Why is it established?---**Mathematical Formula:**
e^(iπ) + 1 = 0
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Euler's identity, e to the i pi plus one equals zero, is widely regarded as one of the most beautiful and profound equations in all of mathematics. This remarkable formula brings together five of the most important mathematical constants in a single, elegant relationship. It connects Euler's number e, the imaginary unit i, pi, the number one, and zero through the fundamental operations of exponentiation, multiplication, and addition. The beauty of this identity lies not just in its simplicity, but in how it reveals deep connections between seemingly unrelated areas of mathematics, bridging analysis, algebra, and geometry in a way that continues to inspire mathematicians and scientists worldwide.
To understand why Euler's identity works, we must first examine each of its components. The number e, approximately 2.718, is Euler's number and serves as the base of the natural logarithm. It appears naturally in exponential growth and decay processes. The imaginary unit i is defined as the square root of negative one, forming the foundation of complex numbers. Pi, approximately 3.14159, represents the ratio of a circle's circumference to its diameter. The number 1 is the multiplicative identity, and 0 is the additive identity. Each of these constants plays a fundamental role in mathematics, and their convergence in Euler's identity reveals deep mathematical relationships.
To understand Euler's identity, we must first grasp Euler's formula: e to the i theta equals cosine theta plus i sine theta. This remarkable formula shows that complex exponentials represent rotation in the complex plane. When we raise e to an imaginary power i theta, we get a point on the unit circle. The real part of this complex number is cosine theta, while the imaginary part is sine theta. As theta varies, e to the i theta traces out points on the unit circle, demonstrating the deep connection between exponential and trigonometric functions. This geometric interpretation is crucial for understanding why Euler's identity holds.
Now we apply Euler's formula to the special case where theta equals pi. Starting with e to the i theta equals cosine theta plus i sine theta, we substitute theta equals pi to get e to the i pi equals cosine pi plus i sine pi. We know that cosine of pi equals negative one, and sine of pi equals zero. Therefore, e to the i pi equals negative one plus i times zero, which simplifies to negative one. This means e to the i pi equals negative one. Rearranging this equation by adding one to both sides gives us the famous Euler's identity: e to the i pi plus one equals zero. The geometric interpretation shows that e to the i pi represents a half rotation, or 180 degrees, around the unit circle, taking us from the point one to the point negative one.
The geometric interpretation of Euler's identity provides the most intuitive understanding of why this relationship holds. On the complex plane, we visualize e to the i pi as a rotation around the unit circle. Starting at the point 1 on the positive real axis, we rotate counterclockwise by pi radians, which is exactly 180 degrees. This half rotation brings us to the point negative 1 on the negative real axis. Therefore, e to the i pi equals negative 1. When we add 1 to both sides of this equation, we get e to the i pi plus 1 equals zero. This geometric perspective reveals that Euler's identity is fundamentally about rotation in the complex plane, connecting the exponential function with circular motion and trigonometry in a beautifully elegant way.