Here is a popular argument: -1 x -1 has to be 1 or -1

If it was equal to -1 then -1 x -1 = -1 x -1 x 1 = -1 x1 and so dividing both sides by -1 we get -1 = 1, which is not a good idea!, hence -1 x -1 = 1

This argument begs so many questions that it is difficult to know where to start.

Here is a much better one, but it does stretch the idea of area a little :

From the diagram (a – 1) x (b – 1) = a x b – a – b + 1

Set a = 0 and b = 0 to get (0 – 1) x ( 0 – 1) = 1, and since 0 – 1 is equal to -1 we get -1 x -1 = 1

This has some connection with evaluating for example 3 x ( 8 – 2) using the distributive law.

The distributive law is a law for a(b + c) and says nothing about a(b – c), but never mind, we go gaily about the common task.

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