Multiplication Notation based on Summation

Question

Just want to ask a question This question becomes a hot topic in my country right now since someone upload a photo about his young brother's homework that marked wrong by his teacher

okay this is the example of the problem $$ 4 + 4 + 4 + 4 + 4 + 4 = .... \times ..... $$ the student answers $4 + 4 + 4 + 4 + 4 + 4 = 4 \times 6 = 24$ (This is considered as a wrong answer by his teacher)

Not the result that becomes a problem in here But the multiplication notation. since the teacher consider the right answer is $6 \times 4 = 24$

there are multiple opinion about this and I just curious about this and want to know about what theorem explain this.

Answer 1

The General Convention::
Multiplication of elements in $\mathbb N$ is most intuitively defined as:

$$ \color{red}{a} \times \color{blue}{b} = \sum_{i=1}^{\color{red}{a}}{\color{blue}{b}} \quad, \space\forall\,a,b\in \mathbb N $$

Meaning:: $$\color{red}{a} \color{olive}{\times} \color{blue}{b} \, \equiv \,\left(\color{red}{a} \color{olive}{\text{ times }} \color{blue}{b} \right) \,\equiv\, \overbrace{\color{blue}{b} \color{olive}{+} \color{blue}{b} \color{olive}{+} \color{blue}{b} \color{olive}{+} \dots}^{\color{red}{a} \color{olive}{\text{ times}}}$$

But why take this convention? Multiplication is commutative ($a\times b = b\times a$) So, does it really matter if you define $a \times b$ as $\left(\sum_{i=1}^{b} a\right)$ or as $\left(\sum_{i=1}^{a} b\right)$ ?

Imagine a 10 year old boy named Ash , only taught the multiplication table, oblivious to the knowledge what the concept of multiplication is. (I bet all my Pokeballs that, at any instance of time, there would atleast exist one child in that situation.)

So, for Ash, multiplication is just a vague function that spits out one number when you give it two. He simply views it as some confusing concept that is beyond him and, in his mind, sees it as:

$$x \times y = \text{mult}(x,y) = \int_x^y??{\Psi}\cdot\text{d}(\mu c\kappa)??\dots$$

Now, although this illustrates the fact that you don't need to know how a car works inorder to drive it, such data abstraction is not desired when understanding fundamental useful ideas developed by cavemen for practical purposes.

It is best to first make the learner understand that the multiplication of two whole numbers is equivalent to the addition of one of them with itself as many times as the value of the other one.

Extending multiplication to $\mathbb R$, $x\times y$ can be seen as the area of a rectangle $x$ high and $y$ wide

Here's the catch: How can you expect Ash to connect the multiplication concept with the binary operation?

He can interpret $2\times 3$ as "two times three" or as "two threes". His interpretation doesn't matter to the result. Even from the perspective of area of a rectangle, height and width are interchangeable notions.

So, again, does it matter how you define it? Actually, it does.
You must interpret it the same way as your teacher inorder to not be screwed over by silly questions in examinations.

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In my opinion, the best and generally most intuitive interpretation of multiplication is as scaling where the multiplicand ($2$) is made as the unit on which the multiplier($3$) is counted.

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^{Apologies for such a chatty and opinionated answer.}

Answer 2

This is a matter of convention, which, perhaps, boils down to three competing interpretations:

(i) $4\times 6$ is four groups of $6$ (which, I think, is more common view in U.S. schools);

(ii) $6\times 4$ is six groups of $4$;

(iii) It doesn't matter.

To me it seems fair that sometimes we may want to make a distinction between the number of groups and the group size (especially to learners first learning multiplication), and so we may want to adopt convention (i) or (ii). In the classroom setting, I suppose the teacher (or perhaps the curriculum) gets to set the convention. In this case, the teacher should be responsible for communicating to the students what convention is being used.