Unveiling the Mystery of Irrational Numbers

Like a figure from Greek mythology, the story of Hippasus is shrouded in mystery and rumored divine retribution. His offense? A groundbreaking mathematical discovery: the existence of irrational numbers. This revelation challenged the very foundations of the Pythagorean worldview, a system built on the belief that all things could be expressed as ratios of whole numbers.

The Pythagorean Worldview and Rational Numbers

The Pythagoreans held a deep reverence for numbers, believing them to be the fundamental building blocks of the universe. Their core tenet, "All is number," implied that everything, from the cosmos to morality, adhered to eternal rules that could be described through numerical ratios. This meant that any number could be written as a fraction. For example:

5 can be expressed as 5/1
1. 5 can be expressed as 1/2
Even infinitely repeating decimals can be expressed as a ratio (e.g., 34/45)

These numbers, expressible as a ratio of two integers, are what we now call rational numbers. However, Hippasus stumbled upon a number that defied this seemingly unbreakable rule.

The Discovery of the Unspeakable: Square Root of 2

The problem arose from a simple geometric shape: a square with sides of one unit. According to the Pythagorean Theorem, the length of the diagonal would be the square root of two. Hippasus attempted to express this value as a ratio of two integers, but his efforts were in vain. Instead of abandoning the problem, he embarked on a quest to prove that it was, in fact, impossible.

Proof by Contradiction: A Mathematical Revolution

Hipassus employed a method called proof by contradiction. He began by assuming the opposite of what he wanted to prove – that the square root of 2 could be expressed as a ratio of two integers, which he labeled p and q. He further assumed that this ratio was in its simplest form, meaning p and q had no common factors.

Here's how the proof unfolded:

Assume: √2 = p/q (where p and q are integers with no common factors)
Multiply both sides by q: √2 * q = p
Square both sides: 2q² = p²
Deduction: Since 2q² is an even number, p² must also be even.
Implication: If p² is even, then p must also be even (because the square of an odd number is always odd).
Express p as 2a: Let p = 2a, where a is an integer.
Substitute into the equation: 2q² = (2a)² = 4a²
Simplify: q² = 2a²
Deduction: Since 2a² is even, q² must also be even, meaning q is also even.
Contradiction: If both p and q are even, they share a common factor of 2. This contradicts the initial assumption that p and q had no common factors.

Therefore, the initial assumption that √2 can be expressed as a ratio of two integers must be false. This groundbreaking proof demonstrated the existence of irrational numbers – numbers that cannot be expressed as a simple fraction.

Visualizing Irrational Numbers on the Number Line

Despite their seemingly abstract nature, irrational numbers can be precisely plotted on the number line. Consider √2 again. By constructing a right triangle with both sides measuring one unit, the hypotenuse will have a length of √2. This length can then be transferred to the number line, marking the exact location of √2.

Similarly, a new right triangle can be constructed using the previous hypotenuse (√2) as the base and a side of one unit as the height. The new hypotenuse will then have a length of √3, which can also be plotted on the number line.

Beyond Ratios: The True Nature of Numbers

It's crucial to remember that decimals and ratios are merely ways to represent numbers. √2 is, in itself, the length of the hypotenuse of a right triangle with sides of length one. Similarly, the famous irrational number pi (π) represents the ratio of a circle's circumference to its diameter. Approximations like 22/7 or 355/113 are useful, but they will never perfectly equal pi.

The story of Hippasus serves as a reminder that the pursuit of knowledge can lead to unexpected and revolutionary discoveries. While the myths surrounding his fate may be dramatic, his contribution to mathematics is undeniable. Embrace the exploration of the seemingly impossible, for it is there that true breakthroughs often lie.