More on, x^2 + y^2 =1

Excerpted from the New York Times, Nov. 25, 2014

The Lives of Alexander Grothendieck, a Mathematical Visionary

“When we write “x2 + y2 = 1,” we wish into existence a perfect circle. Indeed, each solution of this equation is nothing but a pair of coordinates, x and y, of a point of the unit circle on a plane.”

“Right away, one encounters a problem. The above equation gives rise to a circle only if we consider the solutions in the domain of real numbers. But there are many other domains, such as the complex numbers (which involve an imaginary number, the square root of minus 1).

One can show that the solutions of the same equation in complex numbers are points of an entirely different space; namely, a plane with one point removed. For another domain, the space of solutions could be a family of circles of different sizes: Visualize a living and breathing circle evolving in time.”

Alexander Grothendieck who died, Nov. 13, 2014, was a leading contemporary mathematician.


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