# Circular Images

Author(s):

The three circular images on the homepage all have relevance to the title of the magazine:

1.  The lefthand image is of a diagram showing the Chinese method of finding a square root, essentially the method that many of us learned in school before the advent of calculators.  The image shows the determining of three successive digits giving the square root of a five or six digit number.  Note that the succession of answers one gets - and one can continue into decimals if the answer is not an integer - converge to the square root of the given number.

2.  The middle image is of a piece of the graph of the absolute value of Riemann zeta function over the complex plane.  The Riemann zeta function, originally defined for real numbers s greater than 1 as the converging sum of the reciprocals of the s'th powers of all the integers, is extended by analytic continuation to a function on the entire complex plane except for s = 1.  The Riemann hypothesis, one of the great unsolved problems of our time, is that all non-trivial zeros of zeta function have real part 1/2.

3.  The righthand image is a portrait of Augustin-Louis Cauchy (1789-1857), the man most responsible for putting the calculus on a rigorous foundation.  In particular, Cauchy, in his textbooks written for his classes at the Ecole Polytechnique in Paris, carefully defined the term convergence arithmetically, getting away from the previous notion of using quantities infinitesimally small.

"Circular Images," Convergence (April 2004)