Sunday, June 29, 2014
Algebra, Euler, and i
"In the eighteenth century, mathematicians realized that to solve equations no numbers are needed beyond the imaginaries, a result so important it is known as the Fundamental Theorem of Algebra. Every equation written with complex numbers will always produce a solution with complex numbers. The door that Rafael Bombelli walked through to investigate the square roots of negative numbers revealed a solitary room. But what a room it was! The squeamishness mathematicians once had with imaginary numbers has been replaced by joy. The concept of i is now considered a very natural and efficient extension of the number system. For the price of a solitary symbol, mathematicians gained an elegantly self-contained abstract universe. Bargain.
"Imaginary numbers are protagonists of two of the most famous examples of mathematical beauty. One is a picture and one is an equation, known as Euler's identity, which in 2003 was sprayed on the side of an SUV in an ecoterrorist attack on a Los Angeles car dealership. The nature of the graffiti led to the arrest of a physics PhD student at Caltech. 'Everyone should know Euler's [identity],' he explained to the judge. He was correct, but one should nevertheless refrain from daubing it on cars."
-- Alex Bellos in "The Grapes of Math"
(my look at Bellos' latest book is now up at MathTango)