Mathematical Proofs That Satisfy: A Closer Look at a Unique Theorem and a Famous Fallacy

Mathematical proofs are the language through which mathematicians express the elegant and logical underpinnings of our understanding of numbers and space. There is a unique satisfaction that comes from stumbling upon a clever proof that showcases both creativity and rigor. In this article, we explore a particular theorem and its accompanying proof, along with a classic fallacy that has captivated students and mathematicians for generations. Join us as we delve into the intricacies of these proofs and their implications.

A Clever Theorem and Its Proof

Consider the following theorem:

If ax by geq minx y for all x y geq 0, then a 0 and b 0.

The proof of this theorem involves some interesting and advanced concepts. Let’s break it down step by step:

Firstly, if ax by leq minx y for all x y geq 0, then aX bY leq minX Y whenever X Y are nonnegative random variables. Given that the expectation operator is linear and monotonic, we can infer that amathbb{E}X bmathbb{E}Y leq mathbb{E}minX Y. Furthermore, if mathbb{E}X infty or mathbb{E}Y infty, then mathbb{E}minX Y infty. However, if X and Y are independent random variables following an inverse-chi-squared distribution with two degrees of freedom, mathbb{E}X mathbb{E}Y infty, but mathbb{E}minX Y is finite.

This proof, while elegant, requires some non-trivial calculations. Tools like Mathematica can handle these calculations, but even popular web-based tools like Wolfram Alpha may struggle with them.

The theorem’s proof is a testament to how random variables and expectations can be used to solve seemingly abstract problems, underscoring the power and versatility of mathematical rigor.

Visualizing Nicomachus's Theorem

There are also visual proofs that can aid in understanding complex mathematical identities. One such example is Nicomachus’s theorem, which visually demonstrates an interesting identity:

1^3 2^3 3^3 … n^3 (1 2 3 … n)^2

This identity states that the sum of the cubes of the first n natural numbers is equal to the square of the sum of the first n natural numbers. This theorem is beautifully illustrated by visual patterns, where each cube can be broken down into a square of a sum, providing a clear and intuitive proof of the identity.

For a detailed and illustrated explanation, you can refer to the article on Squared Triangular Number on Wikipedia, which provides a detailed and visual breakdown of this theorem.

A Classic Fallacy: One Equals Two

Another fascinating proof, albeit faulty, is the one that famously shows that one equals two. This proof is so famous that it has been featured in many educational articles, from encyclopedias to textbooks. Here’s a step-by-step breakdown:

Start by defining that a is equal to b. Let: a b. Multiply both sides by b: ab b^2. Subtract a^2 from both sides: ab - a^2 b^2 - a^2. Factor both sides: a(b - a) b(b - a). Divide both sides by b - a (common factor): a b (when b - a neq 0). Substitute b a into the equation: b b^2/b. Simplify the right-hand side: b 2b. Divide both sides by b: 1 2.

While the conclusion 1 2 is clearly erroneous, the explanation often comes down to division by zero, which is undefined. The step where dividing both sides by b - a (assuming b - a neq 0) is not valid when b a, as it would involve division by zero.

This fallacy captivates mathematicians and students alike. It serves as a reminder of the importance of rigorous proof and the need to validate each step of an argument, even if it seems logical at first glance. To truly understand this fallacy, it’s recommended to carefully examine the comments and discussion below, as many users have provided detailed explanations.

Whether you find the theorem discussed herein or the fallacy of one equaling two more satisfying, we hope you’ve enjoyed this exploration of mathematical proofs and the satisfaction that comes from understanding and expanding our knowledge within the realm of mathematics.