A recent entry in the visitor’s book of the James Joyce Tower & Museum in Sandycove, Dublin, perplexed the Friends of Joyce’s Tower, the volunteers who run the museum.
The entry, reproduced as the boxed equation in the photograph, seemed as impenetrable as a passage of Finnegans Wake and quite beyond decryption. When invited to elucidate its meaning, the visitor just chuckled. In fact, this equation, familiar to mathematicians and greatly admired for its elegance, is known as Euler’s Identity.
Mathematician Leonhard Euler, born in Switzerland in 1707, spent most of his career in St Petersburg. His work is of singular genius, originality and profundity. His status as a mathematician compares to that of Shakespeare, Rembrandt and Bach in literature, painting and music. Mathematical writer William Dunham observed that if a Mount Rushmore of Maths were carved Euler would certainly feature there.
Euler’s Identity has been selected in several surveys as the most beautiful mathematical formula. It is not the appearance or typographical structure that is prized, but the concepts it embodies and the unexpected links it makes.
Death of Pope Francis - live updates: Funeral to take place on Saturday
CMO Mary Horgan: ‘People often forget how bad infections were; our memories sometimes are short’
Owen Doyle: Chaotic cock-up cost Munster and it reflects badly on a number of people
Dublin-born Irish–American cardinal Kevin Farrell to run Vatican until pope elected
To a mathematician, an equation is a work of art. Aspects of its beauty include simplicity, profundity, utility and the capacity to surprise, often revealing unanticipated connections between different areas of maths.
[ When science becomes a laughing matterOpens in new window ]
The identity unites the five most fundamental mathematical constants. On the right is zero, the origin of the number line and its additive identity: adding 0 to any number leaves it unchanged. Moving to the left, we find the number one, the multiplicative identity: multiplying any number by 1 leaves it unchanged.
The remaining term of the equation is Euler’s number, e, raised to the power of i times p (pi). Euler’s number, the base of the natural logarithms, with a value of about 2.7, is ubiquitous in mathematics and statistics. You will recall from elementary geometry that pi is the ratio of the circumference of a circle to its diameter (perhaps you remember the ratio 22/7 from school). Finally, i is the square root of -1.
“Hold on,” you say, “any real number ¾ plus or minus ¾ multiplied by itself gives a positive result, so no real number can be the square root of -1” Correct!
But the so-called real numbers fill only a single horizontal line in the complex plane, an infinite two-dimensional domain with a multitude of points that are not on the real line but that represent perfectly respectable numbers. They are called complex numbers not because they are complicated but because they require two quantities to specify them, a real part and an imaginary part, for example 4+3i. The simplest of these is the number i, which appears as the point one unit above 0 on the vertical line through the origin.
Euler’s identity is a special case of a foundational equation in complex analysis, Euler’s Formula, which he discovered in 1744. It shows how any complex number can be obtained by rotating a point on the real line around the origin. If the point at +x is rotated through a half turn (pi radians in math-speak), it will land on -x. Euler’s formula is of enormous importance in physics and engineering, enabling us to solve otherwise intractable problems. Nobel laureate Richard Feynman called it “the most remarkable formula in mathematics”.
Why did the visitor write Euler’s identity, and what links it to James Joyce? As Winston Churchill once remarked, that is a riddle wrapped in a mystery inside an enigma!
Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at thatsmaths.com
