Multiplication of negative numbers

[sarah944]

Hello everyone

I have trouble understanding the idea of multiplying two negative numbers. Why is product of such multiplication a positive number? Its so confusing.

 

[dextercioby]

It's the double negative of a positive number. So that (-1)*(-3) = - [(-1)*3]= - (-3) = 3. It follows logically from defining negatives as opposites of positives wrt 0. Therefore, the positive is the opposite of the opposite of itself.

 

[Deveno]

the idea of negatives being "opposites" of positives is probably the simplest explanation to understand intutively.

but what is actually happening is this:

numbers actually lie on a plane. the numbers we "normallly" work with, are on a line running across this plane.

when we have access to the plane, multiplication takes on a new meaning: we stretch by the size, and rotate by the angle.

positive numbers have angle "0". so multiplying two positive numbers doesn't rotate at all, we just stretch.

negative numbers have angle "180 (degrees)". so when we multiply by a negative number, we stretch by the size, and do an "about face".

well if our number was already pointing in the "180 degree direction", when we multiply by another negative number, we (stretch and then) rotate another 180 degrees, and winding up pointing in the "360 degree direction" (which is the same as 0 degees, that is, pointing positive).

 

[Dr. Seafood]

^ That is the coolest explanation I've ever read about this topic. Nice!

 

[nhmllr]

Tell me if the following explanation makes sense:

What does it mean to multiply numbers? 2 x 3, for example?
Well, it means that you add 2, three times.

2+2+2 = 6

Alternatively, you add 3, two times.

3+3= 6

So, what about -2 x 3? Well, you'd add -2, three times.

(-2)+(-2)+(-2)=-6

Now, how about -2 x -3?
Here, you're adding -2, negative three times. But what does it mean to add something negative time? That's the same as subtracting it!

-2 x -3 = -(-2)-(-2)-(-2)

And subtracting a negative number is the same as adding a positive number. (Think about adding and subtracting numbers on a number line if you do not believe me.) So now we know

-2 x -3 = -(-2)-(-2)-(-2) = 2+2+2=6

Which should give us our answer. :)

 

[Grizzled]

This is a very good and correct explanation Deveno but I'm afraid that, without saying so :-), you are moving into the complex numbers field. Might be a bit early for a student still struggling with negatives.

 

[Deveno]

This is a very good and correct explanation Deveno but I'm afraid that, without saying so :-), you are moving into the complex numbers field. Might be a bit early for a student still struggling with negatives.

you are correct, i am foreshadowing them. you see, the integers aren't really a world unto themselves. after we've become comfortable with arithmetic, we leverage that knowledge into "solving equations" with unknowns (i.e., algebra).

so we start with stuff like:

if 2 + x = 5, what is x?

and just using one's fingers, you can reason that x must be 3. of course, switching the 2 and the 5 leads to something a bit stranger:

5 + x = 2.

to solve such an equation, we need something "more" than "counting numbers." if we replace addition by multiplication, we have equations like:

2x = 5,

and trying to solve them leads to "fractional numbers (fractions)".

eventually, we might consider equations like:

x² - 2 = 0, for which we need still "another kind of number".

one might wonder if this process of "enlarging" our concept of number goes on forever (at least in terms of doing algebra). and it turns out that, in one sense, there is a "natural stopping point". thinking of "numbers" as points on the plane, has a certain natural "completeness", in that doing all the types of "enlarging" that lead us to this idea, doesn't get us anything "bigger".

now, i would argue that if one is picturing numbers as lying on a line, anyway, one is already thinking of numbers in a geometric fashion. and in this view of numbers, we have two key pieces of information: size (or distance from the "origin"), and direction (positive and negative). it is no accident that thinking of numbers as a pair (size,direction) leads naturally to points in the plane (horizontal, vertical).

the bigger picture (the plane) lends context to the smaller picture (the line). my comments are meant to suggest that "what happens off the line" influences "what happens ON the line". naively, one can "put blinders on" and just see plus, and minus, and use the rule: opposite of opposite is original (negative times negative is positive). but there's more going on, than this.