Euler's Identity

Euler’s identity is an equality found in mathematics that has been compared to a Shakespearean sonnet and described as "the most beautiful equation." It is a special case of a foundational equation in complex arithmetic called Euler’s Formula, which the late great physicist Richard Feynman called in his lectures(opens in new tab) "our jewel" and "the most remarkable formula in mathematics."

In an interview with the BBC, Prof David Percy of the Institute of Mathematics and its Applications said Euler's Identity was “a real classic and you can do no better than that … It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants.”

"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."

Stanford mathematician Keith Devlin wrote these words about the equation to the left in a 2002 essay called "The Most Beautiful Equation." But why is Euler's formula so breath-taking? And what does it even mean?

First, the letter "e" represents an irrational number (with unending digits) that begins 2.71828... Discovered in the context of continuously compounded interest, it governs the rate of exponential growth, from that of insect populations to the accumulation of interest to radioactive decay. In math, the number exhibits some very surprising properties, such as — to use math terminology — being equal to the sum of the inverse of all factorials from 0 to infinity. Indeed, the constant "e" pervades math, appearing seemingly from nowhere in a vast number of important equations.

Next, "i" represents the so-called "imaginary number": the square root of negative 1. It is thus called because, in reality, there is no number which can be multiplied by itself to produce a negative number (and so negative numbers have no real square roots). But in math, there are many situations where one is forced to take the square root of a negative. The letter "i" is therefore used as a sort of stand-in to mark places where this was done.

Pi, the ratio of a circle's circumference to its diameter, is one of the best-loved and most interesting numbers in math. Like "e," it seems to suddenly arise in a huge number of math and physics formulas.

Putting it all together, the constant "e" raised to the power of the imaginary "i" multiplied by pi equals -1. And, as seen in Euler's equation, adding 1 to that gives 0. It seems almost unbelievable that all these strange numbers — and even one that isn't real — would combine so simply. But it's a proven fact.

Most of modern mathematics and physics derives from work of Leonhard Euler. Who has been described as “the most prolific mathematician of all time” and “the Mozart of mathematics”. But for all his achievements, “many so-called educated people have never heard of him.

His most famous single equation is Euler’s identity. It is said to be the equation that can link all of the constants of mathematics together. The equation combines five of the most important numbers in mathematics.

  • 1 – the basis of all other numbers
  • 0 – the concept of nothingness
  • pi – the number that defines a circle
  • e – the number that underlies exponential growth
  • i – the “imaginary” square root of -1

The numbers all have many practical applications, including communication, navigation, energy, manufacturing, finance, meteorology and medicine. But that’s not all. Euler’s identity also contains the three most basic mathematical operations: addition, multiplication and exponentiation. What appeals to is that this equality connects some incredibly complicated and seemingly unrelated concepts in a surprisingly concise form.

Euler’s identity is amazing because it is simple to look at and yet incredibly profound.

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