Home > Euler's identity

The equation appears in Leonhard Euler's Introduction, published in Lausanne in 1748. In this equation,

The formula is a special case of Euler's formula from complex analysis, which states that

for any real number . If we set , then

and since cos(π) = −1 and sin(π) = 0, we get

1 Perceptions of the identity

Benjamin Pierce, after proving the formula in a lecture, said, "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."

It was called "the most remarkable formula in mathematics" by Richard Feynman. Feynman found this formula remarkable because it links some very fundamental mathematical constants:

• The number 0, the identity element for addition (for all a, a+0=0+a=a). See Group (mathematics).
• The number 1, the identity element for multiplication (for all a, a×1=1×a=a).
• The number is fundamental in trigonometryTrigonometry (Greek: "the measure of triangles") is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine and cosine . It has some relationship to geometry, though there is disagreement on exactly what that relati, is a constant in a world which is EuclidEuclid of Alexandria ( Greek: Eukleides (circa 365 275 BC) was a Greek mathematician, now known as "the father of geometry". His most famous work is the Elements widely considered to be history's most successful textbook. Within it, the properties of geomean, on small scales at least (otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences).
• The number is a fundamental in connections to the study of logarithms and in calculus (such as in describing growth behaviors, as the solution to the simplest growth equation with initial condition is ).
• The imaginary unit (where i² = −1) is a unit in the complex numberThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , whers. Introducing this unit yields all non-constant polynomialIn mathematics polynomial functions or polynomials are an important class of simple and smooth functions. Simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i. they have derivatives o equations soluble in the fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil of complex numbers (see fundamental theorem of algebraThe fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if : (where the coefficients a . a can be r).

Furthermore, all the most fundamental operators of arithmeticArithmetic Arithmetic or arithmetics in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as synonym fo are also present: equality, addition, multiplication and exponentiation. All the fundamental assumptions of complex analysis are present, and the integers 0 and 1 are related to the field of complex numbers.

In addition, the result is remarkable to most students learning it for the first time because it is so highly counter-intuitive. Consider that

while

The simple insertion of i changes the result dramatically.