Euler's Identity: The Most Beautiful Equation in Mathematics

Euler's identity is a remarkable and elegant mathematical equation that connects the five most important and fundamental constants of mathematics, namely, $0$, $1$, $\pi$, $e$, and $i$. It is considered by many to be the most beautiful equation in mathematics due to its simplicity and profoundness.

The Equation

The equation is given by:

$e^{i\pi}+1=0$

This equation is also known as Euler's equation, and it is a special case of Euler's formula:

$e^{ix}=\cos(x)+i\sin(x)$

where $i$ is the imaginary unit, defined as $\sqrt{-1}$.

The Significance

Euler's identity is significant because it shows the close relationship between exponential functions, trigonometric functions, and complex numbers. It is as if these seemingly different areas of mathematics are all connected by this one simple equation.

The left-hand side of the equation, $e^{i\pi}+1$, is a combination of the two most important numbers in mathematics: $e$ and $\pi$. The number $e$ is the base of the natural logarithm and is used in calculus and many other areas of mathematics. The number $\pi$ is the ratio of a circle's circumference to its diameter and is used in geometry and trigonometry.

The right-hand side of the equation, $0$, is the additive identity, which is the starting point of all mathematical operations. It is also the origin of the number line, and the point of reference for all numbers.

The Proof

The proof of Euler's identity involves the use of Maclaurin series, which is a way of representing functions as an infinite sum of terms. The Maclaurin series for $e^{ix}$ and $\cos(x)+i\sin(x)$ are given by:

$e^{ix}=1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}-\cdots$

$\cos(x)+i\sin(x)=1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}-\cdots$

By equating the real and imaginary parts of these two series, we obtain:

$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$

$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$

Substituting $x=\pi$, we get:

$\cos(\pi)=-1$

$\sin(\pi)=0$

Therefore, $e^{i\pi}=\cos(\pi)+i\sin(\pi)=-1+0i=-1$. Hence, $e^{i\pi}+1=0$, which is Euler's identity.

Conclusion

Euler's identity is a beautiful and profound equation that connects some of the most important constants in mathematics. It is significant because it shows how different areas of mathematics are related to each other. The proof of this identity involves the use of Maclaurin series, which is a powerful tool in calculus and analysis. Euler's identity is not just a mathematical curiosity, but it has deep implications in many areas of physics and engineering, such as signal processing and quantum mechanics. It is truly a masterpiece of mathematics, and it is a testament to the power and beauty of pure reason.

Links

オイラーの恒等式[JA]