Unveiling Elegant Proofs in Mathematics: From Geometry to Integrals

Mathematics is a vast field, rich with insightful and elegant proofs that often challenge our intuitive understanding and highlight the beauty of logical reasoning. In this article, we will explore two intriguing mathematical proofs: a simple proof about a geometric problem and a less obvious proof related to an integral.

Not So Obvious but Simple Proofs in Mathematics: Geometry and Distance

The first proof discussed here revolves around a fascinating problem posed in the 1987 Canadian mathematical Olympiad. The problem involves a large flat field on which n people are positioned in such a way that for each individual, the distances to all others are distinct. Each person holds a water pistol and fires at the closest person upon a signal.

The challenge: When n is odd, prove that at least one person remains dry.

Inductive Proof Strategy

Here's the proof:

Base Case: When n 1, the situation leaves that single person dry, as they cannot shoot at anyone. Inductive Hypothesis: Assume that if n people are positioned as described, at least one person remains dry. Inductive Step: Consider the case for n n2. In this scenario, there exists a minimum distance, and the two people at this minimum distance (A and B) will shoot each other. Subcase 1: If no one else shoots at A or B, we can remove them, leaving n people where at least one person remains dry by the inductive hypothesis. Subcase 2: If another person shoots at A or B, then three shots are used for only two people, leaving n - 1 shots to n - 1 people. By the Pigeonhole Principle, at least one person remains dry.

Through the principle of mathematical induction, we conclude that for any odd n, at least one person remains dry upon firing.

Mathematical Intuition and Elegant Proofs: Integrals

Another intriguing area of mathematics involves demonstrating the value of integrals in a simple yet elegant manner. Consider the integral in question: (int_{-infty}^{infty} e^{-x^2} dx).

A Simple but Deceptive Proof

To prove that (int_{-infty}^{infty} e^{-x^2} dx sqrt{pi}), we use a clever transformation and application of polar coordinates.

Start by considering the square of the integral: (left(int_{-infty}^{infty} e^{-x^2} dxright)^2). Convert this double integral into polar coordinates: Integrate over the region of the plane: (0 leq r Utilize the chain rule to find the derivative: (frac{d}{dr} left(-e^{-r^2}right) -2r e^{-r^2}). Compute the final integral step-by-step using substitution and bounds.

By these steps, we find that:

[ int_{-infty}^{infty} e^{-x^2} dx sqrt{pi} />

Thus, we have unraveled a deceptively simple yet powerful proof in mathematics, showcasing the elegance of geometrical demonstrations and the subtlety of integral calculus.

Further Reading and Exploration

For those interested in further exploration, I recommend checking out demonstrations of Proofs without Words. Additionally, consider delving into the book Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler, which compiles some of the most beautiful proofs in mathematics.

Through these resources and explorations, you will find that the world of mathematics is not only vast but also rich with elegance and beauty in its proofs.