Behold Modular Forms the Fifth Fundamental Operation

There are five fundamental operations in mathematics,” the German mathematician Martin Eichler supposedly said. “Addition, subtraction, multiplication, division and modular forms.” Part of the joke, of course, is that one of those is not like the others. Modular forms are much more complicated and enigmatic functions, and students don’t typically encounter them until graduate school. But “there are probably fewer areas of math where they don’t have applications than where they do,” said Don Zagier, a mathematician at the Max Planck Institute for Mathematics in Bonn, Germany. Every week, new papers extend their reach into number theory, geometry, combinatorics, topology, cryptography and even string theory.

Infinite Symmetries

To understand a modular form, it helps Phone Number List to first think about more familiar symmetries. In general, a shape is said to have symmetry when there is some transformation that leaves it the same.A function can also exhibit symmetries. Consider the parabola defined by the equation f ( x ) = x 2 . It satisfies one symmetry: It can be reflected over the y-axis. For instance, f ( 3 ) = f ( − 3 ) = 9 . More generally, if you shift any input x to − x , then x 2 outputs the same value.

The Complex Universe

Functions can only do so much Database USA when they’re defined in terms. Of the real numbers — values that can be expressed as a conventional decimal. As a result, mathematicians often turn to the complex numbers. Which can be thought of as pairs of real numbers. Any complex number is described in terms of two values — a “real” component and an “imaginary” one, which is a real number multiplied by the square root of −1 (which mathematicians write as i ). Any complex number can therefore be represented as a point in a two-dimensional plane.

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