Physics World in 2004, Euler's identity tied with Maxwell's equations (of electromagnetism) as the "greatest equation ever". At least three books in popular mathematics have been published about Euler's identity: Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, by Paul Nahin (2011) A Most Elegant Equation: Euler's formula and the beauty of mathematics, by David Stipp (2017) Euler's Pioneering Equation: The most beautiful theorem in mathematics, by Robin Wilson (2018). Explanations Imaginary exponents Fundamentally, Euler's identity asserts that is equal to −1. The expression is a special case of the expression , where is any complex number. In general, is defined for complex by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition is: Euler's identity therefore states that the limit, as approaches infinity, of is equal to −1. This limit is illustrated in the animation to the right. Euler's identity is a special case of Euler's formula, which states that for any real number , where the inputs of the trigonometric functions sine and cosine are given in radians. In particular, when , Since and it follows that which yields Euler's identity: Geometric interpretation Any complex number can be represented by the point on the complex plane. This point can also be represented in polar coordinates as , where r is the absolute value of z (distance from the origin), and is the argument of z (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of , implying that . According to Euler's formula, this is equivalent to saying . Euler's identity says that . Since is for r = 1 and , this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive x-axis is radians. Additionally, when any complex number z is multiplied by , it has the effect of rotating z counterclockwise by an angle of on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting equal to yields the related equation which can be interpreted as saying that rotating any point by one turn around the