“Euler’s equation reaches down into the very depths of existence. It brings together mental abstractions having their origins in very different aspects of our lives, reminding us once again that things that connect and bind together are ultimately more important, more valuable, and more beautiful than things that separate."

Magazine
Winter/Spring 2002

The most beautiful
equation in mathematics


by Keith Devlin

Bertrand Russell, the famous English mathematician and philosopher, wrote in his 1918 book Mysticism and Logic:

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.”

Mathematics, the so-called science of patterns, is a way of looking at the world, not only the physical, biological, and sociological world we inhabit, but also the inner world of our minds and thoughts. Mathematics' greatest success has undoubtedly been in the physical domain. Yet, as an entirely human creation, the study of mathematics is ultimately a study of humanity itself. For none of the entities that form the substrate of mathematics exist in the physical world. The numbers, the points, the lines and planes, the surfaces, the geometric figures, the functions, and so forth are pure abstractions that exist only in the mind.

At the supreme level of abstraction where mathematical ideas may be found, seemingly different concepts sometimes turn out to have surprisingly intimate connections. There is, surely, no greater illustration of this than the equation discovered in 1748 by the great Swiss mathematician Leonhard Euler.
Euler’s equation



connects the five most significant and most ubiquitous constants in mathematics: e, the base of the natural logarithms; i, the square root of –1; þ, the ratio of the circumference of a circle to its diameter; 1, the identity for multiplication; and 0, the identity for addition.

The number 1, that most concrete of numbers, is the beginning of counting and the basis of all commerce, engineering, science, and music. The number 0 began life as a mere place holder in computation, a marker for something that is absent, but eventually gained acceptance as a symbol for the ultimate abstraction: nothingness. As 1 is to counting and 0 to arithmetic, þ is to geometry, the measure of that most perfectly symmetrical of shapes, the circle — though like an eager young debutante, þ has a habit of showing up in the most unexpected of places. As for e, to lift her veil you need to plunge into the depths of calculus — humankind’s most successful attempt to grapple with the infinite. And i, that most mysterious square root of –1, surely nothing in mathematics could seem further removed from the familiar world around us.

Five different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come together in one glorious, intricate equation, each playing with perfect pitch to blend and bind together to form a single whole that is far greater than any of the parts. A perfect mathematical composition.

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence. It brings together mental abstractions having their origins in very different aspects of our lives, reminding us once again that things that connect and bind together are ultimately more important, more valuable, and more beautiful than things that separate.

Keith Devlin, guest lecturer for the Wabash Center for Inquiry in the Liberal Arts, is executive director of the Center for the Study of Language and Information at Stanford University and is a contributor National Public Radio’s Weekend Edition.

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