I’m currently visualizing negative numbers as holes in the ground. So, for example, 3 holes x 3 holes = 9 holes (‘boxing the line of 3’)= -9. That is correct within the metaphor but wrong in math theory.
What is a better metaphor for negative numbers that would agree with -3 x -3 = 9?
Multiplying a “hole in the ground” by a negative 1 is, in your metaphor. the equivalent of replacing the hole with the bucket of sand that you originally dug out of the hole.
Negative times negative equals positive works well in accountancy and bookkeeping. For example, if you have 10 debts each of $3, and you reduce those debts by 3, you increase your net worth by $9, i.e., (-3) times (-$3) equals (+$9).
Your analogy is wrong. -3 x -3 would be three holes in the ground times -3 (whatever that is). What you’ve done is illustrate -3 x 3.
You’re almost there. " Classroom Resources - National Council of Teachers of Mathematics " is the best simple explanation I can find. You are thinking of a the whole as a “negative space”, but it isn’t really, and that’s confusing the issue. For your metaphor to work, you’d need to think of it as negative wood or something.
Thik of negative numbers as debt. Let us say you have a dedbt of 1 dollar a month. That is, you lose a dollar every month. If someone arranges to pay for your debt for one month, you have financially gained a dollar. Even if you don’t see it, you have one extra dollar on your balance sheet.
It is possible to create alternative mathemathical systems where you would be right. However, doing so creates inconsistencies that render it largely useless. The current system may seem awkward or counterintuitive, but it does work and permits some extremely elegant solutions. I won’t go further into this because I’m not that good.
Negative is, in mathematical terms, a negation. Here, you’re ending up negating the negative itself. That is, as you multiple -3 and -5 (I’m changing your numbers, but it’s the same), you get this…
-3 x -5 = X
(-)(-)(3)(5) = X
(-)(-)(15) = X
(-)-15 = X
15 = X
Now, if that’s confusing, just bear with me, because I’m going to break it down.
See, you can write all negative numbers as (-1)*(the number)! This can be quite handy in proofs, factoring, and calculus.
So, we have
-1 * -1 * 3 * 5 = X
Does this seem easier, now? You can see that the -1’s will reverse themselves (thus making a positive 1)
3 = 3 piles of dirt
-3 = 3 holes in the ground
-3 x 1 = perform the action of digging three holes once
-3 x -1 = perform the ***opposite ***action of digging three holes once (dump three piles of dirt on the ground)
-3 x -3 = perform the opposite action of digging three holes three times(dump nine piles of dirt on the ground)
The way I was first introduced to it was in terms of patterns
3 x 4 = 12
3 x 3 = 9
3 x 2 = 6
3 x 1 = 3
3 x 0 = 0
3 x -1 = -3 = -3 x 1
3 x -2 = -6 = -3 x 2
Now we can apply the same logic to -3, merely following the pattern we have observed.
-3 x 2 = -6
-3 x 1 = -3
-3 x 0 = 0
So, by extension…
-3 x -1 = 3
-3 x -2 = 6
-3 x -3 = 9
These are logically consistent patterns, and helped me understand a lot about what was an abstract idea for a 6-year-old. My maths teacher was quite strong on getting kids to appreciate abstraction in terms of patterns, something I’ve never forgotten.
See if this helps (just another way of stating some of what’s above):
Ground level is zero, positive numbers are piles of dirt, negative numbers are holes in the ground.
Multiplying by 1 doesn’t change anything, 1 x A = A, and the value of A remains positive or negative, whichever it was before multiplying by one. Multiplying by -1 doesn’t change the magnitude of A, but reverses its relation to ground level. If A was a pile, it’s now a hole; if A was a hole, it’s now a pile.
Since -3 = 3 x -1, multiplying by -3 yields three times the magnitude AND reversal of piles to holes, or holes to piles. Thus -3 x -3 is three holes both reversed to piles and increased to 9, equaling 9 piles.
There are certain properties that we would really like our arithmetic to have. One of them is that whenever you have a number x, there’s another number -x such that x + -x = 0. Since -x is a number as well, there must be some number -(-x) such that -x + -(-x) = 0. We can then say that x + -x = -x + -(-x), and conclude that x = -(-x).
One of the other properties we’d like our arithmetic to have is that (x + y)z = xz + yz. If that’s the case, then (-x)y + (-x)(-y) = -x(y + -y). We know that y + -y = 0 and x0 = 0, so it must be the case that (-x)y + (-x)(-y) = 0. However, (-x)y + xy = (-x + x)y, and by the same logic this is also equal to zero. So by the same reasoning as above, we can conclude that (-x)(-y) = xy.
I think the most elementary way to look at this is that multiplying by -1 rotates you by 180º on the real line. So (-1)*(-1) equals 360º and gets you back where you started (and so on and so forth). Why that is so boils down to because it works.
Pedro’s explanation (which is basically Gary T’s as well) is very nice. In fact, continuing along these lines helps explain complex numbers just as easily.
First, take a photograph of me. (This is much easier to do these days because they’ve been building cameras sturdier since my youth. Ah, the wonders of modern materials. I can even have mirrors at my place nowadays.)
Now, take the photo and hag it upside-down. It looks wrong right? Well, of course, it’s a photo of the ugliest guy on the boards, but I mean that it looks wrong because you are seeing my face upside-down.
Now, without changing anything about the photo, look at it again, while hanging upside-down.
That is your -1 times -1!
The way I explain it is in movement. Negative means moving towards the left on a number line, positive means moving right on a number line. When you multiply, the words change meaning: going in the opposite direction (negative) or the same direction (positive).
So, -3 X 3 means going left 3 more steps of -3, or -9. -3 X -3 means going right 3 increments of +3. While this means you literally end up at +6 on the number line, the number of steps is actually 9 if you count it out.
So, the metaphor I could make is a car traveling on a road. If the first number is negative, it is traveling west. If the first number is positive, it is traveling east. If the multiplier is positive, you use “D” on the transmission. If the multiplier is negative, use “R,” and the answer would be on the car’s mileage, not the ending point.
This is the example I use:
“Stand on a flight of stairs, facing downstairs. Now, walk backwards three steps. Going downstairs (to a negative height) in the negative direction (walking backwards) is going upstairs.”
Theory shmeery. Even a smart math guy like Blaise Pascal didn’t think negative numbers exist.
If a talented intellect such as Pascal couldn’t understand it, we should just banish negative numbers from school curriculums.
Seriously, the reason there are many analogies to negative numbers is that it is an invention of the human mind.
It seems like the best analogy is the one based on direction: negative numbers are 180 degrees of positive numbers. I like this because you can re-use this type of conceptualization for imaginary numbers and for vectors. You don’t get as much mileage out of analogies using money (debt) and dirt (holes).
If you owe ten people $3, you have a total debt of $30. A debt is a liability, a negative asset. Your net worth is reduced by the amount you owe. Now if you borrow another #2 from each of those ten people, you now have doubled yoru debt, 10*-$3=-$30; 2(10*-$3)=2(-30) or -60. But if you pay back those debts, you reduce your liability and bring yoru net worth back up. Paying a debt is equivalent to multiplying it by -1; you’re cancelling it out by paying it off. -10(-$3)=$30; you’ve increased your net worth by $30 by paying off the debts.
Bravo!!!
Assume a negative number times a negative is a negative.
Thus, -1 X -1 = -1
Now we know that -1 x 1 = -1
Then it follows -1 x -1 = -1 x 1
Divide by -1: -1/-1 x -1 = -1/-1 x 1
Since a/a = 1: 1 x -1 = 1 x 1
and thus -1 = 1
This is an impossibility, so a negative times a negative must be a positive.
Nice. I haven’t seen a reductio ad absurdum proof of this before.
Nice, but wouldn’t making the assumption that muliplying a negative by a negative makes a negative also suggest that dividing a negative by a negative makes a negative, too? (if someone thinks multiplying a negative by a negative is a negative, why would they think dividing a negative by a negative is a positive? Or, more simply, division is multiplication by its reciprocal. The reciprocal of -1 is -1, so -1/-1 equals -1*-1, which we already decided is -1.)
So, if we assume -1 x -1 = -1
Then we should assume -1/-1 = -1
and our final equation becomes:
-1 x -1 = -1 x 1
and with our initial assumption that a negative times a negative equals a negative:
-1 = -1