# The Intriguing Proof of the Irrationality of √2 via Infinite Descent

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## Chapter 1: Introduction to √2's Irrationality

The mathematical enigma surrounding the square root of two, denoted as √2, has captivated thinkers for centuries. In this article, we will demonstrate its irrational nature using the refined approach of infinite descent.

All illustrations are created by the author. The initial diagram presents the side length of a square, denoted as s, alongside its hypotenuse, labeled d. This serves as the foundation for our proof.

Utilizing the Pythagorean Theorem, we express the square root as a ratio of two integers, d and s, that are in simplest form. The goal is to establish the existence of a smaller right isosceles triangle, where the ratio of the hypotenuse to the side length can be represented with even lesser integers. This leads to a contradiction of the assumption that d:s is in its lowest terms. The diagrams illustrate that this process indeed unfolds. The argument continues indefinitely, with each square diminishing in size as it spirals inward.

Let’s denote the hypotenuse as 2s - d and the side length as d - s.

In this step, we label the hypotenuse of the smaller square as α. Observe that both the blue and orange segments have lengths of s, and when combined, they overlap by the length α.

The original hypotenuse can be calculated as twice the length of those segments, s, minus the hypotenuse of the smaller square.

By rearranging, we derive an expression for α. Since this expression is the difference of two integers, it is also an integer itself. Now, let’s examine the side length.

From the construction above, we can ascertain that the side of the smaller square, which we will call β, has the same length as a portion of the segment s.

In fact, the sum of β and α equals s.

Rearranging again, we find that β is equal to d - s. As this is also a difference of two integers, β is an integer.

Thus, we conclude our proof: we can identify a smaller pair of integers representing the ratio of the hypotenuse to the side length. Therefore, both d and s cannot be integers simultaneously, implying that the square root of 2 is irrational.

This reasoning continues infinitely, spiraling down into the void, ad infinitum.

Photo credit: Giordano Rossoni on Unsplash.

And thus, we arrive at our proof regarding the irrationality of √2. What an astonishing result!

What were your thoughts during this exploration? Feel free to comment below; I’m excited to hear your perspectives!

## Math Puzzles

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