The Discovery of √2

Hippasus is often credited with discovering the irrational number √2. He stumbled upon this peculiar number when attempting to divide a square with sides measuring one unit, or perhaps while experimenting with an isosceles right triangle.

He chose √2 for his calculations because Pythagoras had previously established that the squares of the two shorter sides of a right triangle equate to the square of the hypotenuse. Hippasus examined a case where the height and base were both one unit, leading to a hypotenuse of √2.

Despite numerous attempts to express √2 as a fraction, he ultimately concluded that such a ratio could not exist. To prove this, he employed a method of contradiction by assuming √2 was rational. This method led him to the elegant proof affirming the existence of irrational numbers.

The Pythagorean community was unsettled by this revelation, as it contradicted their rigid beliefs regarding whole numbers. The existence of irrational numbers posed a significant threat to their worldview, which perceived such numbers as chaotic and disruptive.

The Pythagorean Belief System

The Pythagoreans viewed numbers as sacred and believed in the perfection of whole numbers, often expressing this through their reverence for equilateral triangles and other regular shapes. Their motto, “All is number,” was famously etched above their school’s entrance, emphasizing their belief that every number could be accurately represented as a ratio of two integers without common divisors, such as 15 represented as 15/1 or 0.2 as 1/5.

The discovery of √2, which does not have a repeating decimal representation, undermined the Pythagorean ideal of numerical perfection.

Hippasus’s Fate

What should have been a commendation for Hippasus’s groundbreaking discovery instead led to his punishment. The Pythagorean community condemned him to death by drowning, perceiving his challenge to their mathematical system as a grave offense against their harmonious principles.

Rather than embracing Hippasus’s insights, they destroyed his work and vowed to keep the notion of irrationality a closely guarded secret. Some accounts suggest he proved the irrationality of √2 while aboard a ship, prompting his fellow Pythagoreans to cast him overboard, unable to reconcile this new understanding with their rigid beliefs.

As noted in "Fermat’s Last Theorem" by Simon Singh, one narrative recounts how a young Hippasus, while playing with the number √2, realized its irrational nature. This realization filled him with joy, but his master, Pythagoras, was threatened by the implications of such a discovery. Instead of engaging in a meaningful discourse, Pythagoras chose to maintain his authority by executing Hippasus.

Today, we recognize that most real numbers are irrational. Georg Cantor demonstrated that while real numbers are uncountable, rational numbers are countable. Despite Hippasus’s tragic end, √2 remains one of the simplest and most well-known irrational numbers, approximately equal to 1.4142135623...—a non-repeating decimal that continues indefinitely.

Understanding The Pythagorean Theorem And Irrational Numbers explores the foundational concepts surrounding this pivotal discovery in mathematics.

Like the concept of the soul, which is often viewed as eternal and indivisible, irrational numbers such as π are similarly characterized by their inability to be expressed as simple fractions. As Joe Hill eloquently noted, "The soul may not be destroyed... Like the number pi, it is without cessation or conclusion."

In conclusion, while the story of Hippasus may be shrouded in myth, its core message remains significant. The exploration of irrational numbers challenges the very fabric of mathematical belief and invites us to embrace the complexity of our numerical universe.

What was up with Pythagoras? provides a deeper dive into the life and ideas of Pythagoras, shedding light on the philosophical context of his time.

Thank you for engaging with this narrative. I welcome your comments and insights on this fascinating topic!