e^(iπ) + 1 = 0: The Most Beautiful Theorem in Mathematics

The equation above is called Euler’s identity where

• e: Euler’s number, the base of natural logarithms (2.71828 ……)
• i: imaginary unit, i² = −1
• π: pi, the ratio of the circumference of a circle to its diameter (3.14159 ……).

The Euler’s number e is often used in the field of analysis, and i is in algebra, and π is in geometry. Symbols that are usually used in different areas are included in one equation, and the only other numbers are the simple ones 0 and 1. That is why this equation is called the most beautiful.

What does this equation mean?

The Euler’s identity e^(iπ) + 1 = 0 is a special case of Euler’s formula e^(iθ) = cosθ + isinθ when evaluated for θ= π.

So, the next question would be this.

How is Euler’s formula derived?

Why are there the exponential function, trigonometric functions, and the imaginary unit?

First, let’s rewrite the exponential function e^x in a power series.

To those who are not used to this form of expression, no worries. It looks a bit complicated, but a₀, a₁, a₂, … are just coefficients of x⁰(= 1), x¹, x², ….