Is $4 \times 6$ defined as $4 + 4 + 4 + 4 + 4 + 4$ or $6 + 6 + 6 + 6$?

Question

There are long debates among Indonesian netizens about this http://www.globalindonesianvoices.com/15785/is-4x6-the-same-as-6x4-this-primary-school-math-made-controversy-in-social-media/

Answer 1

Are you aware, the multiplication operation is commutative over $\mathbb{N}$.

As such, $6 \times 4 = 4 \times 6$.

And we have also $4+4+4+4+4+4=24=6+6+6+6= 4 \times 6 = 6 \times 4$

EDIT: I forgot the following:

$6+6+6+6=(4+2)+(4+2)+(4+2)+(4+2)=(4+4+4+4)+(2+2+2+2)=(4+4+4+4)+(4+4)=4+4+4+4+4+4$

Thanks to other properties of the additive operation, such as commutativity and associativity...

EDIT2: Changed to $\mathbb{N}$ from $\mathbb{R}$ per suggestion.

Answer 2

Let $A \times B$ mean the addition of $A$ repeated $B$ times.

Since multiplication is commutative in $\mathbb{R}$, that tells us that $A \times B = B \times A$. In other words, the addition of $A$ repeated $B$ times is the same as the addition of $B$ repeated $A$ times.

Thus, $4 \times 6 = 4+4+4+4+4+4 = 6 \times 4 = 6+6+6+6$

Answer 3

Surely, multiplication is commutative in R and I think the teacher was a bit narrow-minded when correction it.

Based on my native language (Portuguese), I would opt for 4x6 = 6+6+6+6, just because this is how I read it.

I can also say 4 objects = (object + object + object +object) and 4 m = 4 * meter, but it would be weird to say (objects x 4) or note a distance as m 4.

Answer 4

Let $a\cdot b$ denote the cardinality of the set $A\times B$, where $A$ has cardinality $a$ and $B$ has cardinality $b$. There is no addition involved in this definition. But of course he have to introduce the notion of ordered pair to speak about $A\times B$, and you may have fun starting your controversy again: Is $\langle x,y\rangle$ a language primitive or defined as $\{\{\{x\},\emptyset\},\{\{b\}\}\}$ or $\{\{a,1\},\{b,2\}\}$ or $\{\{a\},\{a,b\}\}$ or $\{a,\{a,b\}\}$ or ... ?

Answer 5

If you refer to the Peano axiom for multiplication, it says $a \cdot S(b)=a+a\cdot b$, so expanding this we would have $4 \cdot 6=4+4+4+4+4+4$. What definition of multiplication are you using?

Answer 6

If this teacher explicitly defined the notation $a\times b$ to be short for the sum of $b$ terms $a$ and insisted that this convention be enforced, then he was right to count errors for not following the instructions.

But nobody that I know uses this strict and somewhat silly/misleading notation.

Answer 7

When written $4 \times 6$ this is often, in English read as four sixes, which implies $6+6+6+6$ rather than $6 \times 4$ which would be read as six fours which would imply $4+4+4+4+4+4$., but this is a pure linguistic convention and not how we define these things in maths.

Answer 8

Another approach to it, less "mathematical" than my other answer, but linked to the "intuition" of the multiplication.

We are dealing with small children, and multiplication is, for them, repeated addition.

The question is $6+6+6+6=6 \times 4$ or is it $4 \times6$ ?

Some people tend to prefer (they "see") the following reasoning: they count the number of times they will multiply, and then put the number (or the elements, items etc.) they want to "add multiple times". In that case, $6+6+6+6=4 \times 6$.

However, you might want to see that writing $6+6+6+6$, you are dealing with sixes, and that you deal with them four times. So you start by identifying the number at hand ($6$), and then write the number of times you "saw" the number, that is, the number of items added together. In that case, you have $6+6+6+6=6 \times 4$.

Of course, we all know this is the same thing, but intuitively this is different for someone starting with multiplication. Both ways are valid to my eyes...