School math notation - x for multiplication or a variable?

[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?

 

[fresh_42]

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.

 

[Mark44]

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

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.

 

[MichPod]

Since he's only in 5th grade, he isn't studying algebra yet (??),

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.

 

[fresh_42]

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.

 

[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.

 

[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.

 

[MichPod]

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).

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

 

[Nidum]

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

 

[Mark44]

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

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 .

 

[fresh_42]

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 .

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

https://upload.wikimedia.org/wikipe...lSeparator.svg/800px-DecimalSeparator.svg.png

 

[PeroK]

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

https://upload.wikimedia.org/wikipe...lSeparator.svg/800px-DecimalSeparator.svg.png
View attachment 215923

A comma represents a minor pause, whereas a full stop is something more significant so the USA/UK convention seems more logical to me!

 

[fresh_42]

A comma represents a minor pause, whereas a full stop is something more significant so the USA/UK convention seems more logical to me!

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?

 

[PeroK]

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?

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.

 

[PeroK]

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.

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

 

[fresh_42]

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

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.

 

[MichPod]

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

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.

 

[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.

 

[MichPod]

We can already see that ××\times and xxx 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.

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").

 

[fresh_42]

Things are not constant. USSR (where I am from) passed from $\times$ to $\cdot$ 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.

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.

 

[I like Serena]

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").

I did the exact same thing! ... and then I switched back when I started writing both $\times$ and $x$.
To be fair, I still tend to write $x$ as a cross in regular handwriting, but I definitely use the other form in math and physics.

 

[MichPod]

⋅: Gottfried Wilhelm Leibniz - 1698, eventually already in 1694; reasons: risk of confusion with $x$

Yessss!

 

[I like Serena]

For the record, there's plenty more symbols that cause confusion.
$a$ and $\alpha$,
$I$, $l$, and $1$,
$O$ and $0$.
The challenge is to make them all look different.

 

[fresh_42]

For the record there's plenty more symbols that cause confusion.
$a$ and $\alpha$,
$I$, $l$, and $1$,
$O$ and $0$.
The challenge is to make them all look different.

And if one has functions $f,g$ then what is this: $fg=f \cdot g$ or $fg = f \circ g$ or sometimes even $fg= g \circ f$.

 

[I like Serena]

And if one has functions $f,g$ then what is this: $fg=f \cdot g$ or $fg = f \circ g$ or sometimes even $fg= g \circ f$.

Indeed. And another favorite of mine, is f(x+1) or a(x+1) a function call or a multiplication?
And for the record, we can also have $f\cdot g=f \circ g$ or $f\cdot g=g \circ f$ in abstract algebra.

 

[MichPod]

So, to sum the things up, I got the following advices:

• abandon $\times$ and use $\cdot$ as soon as algebra is introduced
• use some special curved way to write $\times$ meaning $x$ ( "with slightly curved ends on the upper left to lower right line" )
• for the variable, just write curved $x$, not cross
• for multiplication, make $\times$ sign very small

And... I got a confirmation that the usage of $\times$ sign for multiplication does cause problems. At least for some people.

 

[fresh_42]

So, to sum the things up, I got the following advices:

• abandon $\times$ and use $\cdot$ as soon as algebra is introduced
• use some special curved way to write $\times$ meaning $x$ ( "with slightly curved ends on the upper left to lower right line" )
• for the variable, just write curved $x$, not cross

• and make the multiplication x very small

 

[I like Serena]

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 .

Or as $ 1 000 001,99 eliminating the ambiguity, which is my personal preference.

 

[Mark44]

So, to sum the things up, I got the following advices:

• abandon $\times$ and use $\cdot$ as soon as algebra is introduced
• use some special curved way to write $\times$ meaning $x$ ( "with slightly curved ends on the upper left to lower right line" )
• for the variable, just write curved $x$, not cross
• for multiplication, make $\times$ sign very small

And... I got a confirmation that the usage of $\times$ sign for multiplication does cause problems. At least for some people.

The $\times$ symbol doesn't ordinarily cause problems, as it is reasonably distinct from $x$. There are times (pun not intended) where the $\times$ symbol is preferable, such is in writing numbers in scientific notation -- e.g., Avogadro's number, $6.022 \times 10^{23}$ or the normalized form of binary fractions $1.011 \times 2^5$.

 

[Mark44]

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

I seriously doubt that the difference in how commas and periods are used as number separators made it into anyone's top 100 reasons for wanting to leave the EU. From my perspective, on the "other side of the pond," the reasons went a lot deeper. I won't elaborate, as this would be 1) off-topic and 2) heading off into politics.

 

[berkeman]

Since he's only in 5th grade, he isn't studying algebra yet (??),

Or vector cross products?

Apologies if this has been mentioned already in the thread (I got some vertigo when the discussion veered into comma space), but just use the star character * for multiplication. It's pretty unambiguous for basic math, IMO.

The "x" symbol has too many meanings to overload it with multiplication, IMO.

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/fqvb.gif

 

[I like Serena]

Can we please please make that $\vec F=q\vec v \times \vec B$?
I'm getting some vertigo myself for seeing an $x$ variable between vectors.

 

[berkeman]

Let me call Hyperphysics...

Oh, nuts, I misplaced their number...

 

[berkeman]

for multiplication, make ×\times sign very small

or *

 

[I like Serena]

Let me call Hyperphysics...

Oh, nuts, I misplaced their number...

From the context it's clear of course that it's a cross product, so in the end there is no ambiguity.
Still, I'm not impressed with the Hyperphysics people, since they are apparently sloppy with their symbols.
It's like seeing spelling mistakes in a curriculum vitae. ;)

Oh, and btw, this cross product is multiplication - just for 3D vectors.
It's just that we distinguish the center dot for the dot product, even though that is not proper multiplication (since its result is not a vector breaking closure).