Euler's identity

Euler's identity is the equation e^{iπ} + 1 = 0, which links the exponential constant e, the imaginary unit i, π, and the basic numbers 1 and 0.

Euler's formula: Euler's formula states that e^{ix} = cos x + i sin x for real x. Setting x to π gives e^{iπ} = −1, the central step behind the identity.
Complex exponential: The complex exponential extends the real exponential to complex inputs and satisfies e^{x+iy} = e^x(cos y + i sin y). At input iπ, its magnitude is one and its angle is π.
Leonhard Euler: Leonhard Euler was an eighteenth-century Swiss mathematician whose work shaped analysis, number theory, and mechanics. The identity bears his name and follows from formulas he developed for complex exponentials.
Mathematical beauty: Mathematical beauty describes qualities such as simplicity, symmetry, and explanatory power in mathematical ideas. The identity is often cited for combining five familiar constants in one short equation.
Complex number: A complex number has the form a + bi, with real numbers a and b and i² = −1. The exponent iπ makes the identity a statement about complex exponentiation.
Unit circle: The unit circle is the circle of radius one centered at the origin in the coordinate plane. The value e^{iπ} is the point reached after a half-turn around this circle.
Introductio in analysin infinitorum: Introductio in analysin infinitorum is Leonhard Euler's 1748 treatise on mathematical analysis and functions. Euler presented foundational relations among exponential and trigonometric functions in this work.
Mathematical notation: Mathematical notation is the symbolic language used to express mathematical objects and relationships. The equation's impact depends on how compactly its symbols encode distinct ideas.
Imaginary unit: The imaginary unit i is a number defined by i² = −1. Its presence in the exponent makes a real exponential constant produce a real result.
Complex plane: The complex plane represents complex numbers as points with real and imaginary coordinates. It makes the rotation encoded by the exponent visible as a point on the plane.

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