Welcome to Numberholic! I’m your host, Sejari, and you’re watching the fourth episode of Proven Yet Hard to Believe. In this series, we uncover fascinating stories behind some of the most significant proofs and constructions in the history of mathematics. Today, we’re diving into a story of contradiction, discovery, and secrecy—a tale that begins in ancient Greece and reveals the unsettling truth about irrational numbers. Are you ready? Let’s get started!
In Prior Analytics, Aristotle tells us the following:
“The diagonal of a square cannot be measured by the common measure as its side. If one assumes they can, then even and odd numbers would become equal.”
What Aristotle meant by this is that the length of the diagonal of a square and the length of its side cannot be expressed as a ratio of two integers. In modern terms, this means that the square root of 2 is irrational.
The idea of a “common measure” refers to the possibility of expressing two quantities as integer multiples of a shared unit. For example, the numbers 3.2 and 4 can be measured using 0.8 as a common unit: 3.2 equals 0.8 multiplied by 4, and 4 equals 0.8 multiplied by 5. Thus, 3.2:4 is equal to 4:5. This reflects the Greek, and particularly the Pythagorean, way of thinking about ratios.
But when it comes to the side and diagonal of a square, no such common measure exists.
The Greeks did not have our modern notation for square roots. The symbol for square roots first appeared in 1525.
Instead, they referred to quantities that could not be measured by a common unit as “incommensurables.” Aristotle, while discussing valid methods of reasoning in his work, used the proof by contradiction to demonstrate this unmeasurability. His example—the diagonal and side of a square—shows that the Greeks had already widely accepted the irrationality of square root of 2 by his time. This proof, often taught in middle school, is more than 2,000 years old.
Here is the outline of the proof by contradiction:
- Assume the side and diagonal of a square can be measured using a common unit. Represent the ratio of their lengths in its simplest form as a:b.
- From the Pythagorean Theorem, “b” must be even.
- Since and are coprime (having no common factors other than 1), “a” must be odd.
- Let b=2c. Substituting this into the equation gives “a” squared is two times of “c” squared. Thus, “a” must also be even.
- But “a” cannot be both odd and even. This contradiction proves that square root of 2 is irrational.
Plato, in his dialogue Theaetetus, writes that around 400 BCE, Theodorus of Cyrene had demonstrated the irrationality of square roots of numbers like 3, 5, and up to 17. This further cements the ancient understanding of irrational numbers.
The first person credited with discovering irrationality, however, is Hippasus. He was a member of the Pythagorean school, and his discovery reportedly caused a great upheaval among the Pythagoreans.
Pythagoras himself is said to have coined the term “cosmos” to describe the universe as an orderly, harmonious system. For Pythagoras and his followers, numbers and their symmetries were the foundation of this cosmic order. Numbers were thought to govern all attributes of existence and were even considered the cause of all being. This numerical harmony was called “kosmos.”
But ironically, the Pythagoreans’ greatest intellectual achievement—their discovery of irrational numbers—shattered their vision of a perfectly ordered universe.
Some historians speculate that Hippasus’s original proof of irrationality might not have been algebraic but geometric.
For example, when we reduce any rational number to its simplest form, it means that the numerator and denominator share no common factor other than 1. For instance, 1.4 can be written as the fraction 14/10, which simplifies to 7/5. This is an irreducible fraction, and no smaller integers can represent 1.4. Now, suppose square root of 2 were rational and could be written as an irreducible fraction. Constructing a right isosceles triangle based on this assumption would lead to an infinite regress of smaller triangles with integer sides. This geometric impossibility demonstrates the irrationality of square root of 2.
Some mathematicians argue that Hippasus may have first discovered irrationality not in square root of 2 but in the golden ratio. The golden ratio, a hallmark of Pythagorean pride, is also irrational. Euclid’s definition of the golden ratio, given centuries later, directly shows its incommensurability. The very definition of the golden ratio reveals its incommensurability in two distinct ways. If the whole and the greater segment can be expressed as a ratio of integers in their smallest possible terms, what then of the ratio between the greater and the lesser segments, which would also need to be integers?
Can you see the paradox?
Shh!
Even if you do, keep it to yourself—lest the wrath of the Pythagoreans rises from their graves to haunt you!
For the Pythagoreans, this discovery was nothing short of heretical. Legend has it that Hippasus broke an oath of secrecy by revealing the existence of irrational numbers. In response, the Pythagorean community, furious at this betrayal, is said to have executed him by drowning—a grim end for a truth-teller.
The discovery of irrational numbers challenged the very foundation of Pythagorean philosophy and redefined our understanding of mathematics. From the unmeasurable diagonal of a square to the golden ratio’s elusive properties, these revelations forced humanity to confront the infinite complexities of the universe. Though the Pythagoreans resisted the idea, their discoveries laid the groundwork for centuries of mathematical progress.
Thank you for joining me on this journey through history and proof. Stay tuned for the next episode of Proven Yet Hard to Believe, where we’ll explore another astonishing story of mathematical discovery. Until next time, keep questioning, keep exploring, and keep being amazed by the power of numbers!