Why does a negative number times a negative number equal a positive number?
Why does a negative number times a positive number equal a negative number?
Anyone? Anyone?
Bueller?
Well, there’s lots of ways to try to explain it. The short answer is it’s the only thing that makes sense; doing it an other way breaks some other fundamental property of multiplication.
For somewhat longer answers, Negative x Negative = Positive, Why Does a Negative Times a Negative Equal a Positive?
There are different ways to describe or justify this, I like thinking of it in a geometric way.
If you think of the number line, multiplication by a positive number basically stretches (or compresses) the number line by that factor. Multiplication by -1 flips the number line at the point zero. If you do that twice, you’re back where you started, so (-1)*(-1) = 1.
And there’s actually more to it than that; it’s a little hard to describe without a picture, but I’ll try.
If you’re familiar with the complex numbers (a + ib where a and b are real and i is sqrt(-1)), then we don’t have just a real number line anymore, we’ve got a complex plane. Just like before when I said multiplication stretches (or compresses) the real line, and flips it if the number is negative, I can also describe what multiplication does to the complex plane. It still stretches/compresses the plane, but it also rotates the plane, instead of just flipping it. Multiplying by -1 rotates the plane by 180 degrees, which is the same as flipping all the lines in the plane at the point zero (well, all the lines that go through zero, anyway). So that “flip” in the real number line before was actually a rotation in the complex plane.
I don’t know if this was the kind of answer you were really looking for, let me know if it wasn’t, and I really don’t know if that will make much sense without a picture, if I could draw a picture it’d be much easier to see how it works. I’ll see if I can find a picture describing complex multiplication.
Not official, but these work for me:
Assume a negative times a positive is a positive.
Thus 1 x -1 = 1
But 1 x 1 = 1
Therefore 1 x 1 = 1 x -1
Divide both sides by 1 and you get 1 = -1
This is an obvious contradiction.
Therefore, a negative times a positive is not a positive.
Since the only other option is for it to be a negative, it must be a negative.
Same thing. (This is how I explained it to my daughter.)
Assume a negative times a negative is a negative.
Thus -1 x -1 = -1
1 x -1 = -1
Therefore -1 x -1 = 1 x -1
Divide by -1 and you get -1 = 1
This is impossible, so a negative times a negative must be positive.
There’s a picture describing the geometry of complex multiplication at the bottom of this page:
But if 1 * -1 = 1, wouldn’t 1 * 1 = -1?
and therefore 1 * 1 would not equal 1 * -1.
Consider this (basically what RealityChuck said:
Let A = -1[sup]2[/sup]
Thus A - 1 = -1[sup]2[/sup] -1
Thus (by distributive property) A - 1 = -1 x (-1 + 1)
Thus A - 1 = 0
Thus A = 1
So, -1[sup]2[/sup] = 1
QED
I’m assuming by -1[sup]2[/sup] you mean (-1)[sup]2[/sup]…
But in a universe where -1 * 1 = 1, distributing
A - 1 = -1[sup]2[/sup] - 1 would make
A - 1 = -1 x (-1 - 1) rather than
A - 1 = -1 x (-1 + 1)
because -1 / -1 would give -1, just like -1[sup]2[/sup]/-1 gives -1.
jbird3000 said …
I haven’t followed the links, so maybe it’s mentioned there, but the system you describe isn’t one that would ‘work well’. To have an arithmetic operation ‘work well’, as I put it, you need to have an indentity, defined as some number a such that a *x* = *x* (where is any operation you choose to define).
So for addition ‘0’ is the identity since any number + 0 = that number.
For multiplication, ‘1’ is the identity and thus any number * 1 = that number.
I believe this is axiomatic, so you could have a system defined as above, only you can’t expect it to behave like familiar arithmetic (it may even be that the distributive property wouldn’t hold, but I’m not completely sure of that at this late hour.)
panama jack
Preview … submit … they’re so close together.
That should say, a system as you described above (i.e. in your alternate universe)
To think of it without algebraic proofs, think of this: A negative sign works similar to the word “Not” in the english language. If you -2 groups of -4 items, you have 8 items. Similarly, if I say “I am not not excited to be here”, it means “I am excited to be here.” The two 'not’s cancel each other out…same with the - sign. You can think of a negative as owing someone money. If I have -10 dollars, it means I owe someone 10 dollars. Now, suppose I owe -2 people 10 dollars. (meaning, 2 people owe ME ten dollars.) Then that is mathmatecally equivalent to saying (-2 people) x (-10 dollars) = 20 dollars for me. Hope this helps.
Jman
I like to think of it etymologically. What does “times” mean? Seven times eight means 8+8+8+8+8+8+8, right? One times eight means just 8.
So, what would negative one times eight mean? What could it possibly mean? Has to be negative eight, right? Negative one times eight plus seven times eight would have to be six times eight.
So, a negative times a positive is a negative.
Oh, it’d work fine to say that 11=-1, and -11=1, and it’d even look a lot like the arithmatic we have now. All you’ve done is switch the meaning of the symbols “-1” and “1”. In this new arithmatic, the symbol “-” means that a number is positive, and the lack of that symbol means that it’s negative.