Why're two negs mult pos?

If a and b are two negative integers they can be written as two positive integers c and d such that

[itex] ab = (-1)c(-1)d = (-1)(-1)cd = 1cd = cd [/tex] where a and c are equal in magnitude, as are b and d.

edit: itex is prettier.

 

Perhaps this may help

 

read the page he sent you.

 

It contains multiple examples to make it 'understandable' as well as some mathematical details I believe.

 

-1x=opposite of x. -1(-x)=opposite of a negative, a positive.

 

The integers are designed so that they form what is known as a ring. A ring must have (among other things) an additive identity and an additive inverse. The additive identity is the number we call 0, which has the property that 0 + x = x + 0 = x. Next we define the additive inverse of 1 as that element which when added to 1, gives the sum 0, or if x + 1 = 0, then x is the additive inverse of 1. We use the symbol "-1" to represent this number. Next we make use of the distributive property and the definition 0 = 1 + (-1) to write
0 = 0*(-1) = {1 + (-1)}*(-1) = 1*(-1) + (-1)*(-1)
Since 1 is the multiplicative identity, we know that 1*x = x*1 = x, for all x, and hence, 1*(-1) = -1. So we have 0 = -1 + (-1)*(-1). But we know that 0 = -1 + 1, therefore, from the uniqueness of addition (-1)*(-1) = 1.

Note : The operation of subtraction is merely a shorthand for adding a negative number.

 

Or the other way around from the page is to accept the field axioms, then prove yourself that (-1)(-1)=1 using them. You just need to use more precise terms of what a negative refers to and what rules multiplication follows.

 

We debated this a while back on this thread https://www.physicsforums.com/showthread.php?t=82997.

My approch was to use English grammer exmaples to explain why -- = +. Basicly, any negative statment such as "not" or "didnt" counts as a -1.

[Quote = eNathan]Well that kinda gets into the very logic of what a negative is. Think of it as logic "not" operations. Think about using these in sentences and you will get the idea.

-1*-1=1 ; No + No = Yes
1*1=1 ; Yes + Yes = Yes
-1*1=-1 ; No + Yes = No
1*-1=-1 ; Yes + No = No

For instance, if I said...
I did not not go to the store. That really means, you DID go to the store. -1 * -1 = 1 ; No + No = Yes ... and so forth.

You kinda see how there is a relationship between mathematics and grammer? lol Im not sure if this is an official way to present it, but I just though it up a while back.[/quote]

 

You've almost got the geometric picture, methinks. It's a nice one, and is important to know when you deal with the complex numbers.

In the real case, multiplication by a positive means you leave the direction unchanged, and multiplication by a negative means that you flip the direction. So, if you have a negative, and multiply by another negative, the result is a positive.

In the complex case, numbers can be seen as having a magnitude and a direction in the complex plane. Multiplying by a complex number that lies at an angle θ means rotation by θ. (Angles are measured counterclockwise from the positive x axis)

 

However...

... multiplication is nothing more than a shorthand for repeated additions and in that light, there's no way to justify -1 * -1 = 1. Despite the convoluted logic the conventional explanation for multiplying two negative numbers amounts to this

for negatives
-1^2 = -1 + 2
-2^2 = -2 + 6
-3^2 = -3 + 12
-4^2 = -4 + 20

whereas for positives
1^2 = 1 + 0
2^2 = 2 + 2
3^2 = 3 + 6
4^2 = 4 + 12

So we're supposed to buy the explanation that, through some magical process, the interval between a negative and it's square is always larger then the interval between the positive of the same number and it's square?

Doesn't smell right.

 

In the sentence examples you stated, the logic follows, but your logic table could also equal this:

-1*-1=1 ; No + No = Emphatic No
1*1=1 ; Yes + Yes = Emphatic Yes
-1*1=-1 ; No + Yes = Terminating Yes
1*-1=-1 ; Yes + No = Terminating No

 

Well, what is the better question?? We're waiting!

 

That loses meaning when one looks beyond integers. What does it mean to repeat addition 1/2 times, or Pi times ? i times ?
If you just want to stay within the integers, you first have to define what a negative integer is. In that case, it is the number that when added to the corresponding positive integer, returns 0 (called the additive inverse). Multiplication defined your way makes sense for positive multiples. A negative multiple would have to be translated to a positive multiple multiplied by -1, in which you would apply your repeated addition definition, then have the negative sign applied. A negative multiplied by a negative gives us the following problem then: (-a)*(-b) = (-1)*(-1)*a*b, so we now have the problem of figuring out (-1)*(-1). Now, we know from our definition that 1 + (-1) = 0, so we have (-1) + (-1)*(-1) = 0. It is easy to prove that each integer has only one additive inverse (uniqueness). Thus, (-1)*(-1) must be the additive inverse of -1, which is 1.

 

That's because you are not following the rules!

It would be like me saying,

1 + 1 = 2 + 0

but

2 + 2 = 5 - 1

And then declaring that I have shown something smells fishy.

In reality, however, you have established a false opposition. It is not as if
-1^2 = -1 + 2
in contrast to
1^2 = 1 + 0

To prove it, I could just as easily say:
-1^2 = 1 + 0
and
1^2 = -1 + 2

The fact is, -1^2 = 1^2

Why? Because multiplication by -1 is like 180 degree rotation (flipping) -- note that 90 degree rotation would be multiplication by i. Someone has already pointed us in this direction.

 

For k>0
neg interval = (-k)² - (-k) = k² + k
pos interval = k² - k

So, neg interval = pos interval + 2k, and since k>0, neg interval > pos interval.

See ? No magic ! No bad smell !

 

I feel the need to comment that -1^2 = -1. Exponents before multiplication.
:tongue2: (Of course, I know both of you meant (-1)^2)

 

a) False
b) Correct, your assertion in a) stinks.