The Secret Society Who Discovered Irrational Numbers

The discovery of irrational numbers stands as a pivotal moment in the history of mathematics. These numbers, unlike rational numbers, cannot be expressed as a simple fraction. Their existence challenged the very foundation of the Pythagorean belief system, leading to both intellectual triumph and a touch of mystery.

The story is often attributed to the Pythagoreans, a secret society in ancient Greece obsessed with the harmony of numbers and their connection to the cosmos. While details are fuzzy, credit for the discovery is usually given to Hippasus of Metapontum, a Pythagorean philosopher.

Hippasus' revelation likely stemmed from the study of perfect squares – integers that can be obtained by squaring an integer (e.g., 4 is a perfect square because 2 x 2 = 4). The Pythagoreans believed that all lengths could be expressed as ratios of whole numbers. However, Hippasus, while investigating the diagonal of a square relative to its side,demonstrated that this wasn't always true.

Imagine a square. The length of a side is a simple, whole number (let's say 1). According to the Pythagorean theorem, the diagonal of this square would be equal to the square root of 2 (√2). However, try as they might, the Pythagoreans couldn't express √2 as a fraction with a whole number numerator and denominator. They reached a logical contradiction: if the side length is rational and the Pythagorean theorem holds true, the diagonal must also be rational. But their attempts to express √2 as a fraction kept failing.

This discovery of irrational numbers, numbers that were "incommensurable" with rational numbers, was a major blow to the Pythagorean worldview. Their belief system was built on the idea that everything could be neatly explained by whole numbers and their ratios. The existence of irrational numbers threw a wrench into their philosophy, highlighting the limitations of their understanding of mathematics.

The legend surrounding Hippasus' discovery adds a dramatic flair. Some accounts claim he was ostracized or even drowned at sea by his fellow Pythagoreans for revealing this unwelcome truth. While the details may be embellished, the story underscores the profound impact of this discovery.

The revelation of irrational numbers forced mathematicians to re-evaluate their understanding of numbers themselves. It opened doors to new areas of mathematical exploration and paved the way for advancements in geometry, algebra, and beyond. Although the exact details surrounding the discovery remain shrouded in the mists of time, the story of irrational numbers serves as a reminder of the power of challenging established beliefs and the ongoing quest for mathematical knowledge.  Let me know what you think, I'd love to hear.  Have a good day.