I do not understand the reason that, when 2 negative numbers are multiplied, the result is a positive number.
Any thoughts?
Thanks.
This was discussed back in March, 2004. It seems no one could come up with a concrete example of why 2 negative numbers equal a positive when multiplied.
It’s pretty much a simple consequence of the way addition and multiplication are defined. Basically, you want the properties we’re used to:
- a + b = b + a
- a + (b + c) = (a + b) + c
- a + 0 = a
- For every number a, there’s a number (-a) such that a + (-a) = 0
- a * b = b * a
- a * (b * c) = (a * b) * c
- a * (b + c) = a * b + a * c
One of the first things you can deduce from this is that (-a) is unique; that is, if a + (-a) = 0 and a + b = 0, (-a) = b. More generally, if a + b = a + c, then b = c. It’s also easy to show that a * 0 = 0, which is a useful fact.
Consider a * a + a * (-a). By 7), this is equal to a * (a + (-a)), which is equal to 0. Now consider (-a) * (-a) + (-a) * a. By the same properties, this is equal to -a * ((-a) + a), which is also equal to 0.
So we have now that a * a + a * (-a) = (-a) * (-a) + (-a) * a. By 5) above, we can write this as a * a + a * (-a) = (-a) * (-a) + a * (-a), and immediately conclude that a * a = (-a) * (-a). Do you follow?
Note that we don’t actually need 5), but it makes the proof much simpler, so I’ve left it in.
On preview: There’s no need for a concrete example, because it’s a theorem that follows from properties that everyone agrees integer arithmetic should have.
I always though of it as a negative number is “opposite” a positive. The opposite of an opposite is the normal. Thus, a negative times a negative is a positive.
ultrafilter: How would a=0 affect your examples above?
Not at all, because the proof doesn’t require that a not be 0. I probably should have proved that a * b = (-a) * (-b), but the idea’s exactly the same, so there’s no need to walk through that proof.
Thanks. I guess I just had the question pop into my head when I saw the word unique in the description. You’re right, of course. Affects it not at all.
An heuristic, geometric answer. Can agree that multiplying a number by a negative number reverses the direction in which the answer is measured on the number line? If so then if I start with a negative number measured to the left of zero on the number line and multiply it by a negative number I reverse the direction and the answer is measured in the positive direction on the number line.
wolf_meister writes:
> It seems no one could come up with a concrete example of why 2 negative
> numbers equal a positive when multiplied.
I thought there were a number of good examples in that thread. The only ones that don’t really work are those that claim that in language, a double negative is a positive. Well, sometimes it is and sometimes it isn’t. It depends on the language, the dialect, and the particular sentence. That isn’t a good example then of the product of two negatives being a positive, so please don’t anyone give such examples, since we’ll pick them apart immediately.
A better concrete example is this: Suppose you had a bunch of credit and debit slips. They say something like:
A owes me $10.
B owes me $15.
I owe C $5.
D owes me $20.
I owe E $25.
So the amount of money you have (for these slips) is 10 + 15 + (-5) + 20 + (-25) = 15. Suppose every slip is doubled in value. This is multiplying by 2. Then the amount of money you would have is (2 * 10) + (2 * 15) + (2 * (-5)) + (2 * 20) + (2 * (-25)) = 20 + 30 + (-10) + 40 + (-50) = 30. Suppose every slip is doubled in value and inverted as to who owes who money. This is multiplying by -2. The amount of money you would have is ((-2) * 10) + ((-2) * 15) + ((-2) * (-5)) +((-2) * 20) + ((-2) * (-25)) = (-20) + (-30) + 10 + (-40) + 50 = -30.
Fully explains negative numbers and mathematical operations involving same.
Think of the real numbers as they are laid out on a real line:
<-------------------------------0------------------------------------>
Signed numbers are viewable as very simple vectors, where said vectors can have only two directions: +/Right; -/Left.
In this setting, real numbers have both a magnitude component (absolute distance from 0), and a direction (+,-).
We can then understand the simple product as scaling and flipping:
ab, a,b>0 is simply stretching a vector of length b into a vector of length ab.
ba, a,b>0 is simply stretching a vector of length a into a vector of length ba.
In simple product notation, -1 can be viewed as a “flip”, as a simple reversing of direction.
Then ab = ab when a,b>0; “Single Stretch”
ab = (-1)|a|b when a<0 and b>0; “Single Stretch, Single Flip”
ab = (-1)(-1)|a|*|b|; “Single Stretch, Double(No) Flip”
Think of a as being a when a>0 (|a|=a) and as (-1)*|a| when a<0.
The best explanation-by-example I can think of:
Somebody gives you a $20 check: +20
Somebody gives you a $20 debt: -20
Somebody gives you two $20 checks: (+2) * (+20) = +40
Somebody gives you two $20 debts: (+2) * (-20) = -40
Somebody takes two $20 checks away from you: (-2) * (+20) = -40
Somebody takes two $20 debts away from you: (-2) * (-20) = +40
I pay £50/month electricity bills.
One month from now, I will owe £50:
1 x -50 = -50
Two months from now, I will owe £100:
2 x -50 = -100
A month ago, I had £50 more than now:
-1 x -50 = 50
Assume multiplying two negative numbers gives a negative result.
If so
-1 x -1 = -1
But -1 x 1 = -1
So it follows that -1 x -1 = -1 x 1
Remove the -1 from both sides of the equation:
-1 = 1
Obviously, this is untrue. Therefore the original assumption is invalid and two negatives multiplied cannot give a negative number. They must give a positive number.
That is brilliant, possibly the best illustration I’ve seen yet of the concept.
This is getting technical, but to really conclude that from your argument, you’d need to deal with arbitrary number c rather than a specific one. Otherwise, it’s a nice argument.
You can read an explanation of this on Google Book Search in “Where Mathematics Comes From” by Lakoff & Nunez. It may not make sense unless you at least read the whole chapter. (I managed to read the whole book. I like pain)