The equation which has become known as “Euler’s Identity” has been rated by mathematicians and laymen alike to be one of the most beautiful expressions in the history of

In his 1748 work, Introduction in Analysis Infinitorum, Swiss mathematician and physicist Leonard Euler first took steps toward deriving the equation, which to some would prove to be one of the most beautiful mathematical expressions ever produced – a mysteriously perfect statement with profound implications.

As Euler originally penned it, however, the equation now known as Euler’s Identity (or Euler’s Equation) looks very little like its present-day simplified form. Euler’s retained a strict trigonometric form:

eix = cos x + i sin x

Today, we know that this same equation can be written much more simply as:

eip + 1 = 0

## What Does it Mean?

Euler’s formula, in a sense, demonstrated that, though the reasons for this have never been fully clear to mathematicians, there is an intrinsic connection between complex exponential functions and trigonometric transformations.

When viewed in light of Euler’s original expression (which is, in essence, mathematically equivalent to the simpler version), one can view the expression as showing a path on a standard trigonometric unit circle (a unit circle is merely a circle drawn onto a standard Cartesian coordinate system with a radius of one unit).

Beginning at point (1,0) on the unit circle (which is where the necessity of the “+ 1” in the expression arises), the transformation follows around the circle’s circumference to (-1,0), then returns back to the origin (hence the result of the equation being “0”). From my own experience

### Why Does this Matter?

The beauty of Euler’s Identity arises from a number of directions. Most evidently, there is the sheer number of mathematical ideas being used in a single expression, resulting in the simplest possible answer – 0.

Within the equation one finds themselves utilizing the standard algebraic ideas of addition (or subtraction depending on how it is written), multiplication, powers and mathematical equivalence. In addition, five of math’s most fundamental constants find their values working together in a mysteriously interconnected way:

*e is Euler’s number, with a value of roughly 2.718.*

*i is the Imaginary number, equivalent to the idea of the square root of negative one.*

*p is the well-known constant which refers to the simple, yet irrational relationship between a circle’s circumference and diameter, with a value of approximately 3.14159.*

The other two constants at use here are 0 and 1, which serve double purposes as both standard integers and constants. The number 1 serves as a numeral under addition and subtraction, but as a constant under multiplication, addition, or exponentiation. This latter idea can be understood by realizing that any number or variable can be defined as itself multiplied by 1.

0 on the other hand is a very useful, multipurpose constant which can be used in countless algebraic capacities.

The true beauty of this formula comes from the fact that, while the true nature of several of these constants continues to remain a mystery to mathematicians, within the confines of this equation they all work together in such a way that they interlock like pieces of a mathematical jigsaw puzzle, the end result of which has the mathematical traveler ending up right back where he began – at the journey’s origin.

It is not any sort of true practical value which has forced this equation near the top of almost every list of the greatest equations in history; rather it is the fact that it demonstrates the beautiful complexity of mathematics itself which has left humanity in awe of it for almost three hundred years.

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