiplication or a variable?

1. Nov 30, 2017

### MichPod

I got my education out of the American continent, yet currently I live in Canada and my son goes to school here (5th grade + some additional math training)

What puzzles me is that currently he may write things like
$$5\times-1$$

meaning $5\cdot(-1)$, which I can only read as $5x-1$ (i.e. an expression with x-variable).

Well, may be it is generally a bad idea to write $"-1"$ without brackets after a multiplication sign (I never did), but what bothers me more is how he is going to distinguish in the future the $"\times"$ multiplication sign from the $"x"$ sign for unknown variable in his writings. I personally was taught to use a dot $"\cdot"$ for multiplication so I never had such a trouble and I have no idea how to resolve such a conflict.

Many people here must have studied to use $"\times"$ sign for multiplication in schools so they must know if any problem happens when unknown $x$ is introduced. Any help with this?

2. Nov 30, 2017

### Staff: Mentor

It is usually only a problem in handwriting, since books should use a dot. A handwritten dot is hard to see, that's where the x comes from. I used to write it very small so there wasn't a risk of confusion. The x has by the way another meaning, namely the (direct) product of sets:
$$S \times T = \{\,(s,t)\,\vert \,s \in S\; , \; t\in T\,\}$$
This means in return, that x to replace the dot shouldn't be allowed, or written very small.

3. Nov 30, 2017

### Staff: Mentor

Since he's only in 5th grade, he isn't studying algebra yet (??), so the $\times$ symbol is unambiguous in the context. When I was going to school, algebra didn't come until 9th grade, so I'm assuming that they haven't moved it so far forward into the elementary grades. After variables represented by letters are introduced into the mix, then using $\times$ to indicate multiplication should be discouraged.

Regarding $5 \cdot -1$, parentheses around -1 aren't strictly needed. They certainly aren't in most programming languages, which have a better defined set of precedence rules than does mathematics. In the expression above, the '-' is a unary operator, so it operates on 1 before the multiplication is performed. So $5 \cdot -1$ and $5 \cdot (-1)$ evaluate to exactly the same thing, -5.

Last edited: Dec 1, 2017
4. Nov 30, 2017

### MichPod

Not yet, I am just worried for the future (which may be much sooner than 9th grade). Actually, he already knows how to solve some simple equations, yet to write them with $"\times"$ sign makes a complete mess.

5. Nov 30, 2017

### Staff: Mentor

Another way out is to rename the variable x and write it with another letter. But I see that t for time might cause problems with the plus sign. It is more important to know what is meant than how it is written. A clean way is not to write a multiplication with an x. Often it just can be omitted, if it is not between two numbers.

6. Nov 30, 2017

### The Bill

When I started running into situations where I might want to use both symbols, I began writing the letter x with slightly curved ends on the upper left to lower right line similar to the typeface in many books. I would write the $\times$ sign perfectly straight. This made it easy enough for me to distinguish the two in my notes, and for my professors when grading my work. At this point, I was using $\times$ for the vector cross product or for Cartesian products, depending on the course material and problems.

I think I learned this writing convention from the blackboard style of one of my physics professors.

7. Dec 1, 2017

### gmax137

It is a pretty rare notation that is completely unambiguous. Take the "dot" for multiplication. How is that not confused with the decimal point? (Example: with sloppy handwriting, 235.23 could be thought to represent 5405, as 235 * 23 = 5405). Humans have a pretty sophisticated tolerance and appreciation of ambiguity. We are very good at evaluating situations based on context.

8. Dec 1, 2017

### MichPod

Interestingly, I was taught in school to use comma for decimal point. I.e. I would write 235,23
Now I see why.

9. Dec 1, 2017

### Nidum

For my own notes I often use the asterisk symbol * to denote multiplication . Comes from using programing languages like BASIC many years ago .

Last edited: Dec 1, 2017
10. Dec 1, 2017

### Staff: Mentor

In many (all?) European countries, the separator between the integer part and the fractional part of a number is the comma, and a period is used to separate groups of three digits in the integer part. For example, in the U.S. we would write $1,000,001.99, but in Europe, it would be written as$1.000.001,99 .

11. Dec 1, 2017

### Staff: Mentor

Wikipedia lists UK and Eire as dot countries, too, what a surprise. In Switzerland are both in use, as is in Canada.

12. Dec 1, 2017

### PeroK

13. Dec 1, 2017

### Staff: Mentor

I think the major point is the better visibility of the comma, which is important. "More logical" is a strange criterion. One could also say: A comma separates the major from the minor part of a sentence, resp. number. I don't really care except on handwriting, because of the visibility. Dots tend to get lost. It's only annoying when you cannot use the separator of the number block on a keyboard or - which I had to do once - reformat the number of elements of the monster group. On the other hand, this is a great example of the standard software question: Shall I use an editor (program), or is it faster to do manually?

14. Dec 1, 2017

### PeroK

One of the things I never achieved was to get 100% in any exam. Once I got everything correct in an "arithmetic" exam except one question where I interpreted "x" as $x$.

Forty years later it still rankles.

15. Dec 1, 2017

### PeroK

Sadly, these are the sort of cultural differences that have led to Brexit!

16. Dec 1, 2017

### Staff: Mentor

Sadly is right. I always considered our cultural differences as an enrichment rather than a disadvantage. They make Europe interesting. In part it makes sense to find a common system, I mean, not that long ago we had one country - one mile.

17. Dec 1, 2017

### MichPod

Things are not constant. USSR (where I am from) passed from $\times$ to $\cdot$ in school notation somewhere in 1950-1970. My mother once told me she was using $\times$ in her school, now my son is using it again in Canada to my surprise and some disappointment. I kinda feel having fallen into the past.
Besides, I learned brackets and equations in grade 1, my son is supposed to learn the brackets somewhere near grade 7.

Last edited: Dec 1, 2017
18. Dec 1, 2017

### I like Serena

Both denote multiplication and I use both of them. Both were taught in school as well (Netherlands).
I'd write for instance $1.3{}_{{}^\times} 10^{3}\cdot 2.1{}_{{}^\times} 10^2$ to make it easier to read (compare with $1.3\cdot 10^{3}\cdot 2.1\cdot 10^2$ that tends to get messed up with all those dots).
As for the problem with handwriting, I recommend to learn how to write every symbol and letter in such a way that there is no ambiguity.
We can already see that $\times$ and $x$ look very different. The first has straight lines that cross each other, and the second has rounded curves that don't actually cross in the middle. I recommend to learn to emphasize those differences in handwriting.

19. Dec 1, 2017

### MichPod

Hmm... as for me personally, in my childhood I used to write $x$ the way you propose but after a while I found that not very clear (it sometimes looked like two letters "c" (first mirrored)), so I thoughtfully abandoned that and started to use the cross ("x").

20. Dec 1, 2017

### Staff: Mentor

I haven't checked the sources, but it is interesting anyway, what Wikipedia has about the historical origins:
$\times \, : \,$ William Oughtred - 1631, eventually already in 1618
$\cdot \, : \,$ Gottfried Wilhelm Leibniz - 1698, eventually already in 1694; reasons: risk of confusion with $x$
$\ast \, : \,$ Johann Rahn - 1659
However, it doesn't mention specific Russian uses.