Science Bytes: Euler’s Gem

Leonhard Euler (Swiss, 1707-1783) is undoubtedly one of the greatest mathematician of all time. His works (which number about nine hundred papers) span an enormous range of subjects, from number theory to graph theory, topology to logic. Many of his abstract insights spill over to various fields in applied mathematics, including physics, astronomy and finance. No wonder Euler has been called the ‘Mozart of mathematicians”. The French mathematician, Francois Arago said of Euler: “He calculated without any apparent effort, just like men breathe and eagles sustain themselves in the air.”

Arguably, the jewel in Euler’s crown is the identity that is named after him:

The eminent physicist, Richard Feynman calls this identify “a most remarkable formula in mathematics.” It’s not hard to see why.

First, in one little equation, we have five of the most fundamental numbers in mathematics seated side by side:

1 – the basis of all other numbers

0 – the measure of nothing

\pi – a number that appears everywhere in nature e- the base of natural logarithms and the number that measures exponential growth

i – the ‘imaginary’ square root of -1, and the basis of complex numbers that play a central role in quantum mechanics (I have more to say about i in another post)

Moreover, three of the most basic mathematical operations: addition, multiplication and exponentiation are in the identity as well. The sheer compactness of Euler’s identity has an aesthetic beauty that is akin to the best art works. Indeed, neural science research shows that Euler’s identify excites the same areas of the brain as a great piece of music or art [1].

There is a surprising connection between Euler’s identity and the mathematics of waves. This branch of mathematics makes heavy use of trigonometric functions like sines and cosines. Euler’s identity is a special case of Euler’s trigonometric form

where x is any real number. If you replace x by \pi and apply some elementary trigonometry substitutions, viola!, this formula becomes Euler’s identity.

Beauty aside, Euler’s formula has an impressive range of practical applications. Every major electronic device in the modern home, including the washing machines, air-conditioners and fridges has the formula lurking behind it. A bit more technically, modern electronic devices are designed around alternative current or AC. Engineers model AC as complex waves to simply the math. The equation of a complex wave can be written as f(x,t) = e^{ik(x-ct)} where k is the wave number and c, the angular frequency. It can be shown that this equation is a special solution – in propagating wave form – of e^{i \theta} = \cos \theta + i \sin \theta where theta is angle (in radians). This equation clearly evaluates to Euler’s identity when \theta = \pi !


[1] Zeki, S., J.P. Romaya, D.M.T. Benincasa and M.F. Aitiyah (2014), “The experience of mathematical beauty and its neural correlates”, Fronties in Neuroscience.

[2] For an entertaining read on the Euler equation, see David Stipp, A Most Elegant Equation, Basic Books, New York, 2017.

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