Sum Equal to Product

When I was exploring the tangent identity a couple weeks ago, I thought that one of the more peculiar aspects of it was that the sum of three numbers was also equal to the product of those three numbers. After all, among the integers this is only true of (0, 0, 0), (1, 2, 3), and (-1, -2, -3).

But as it turns out, this is so common among the real numbers that given any two real numbers x and y, there is a third number z such that x + y + z = xyz as long as xy ≠ 1. This is fairly easy to show with some algebra.

What does a plot of all these points (x, y, z) look like? Let’s turn to WolframAlpha to give us a 3d graph:

I expected the above to be a little more interesting until I realized the boundaries were too narrow. Here’s a better look:

Interesting!

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