The expression "2πi" is a complex number where "i" represents the imaginary unit, defined as the square root of -1. In mathematics, "π" (pi) is a constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter. The combination of these elements is significant in various fields, particularly in complex analysis and number theory. In the context of Euler's formula, e^ix = \cos(x) + i\sin(x) , substituting x = 2\pi yields e^2\pi i = 1 . This illustrates a profound relationship between exponential functions and trigonometric functions, highlighting the beauty of mathematics and its interconnected concepts, such as complex numbers, trigonometry, and Euler's formula.