Pi Day: 5 Greatest Mathematical Discoveries in History

Thursday is Pi Day, and in celebration of 3.14 and the irrational transcendent number that is pi, take a moment to grab some dessert and hunker down to this highly subjective list of the five greatest mathematical discoveries in history:

1. 


This equation, known as Euler's Identity, is both stunning in its beauty and deceiving in its simplicity. For those unfamiliar with the symbols, here  refers to Euler's Number, the base of the natural logarithm and equal to approximately 2.718, and  refers to the imaginary number . Even if this equation looks like nonsense, take a moment to revel in the fact that with just seven symbols Euler's identity manages to link five of the most important yet seemingly disparate constants in mathematics. It is beyond me to adequately describe in words the profundity of the connections within mathematics that this identity encapsulates so I leave it to those much more qualified. The great physicist Richard Feynman called the identity "one of the most remarkable, almost astounding, formulas in all of mathematics" and mathematics professor Keith Devlin said like 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." Now if that doesn't send chills up your spine I'd check your pulse.   

2. Fast Fourier Transform


To understand the importance of the discovery of the fast Fourier transform (FFT) on the modern computing age, it's first necessary to understand the purpose of the discrete Fourier Transform (DFT). The DFT is a transform first introduced by Fourier in the early 19th century that, put simply, has the ability to break down signals (think sound waves or wireless signals) into their component frequencies. Click here to read a more detailed explanation of the DFT. Once a signal is transformed into its frequencies, often times it can be manipulated in a much easier fashion. For example, a sound decomposed into its frequencies can have its high-frequency noises (which should be unnoticeable) filtered out thereby decreasing the noise and size of the signal without harming the quality. This is just one of a large amount of DFT applications which range from data and image compression (by being able to discard the least noticeable frequencies) to Magnetic Resonance Imaging and many fields in between. Now all this is well and good from a theoretical standpoint, but the DFT and its inverse suffer from requiring a largely impractical amount of time to compute. Were it not for the invention of a FFT by J.W. Tukey and John Cooley in the 1960s, the DFT might have remained a footnote in history. However, their algorithm drastically reduced the time needed to compute the DFT and led to the ubiquity of its application across engineering and mathematical fields.

3. Gödel's Incompleteness Theorems


At the beginning of the 20th century, mathematician David Hilbert presented a list of 23 of the most important unsolved problems in mathematics. Second on his list was to prove that the axioms of arithmetic are consistent i.e. free from internal contradictions. It may appear obviously true, yet in a groundbreaking 1931 paper that unified logic, mathematics and philosophy, Gödel is widely believed to have proven Hilbert's problem in the negative. In fact, his incompleteness theorems went so far as to show that in any axiomatic system, such as arithmetic, there exist statements that are undecidable. An imperfect but useful analogy is found within the liar paradox. In this paradox we begin with a machine that can be fed any statement and outputs whether that statement is true or false with unfailing accuracy. Now consider inputting the statement:"this statement is false". The machine could output neither "true" nor "false" without producing a contradiction - the equivalent of an undecidable proposition. The results from Gödel 's theorems still resonate within many areas of mathematics and philosophy and have been extended, perhaps speciously at times, to speculate on the philosophical limitations of computational systems and even the human mind itself.

4. Fermat's Last Theorem


An amateur mathematician, a scribble in a margin, and a 350 year old mystery. What could be a movie script for an exciting mathematical thriller (at least I'd go see it) is actually the real life story behind Fermat's Last Theorem. Pierre de Fermat was a French amateur mathematician who made numerous contributions to analysis and number theory in the early 17th century; however, his most enduring legacy involves a small note that he transcribed on the inside of Diophantus' Arithmetica. Fermat was examining Diophantine  equations of the form  for integers , and . When  the solutions to the equation simplify into the Pythagorean triples that most of us learned in Algebra 1 (for instance, ), but Fermat had the insight to conjecture that for  there are no integer solutions for, and that satisfy the equation. In fact, in the margin he actually transcribed "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." It was this proposition of a marvelous proof that cemented Fermat's legacy, as despite the wide publicity of the problem and efforts of thousands of mathematicians, the proof of his conjecture went unsolved for over 350 years. It wasn't until 1996 and using techniques entirely unavailable to Fermat that Sir Andrew Wiles at last proved Fermat's conjecture in the affirmative, submitting a proof that surpassed 100 pages. Did Fermat truly discover a simple elegant proof as he had claimed? It appears unlikely, but regardless, it's clear that his work helped inspire push the boundaries of mathematical insight for hundreds of years after his death.

5. Euclid's Elements


No list of mathematical achievements would be complete without the inclusion of the most seminal and influential mathematical work to come out of Greek antiquity. Written around 300BC, Euclid's work built the foundation for modern mathematics by introducing a set of axioms and proceeding to demonstrate by mathematical rigor a collection of theorems that naturally followed. Covering subjects ranging from algebra to plane geometry (also now known as Euclidean Geometry), Elements remained a cornerstone of mathematical teaching for over 2,000 years following its creation. Elements influenced the thinking of great minds ranging from Dostoevsky to Einstein, and Abraham Lincoln's inclusion of the phrase "dedicated to the proposition" in his Gettysburg address is often attributed to his readings of Euclid.

Honorable Mention: Public Key Encryption, Calculus

What should I have included? What should I have left out?  Let me know in the comments and don't forget to check back in two years on 3.14.15 when math lovers all around the world celebrate the most exciting day of the century.