How to make sense of multiplication in the case of negative times positive?

Question

Multiplication, most fundamentally, means that when there are two or more equal numbers to be added together, the expression of their sum can be abridged:

$2+2+2+2+2+2$ can be abridged as $6\times 2$ (which essentially means the repeated addition of $2$ for $6$ times)

$(-8)+(-8)+(-8)+(-8)+(-8)$ can be abridged as $5\times(-8)$ (which essentially means the repeated addition of $-8$ for $5$ times)

Conversely one can conclude from $4\times 2$ the repeated addition of $2$ for $4$ times $(2+2+2+2)$ and from $2\times 4$ the repeated addition of $4$ for $2$ times $(2+2)$ and one can further discover the commutative property for the multiplication.

Till this things make sense but how to make sense of $(-3)\times 4$ (repeated addition of $4$ for $-3$ times!) and also how to establish the commutative property for the same case?

I don't know if this is really the sort of thing you want, but you might find some of the ideas in the answers here helpful. — Stahl, Dec 22 '18 at 22:07
Multiplication isn't repeated addition. If you are having trouble making sense of multiplication by negative numbers, the right course of action is almost certainly to think in terms of a better model of multiplication (personally, I tend to think of it as a scaling, where the sign indicates an orientation). — Xander Henderson, Dec 22 '18 at 22:19
Possible duplicate of Why is negative times negative = positive? — Xander Henderson, Dec 22 '18 at 22:19
Hmm, I could have sworn we had a question for the more specific case of "negative times positive", but the search function is uncooperative. — hmakholm left over Monica, Dec 22 '18 at 22:33

score 2 · Answer 1 · answered Dec 22 '18 at 23:02

2

Repeated addition of $4$ for $-3$ times means repeated subtraction of $4$ thrice.

answered Dec 22 '18 at 23:02

Martund

14,826
2
14
33

score 1 · Answer 2 · edited Jan 27 '19 at 00:01

As noted elsewhere, you could consider the last example as a case of multiple subtraction.

Another approach is a graphical one. Imagine that multplying by a positive number stretches the number line. Multiplying by a negative number rotates the number line 180 degrees in addition to the stretch.

So $4 \times (-3)$ would see you start on $-3$ and stretch the number line by a factor if 4 so you would end up at $-12$. $(-3)×4$ would have you start at $4$, undergo the stretch, then rotate the number line to end up at $-12$.

This geometric interpretation may seem forced, and unnecessarily complicated, but you'll be truly thankful for it when you see how immensely it simplifies working with Complex numbers. Then, you will be rotating the number line through any amount of arc, and plotting numbers on the plane, and not only the number line.

How to make sense of multiplication in the case of negative times positive?

2 Answers2