Elementary maths gyan!

Digging more things from school maths, I thought to write some information about number system.

If you consider any number system - Natural numbers, Integers, rational etc., the basis of all of them is a set of axioms or laws that we have assumed to be true. These law, actually, help us carry out operations such as addition, multiplication etc. Anything, that we want to prove should be from these laws. For example, consider these laws -  commutative, associative and distributive laws - it is must be that we need to use these definitions to prove anything in any number system.

Further, if we have any new definition, then it must as well preserve these laws. Coming to the point of this post:

Why is (-1)(-1) = 1?

Yes, it is follow -up post on my previous post. Can we find any proof for it? No, we can only convince ourselves that it is true by showing few examples that otherwise would go haywire if it were not true.

History has it, even the great Euler tried arguing why this equation must be true - but, alas it was unconvincing. The reason is simple - (-1)(-1) = 1 is, actually, a definition, rather than a statement that we wanted to prove.

Suppose, i we had defined : (-1)(-1) = -1, then consider the distributive law : 

a(b+c) = ab + ac.

Now let us substitute:

a = -1

b = 1

c = -1

Then RHS = (-1)(1-1) = 0.

Whereas, LHS = (-1)(1) + (-1)(-1) = -1 -1 = -2.

But if we set (-1)(-1) = -1, then everything just sits properly - no crazy things happen. It took mathematicians a very long time to realise the “rule of signs” cannot be proved and hence, they were created by us to preserve fundamental laws.

source: what is maths