prove practically and theortically that -1x-1=+1
Oh but it is! By someone proving it to be correct (or incorrect as in this case), we can all take the solution for granted. This is a fundamental characteristic of Mathematics and Science.Lets not prove this. It is not of any use
Yes I could have been clearer, I am merely demonstrating that if we assume -1*-1 = -1, then the distributive property doesn't work, so I have arbitrarily chosen:Mr Dave plz tell
(-1)(1 + -1) = (-1)(1) + (-1)(-1)
why u have taken "1" as positive (which is bold above).
But the question is to prove that -1*-1 = +1. In your above assessment you are taking that -1*-1 = +1 is a given.It's mostly by definition that 1 * Anything equals itself, ie... the Identity Principle
it's also by definition that -1 * Anything changes the sign, equals the additive inverse.
-1 * -1 = 1 is by definition.
No amount of manipulation is going to get you anywhere without evaluating -1 * -1 at some point which is what you are trying to prove. Subtracting 1 from both sides will give you:
(-1 * -1) - 1 = 0, even if you try to prove the assumption that -1 * -1 <> 1, you still need to eventually evaluate -1 * something.... without doing the evaluation you will only go in loops.
Once you have done the evaluation you have resorted to the definition...
Dave, you're assuming that -1 * -1 = -1. If you arrive at a contradiction (which you did, but I am omitting), all you have established is that your assumption turned out not to be true. IOW, that -1 * -1 != -1. That in no way implies that -1 * -1 = 1.The only way I can think of proving this off hand is by disproving it:
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But for arguements sake lets say: -1*-1 = -1 (Which we know is wrong)
I understand what you are saying, however I think we are approaching the problem from a different angle. I have taken the liberty to assume that we know the multiplicative identity of the number 1 for x = 1, namely:Dave, you're assuming that -1 * -1 = -1. If you arrive at a contradiction (which you did, but I am omitting), all you have established is that your assumption turned out not to be true. IOW, that -1 * -1 != -1. That in no way implies that -1 * -1 = 1.
For a proof by contradiction to work you have to start with an assumption that, if proved false, incontrovertably led to the statement you really wanted to prove.
In this case, you could start by assuming that -1 * -1 != 1. If you arrive at a conclusion that shows this assumption could not possibly be true, then you will have established that -1 * -1 = 1.
I concur with the proof you offered for the case -1 x -1 = 1.We know -1*-1 = +1 (I have used the number "1" here but to be fair this could be any numbers and hence we can extrapolate this to be a general case for all negative numbers).
But for arguements sake lets say: -1*-1 = -1 (Which we know is wrong)
Considering the distributive property:
(-1)(1 + -1) = (-1)(1) + (-1)(-1)
(-1)(0) = -1 + -1
0 = -2
This is wrong, hence we can say that a negative multiplied by a negative must be a positive.
The phrase "by definition" indeed sounds nice, but unfortunately doesn't hold but for a very limited amount of cases. Here is the proof provided by a friend of mine, who studies Mathematics in the University of Athens, Greece:-1 * -1 = 1 is by definition.
0 = -2. Yes it's wrong, however that too needs to be proved. It's not as obvious as you might think. (In fact if you do not assume the axioms of order it cannot be proved that (-1) + (-1) does not equal 0. What's more shocking is that you cannot prove that 1 does not equal -1 either!!! let alone define (-1) + (-1) as -2). What I am saying here is that you have merely substituted the statement we want to prove with another, which may be obvious but also needs to be proved/disproved.Originally Posted by (-1)(0) = -1 + -1
0 = -2
From what I have said above you realize of course that you have not proved such a thing...hence we can say that a negative multiplied by a negative must be a positive.
True but not by definition.. it's one of the axioms of the real numbers. There exists a real number, which we denote 1, not equal to 0, for which it holds 1x = x for every real x.It's mostly by definition that 1 * Anything equals itself, ie... the Identity Principle
There is no such definition. However it can be proved that -1x = -x for every real x. But thats the problem.. it needs to be proved (it's easy of course).it's also by definition that -1 * Anything changes the sign, equals the additive inverse
Why do you insult mathematics? It is not by definition-1 * -1 = 1 is by definition
That's even worse..Divide both sides by -1
Prove it if you can! As I said above this is not that obvious (correct still). To explain things further.. if you accept only the algebraic axioms i.e the: associative, commutative laws, the distributive law, the identity axioms and the inverse axioms (fairly simple and self evident properties of the real number system you must agree) there is no way you can prove that.. (for example you can prove that 1 + 1 does not equal 1 but you can't prove that 1 + 1 does not equal 0!!!).there is no multiplicative combination of the numbers +/-1 that will yield an answer other than +/-1.
No they did not.. in fact you only have to accept 9 simple and self-evident axioms (along with the operations of addition multiplication and the order relation) to derive everything else (a 10th axiom is required if one wants to study analysis). Emm.. everything else you say is.. I dont know what it is.. but surely it's not mathematics.TO CUT A LONG STORY SHORT,lets just say that mathematicians just DEFINED these operations and their results
We agree that I can say -1 + -1 = 0 is acceptable, because the "extremely rigourous" proof that meets your standards provided by mavromap makes the assumption 1 + (-1) = 0. So this one isn't an issue is it.0 = -2. Yes it's wrong, however that too needs to be proved. It's not as obvious as you might think. (In fact if you do not assume the axioms of order it cannot be proved that (-1) + (-1) does not equal 0. What's more shocking is that you cannot prove that 1 does not equal -1 either!!! let alone define (-1) + (-1) as -2). What I am saying here is that you have merely substituted the statement we want to prove with another, which may be obvious but also needs to be proved/disproved.
Again I state that I have taken the liberty that we know the multiplicative identity of the number 1 for x = 1, namely: 1 x 1 = 1, something I state. There is a proof for this elsewhere - again I state, engineers don't reinvent the wheel.Prove it if you can! As I said above this is not that obvious (correct still). To explain things further.. if you accept only the algebraic axioms i.e the: associative, commutative laws, the distributive law, the identity axioms and the inverse axioms (fairly simple and self evident properties of the real number system you must agree) there is no way you can prove that.. (for example you can prove that 1 + 1 does not equal 1 but you can't prove that 1 + 1 does not equal 0!!!).
You are probably right, what is good for the engineer is not good for the mathematician; and probably vice versa - thank goodness the job of making the real world work is left to the engineers.The proof mavromap gave is extremely rigorous .
What I tried to point out is that none of the arguments used with the exception of the proofs mark44 and mavromap gave (btw who gave you that proof I wonder ) is rigourous i.e not good enough for mathematicians (maybe for engineers more practical than theoritical as you are..).
Search google: "axioms of real numbers" to clarify things..
by Steve Arar
by Gary Elinoff