If the same thing which is done to top and bottom is multiply or divide, it is ok. But if this thing is add or subtract, it is not. IOW, 4/14 is certainly not the same as 2/12 (subtracting 2 from top and bottom). How do we know that it is the same as 2/7 (dividing top and bottom by 2)?
Me too. But I remember that it bothered me, because how ONE piece of pie can be EQUAL TWO pieces of pie? Sure, they weigh the same, but they are different in so many ways...
If we accept that -1 and +1 are additive inverses, and if we want to keep the distributive property of multiplication, then it appears that we don't have a choice but make (-1)×(-1)=+1. Here it goes:
(-1)×(-1)=(1-2)×(-1)=1×(-1)-2×(-1)=(-1)-(-1)-(-1)=(-1)+(+1)+(+1)=+1
QED
What is there to consider? What does make this scenario interesting or non-trivial? It is just a graph of \(x=x(t)\) function with \(x\) and \(t\) axes flipped.
The horizontal coordinate on the graph is projection of the asteroid position on the direction of the major axis.
A geodesic is a line in spacetime, i.e., in four dimensions. To illustrate it in two dimensions, one needs to take a projection.
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