Why the Equation eπ1 is Incorrect and the Truth Behind e^iπ -1

Introduction

It is a common misconception that the equations e π1 and e^iπ -1 are equivalent. However, it is clear that e π1 is incorrect, while e^iπ -1 is a well-known identity in mathematics, specifically Euler's identity.

Why eπ1 is Incorrect

The base of the natural logarithm, e, is approximately 2.71828, while π is approximately 3.14159. Therefore, when we insert the values, we find that e is far from being equal to π1, which is approximately 4.142. This makes it evident that the equation e π1 is not true.

The Correct Identity: e^iπ -1

What truly matters is the equation e^iπ -1, a famous identity derived by Leonhard Euler. This identity connects the fundamental constants e and π, along with the imaginary unit i. To understand why this equation holds true, let's explore the underlying mathematics and the Taylor series expansion.

1. Euler's Formula and Taylor Series Expansion

Euler's formula states that for any real number x, in radians, we have the following series expansion:

e^ix cos(x) isin(x)

This formula can be derived by using the Taylor series expansions for the exponential, cosine, and sine functions.

The Taylor series for the exponential function is:

e^x 1 x x^2/2! x^3/3! ...

The Taylor series for the cosine function is:

cos(x) 1 - x^2/2! x^4/4! - ...

The Taylor series for the sine function is:

sin(x) x - x^3/3! x^5/5! - ...

When we substitute ix into the exponential series, we get:

e^ix 1 ix (ix)^2/2! (ix)^3/3! (ix)^4/4! ...

Grouping the real and imaginary terms, we end up with:

e^ix (1 - x^2/2! x^4/4! - ...) i(x - x^3/3! x^5/5! - ...)

Which simplifies to:

e^ix cos(x) isin(x)

2. Applying Euler's Formula to π

Let's set x π and substitute it into the formula:

e^iπ cos(π) isin(π)

Knowing that cos(π) -1 and sin(π) 0, we get:

e^iπ -1 i(0) -1

This remarkable result, known as Euler's identity, beautifully ties together the most important numbers in mathematics: 0, 1, e, π, and i, the foundation of complex numbers.

Conclusion

It is important to understand the distinction between the incorrect equation e π1 and the correct identity e^iπ -1. The latter, Euler's identity, is a cornerstone in mathematics, revealing a deep and elegant connection between different fundamental constants. The base of the natural logarithm e and the ratio of a circle's circumference to its diameter, π, are indeed connected through complex numbers via Euler's formula and the Taylor series expansions.

References:

[1] Euler's Identity - Wikipedia

[2] Taylor Series - Wikipedia