I borrowed this from http://fivetriangles.blogspot.com/
184. *** Overlapping sectors
In the diagram is rectangle ABCD with height 10 cm. An arc with centre at point B is drawn from point A to side BC. An arc with centre at point C is drawn from point D to side BC. Given that the shaded (coloured) regions ⓐ and ⓑ have equal area, determine the length of BC.
The extension is “and where do the quarter circles cross?”
It could be a two line calculation !
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.