It is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth. – 1800s mathematician Benjamin Peirce
In 1773, Denis Diderot, a leading figure of the French enlightenment and, in his time, considered a universal genius: philosopher, playwright and, most notably, editor of the famous French Encyclopedie, was invited to Russia by Czarina Catherine II. Diderot came to Russia, preaching the new scientific gospel of rationalistic atheism. Catherine turned to another of her favorites, mathematician Leonard Euler, in the hope that he could silence Diderot. She had her courtiers circulate a rumor; intended for Diderot’s ears, that Euler had discovered a mathematical proof for the existence of God. Impressed by all things scientific and, especially, mathematical, Diderot asked to see Euler and hear his magnificent proof. Standing before all of Catherine’s court, Euler solemnly proclaimed a mathematical equation that any scientist (let alone any mathematician) would recognize as nonsense. Euler concluded his address to Diderot with: “Hence God exists; reply.”Awed by Euler’s confident statement of something that might have been Chinese for all he understood it, Diderot stood in silent embarrassment. The Russian court roared at seeing the uppity Frenchman finally get his comeuppance. Shamefaced, Diderot asked for Catherine’s permission, quickly granted, to return to France. Mathematics, or rather pseudo-mathematics, had triumphed over rationalism.
eiπ + 1 = 0
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. – Keith Devlin
Euler Identity is considered to be the most beautiful mathematical equation, which brings 0, 1, π, i and e into one simple equation.
0 and 1 are not just the numbers, they are the additive inverse and the multiplicative inverse.
π, in flat (Euclidean) geometry, is the ratio of the circumference to the diameter of a circle. It’s the most used mathematical value.
The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm. It is the limit of (1 + 1/n)n as n approaches infinity. In contemporary internet culture, individuals and organizations frequently pay homage to the number e.
i is defined as the square root of -1. i Expands our thoughts from one dimension to two dimension.
The first glance of the Euler’s identity will make you think something is wrong with the equation, and makes you wonder. Upon closer you realize you are looking at something so truly amazing and so perfect that it requires someone like god to create the equivalence.
It isn’t there to serve a particular purpose. It isn’t there to solve a problem It’s simply there, representing the perfect and elegant unification of the five most important, yet seemingly unrelated mathematical constants of our universe. It just exists, to be admired. It’s unjudging, irrefutable, incorruptible, unique, elegant, not open to interpretation. It does not care whether or not you are able to appreciate it, as it has existed long before the rise of humanity and will continue to exist past the death of humanity. It shares many characteristics with god. It’s like Earth, Fire, Wind, Water, and Heart melding together to form a Planet. Through equations like these God gives us a hint of his existent.
Uses of Euler’s identity:
In a sense, this formula is divulging the secrets of complex space, and gives us a window through which we can peer into the other side.
Logarithms of Negative Numbers: Bizarre idea, which comes to reality using the Euler’s Identity.
The Euler’s Identity gives us the power to compute the Complex powers, one astonishing result:
ii = e (–x/2) ~ 0.2078796
Even more astonishing concept would be 1 raised to irrational powers and it having complex values.
It is implemented in linear, time-invariant function input-output machines, otherwise known as LTI boxes. The context of this equation is bandwidths mainly in radio station wavelengths. Another real-life example of Euler’s equation being applied is in Newton’s gravitational law. Another astronomical phenomenon that is explained best with the use of complex exponents is planetary retrograde motion. Euler’s identity made ship building, cannon ballistics, fluid dynamics, lunar orbit theory, and other mechanics easier and more accurate to calculate and understand.
Design patents and patents pertaining to telecommunication, radar technology, geometrically modifying electromagnetic radiation, signal processing (especially systems involving Fourier transformation), wireless communication, transversal filters etc are heavily dependent on Euler’s Identity. As many as 700 patents refer to Euler’s identity. Euler’s identity can be seen in various walks of the life. It’s nothing short of a miracle to link several important numbers in an unexpected way.
(Featured image source: https://en.wikipedia.org/wiki/Rhombic_dodecahedron)