Math Question: Why is -1 X -1= 1?

[Sui Generis]

I know that when you multiply a negative and a negative, you get a positive, but why?

And yes, I know absolutly knowthing about math, lol

 

[Abaddon]

oh dear....

 

[The Last Conformist]

I know that when you multiply a negative and a negative, you get a positive, but why?

And yes, I know absolutly knowthing about math, lol

Do you know what you mean by "why"? Are you looking for a proof from first principles, or for an intuitive justification, or what?

 

[Bigfoot3814]

http://forums.civfanatics.com/showthread.php?t=288978

 

[Sui Generis]

Do you know what you mean by "why"? Are you looking for a proof from first principles, or for an intuitive justification, or what?

I don't know, I just want to know the reason for -1 X-1=1

 

[The Last Conformist]

I don't know, I just want to know the reason for -1 X-1=1

Then I expect you are doomed to disappointment.

 

[raketooy]

Because.


....

 

[Luckymoose]

I know that when you multiply a negative and a negative, you get a positive, but why?

And yes, I know absolutly knowthing about math, lol

Because the people teaching you say so. That should be enough of a reason. I remember the first time someone told me about variables and I thought it was a lie.

 

[warpus]

Look at this pattern

3 x -3 = -9
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = 3
-2 x -3 = 6

 

[Evie]

Because it's useful when calculating debts and that sort of things.

Think of it this way. You have 100$ in your wallet You owe some guy one dollar. How much total money do you have.

1 x -1 = -1. You owe a dollar, so you really have 99$

You owe two guys a dollar.

2 x -1 = -2. You owe two dollars, so you really have 98$.

You owe two guys a dollar, but three guys owe you a dollar.

2-3 x -1 = 1. You are OWED a dollar, so you really have 101$

 

[Sui Generis]

I am not denying it is useful, I just want to know "why"

 

[ParadigmShifter]

1x(-1) = -1

I'm sure you would agree.

(-1)x1 = 1 by commutativity
so (1x(-1))x(-1)x1 = 1 substituting for -1 from the the first equation
but, multiplication is associative, so we can rearrange this as
1x1x(-1)x(-1) = 1
but 1x1 = 1
so 1x(-1)x(-1) = 1
but 1 times anything is the original number, so
(-1)x(-1) = 1

QED?

EDIT: Added some brackets

 

[Sui Generis]

1x(-1) = -1

I'm sure you would agree.

(-1)x1 = 1 by commutativity
so (1x(-1))x(-1)x1 = 1 substituting for -1 from the the first equation
but, multiplication is associative, so we can rearrange this as
1x1x(-1)x(-1) = 1
but 1x1 = 1
so 1x(-1)x(-1) = 1
but 1 times anything is the original number, so
(-1)x(-1) = 1

QED?

EDIT: Added some brackets

I think I get what you are saying. Math breaks down if -1 x -1=1 is not right, yes?

 

[ParadigmShifter]

Well, we ASSUME that multiplying whole numbers is associative (i.e. ax(bxc) = (axb)xc and commutative (i.e. axb = bxa).

But good luck finding a counter example...

That's all that the above proof relies on. Oh, and that 1 multiplied by (x) equals (x).

EDIT: I'm sure there's a simpler proof, that's one I just knocked up while typing.

 

[Tomoyo]

Math breaks down if anything that is true isn't true.

What ParadigmShifter did was prove that (-1)(-1) = 1.

 

[Narz]

Because it's useful when calculating debts and that sort of things.

Think of it this way. You have 100$ in your wallet You owe some guy one dollar. How much total money do you have.

1 x -1 = -1. You owe a dollar, so you really have 99$

You owe two guys a dollar.

2 x -1 = -2. You owe two dollars, so you really have 98$.

You owe two guys a dollar, but three guys owe you a dollar.

2-3 x -1 = 1. You are OWED a dollar, so you really have 101$

This is assuming you live in a perfect world where everyone pays up.

 

[ParadigmShifter]

Yikes, I've been drinking, my proof doesn't look right at all now I look at it again. I'll try again

 

[Defiant47]

I think I get what you are saying. Math breaks down if -1 x -1=1 is not right, yes?

Math breaks down if anything that is true isn't true.

What ParadigmShifter did was prove that (-1)(-1) = 1.

No further comment necessary.

 

[cubsfan6506]

Y did no one point out that (-1)(X)(-1)=X not 1

 

[nihilistic]

1x(-1) = -1

I'm sure you would agree.

(-1)x1 = 1 by commutativity
...

Apparently no one noticed that you got it wrong on the first step. The rest is of course, not worth reading. Now for the real proof:

(-1) + 1 = 0 additive inverse
(-1) ((-1) + 1) = (-1) 0 = 0 i'll provide proof of a*0 = 0 in another post if you want
(-1)(-1) + (-1)1 = 0 distribution of * over +
(-1)(-1) + (-1) = 0 multiplicative identity
((-1)(-1) + (-1)) + 1 = 0 + 1 = 1 additive identity
(-1)(-1) + ((-1) + 1) = 1 associativity of +
(-1)(-1) + 0 = 1 additive inverse
(-1)(-1) = 1 additive identity