HIPPASUS : AN UNFORTUNATE MATHEMATICIAN

This is my first blog on mathematics — the science of numbers, patterns, and logical reasoning. Mathematics is a fundamental tool used to understand and explain the world around us. It provides a systematic way to study quantities, shapes, structures, and changes, forming the backbone of fields like physics, engineering, economics, and computer science.

At its core, mathematics helps us measure, calculate, analyze data, and solve problems — from simple everyday tasks like managing money to complex scientific research exploring the universe. It is often called the language of science because it allows scientists to express natural laws and physical phenomena with precision.

 ''Mathematics is the Queen of all Sciences and Arithmetic is the Queen of Mathematics''

-Carl Friedrich Gauss

With the help of set theory, if the consider the universal set (U) as 'number' denoted for simplicity as S, i.e. U= S, then I can denote the elements in S in the form of a set as given below.

S={0,1,⅔,5i,√2,-3}

Within the set $S$, we encounter different types of numbers. To better understand them, numbers are classified into distinct groups based on their properties. Below is an introduction to these classifications, along with examples selected from the set $S$:

• Natural Numbers: Example - 1
• Whole Numbers: Examples - 0, 1
• Integers: Examples - 0, 1, -3
• Rational Numbers: Examples - 0, 1, $\frac{2}{3}$, -3
• Irrational Numbers: Example - $\sqrt{2}$
• Real Numbers: Examples - 0, 1, -3, $\frac{2}{3}$, $\sqrt{2}$
• Complex Numbers: Examples - 0, 1, -3, $\frac{2}{3}$, $\sqrt{2}$, 5i

In this blog, I would like to introduce you to a mathematician whose deep passion for numbers ultimately led to his tragic end.

Hippasus was a brilliant student who studied under the renowned Greek mathematician and philosopher, Pythagoras. Pythagoras is a name most of us recognise, as we have encountered his contributions extensively in our mathematics textbooks. In particular, we are all familiar with the Pythagorean Theorem and Pythagorean Triplets, two fundamental concepts closely associated with his legacy.

Before we move forward, let us briefly recall these important mathematical ideas:

Pythagoras Theorem states that, 

''In a right angled triangle, the sum of the squares of any two sides of the triangle is equal to the square of the third side (the longest side) which we call the hypotenuse''

                                                                              

                                                                            

Figure 1: A representative figure to explain Pythagoras theorem

In this figure, if we apply Pythagoras theorem we find that, a² + b² = c²

Now what do we mean by a Pythagorean triplet. It is nothing but the set of three numbers which make up the three sides of a right triangle such that the three numbers are in accordance with the Pythagoras Theorem.

For example, {5,12,13} is a Pythagorean Triplet

Because, 5²+12²=13²=169

Similarly, {1,4,√5} is also a Pythagorean Triplet.

Now let us go into the main topic,

                                                                   

Figure 2: Bust of Pythagoras

During the time of Pythagoras, a philosophical and mathematical school of thought known as Pythagoreanism emerged. As the name suggests, this school was built entirely upon the teachings and beliefs of Pythagoras and his followers.

At the heart of Pythagoreanism was a strong belief that the entire universe could be explained through whole numbers and their ratios — in other words, through rational numbers. According to this philosophy, every measurable quantity in the universe, from the length of a line to the motion of the stars, could be represented using fractions — the ratio of two integers.

However, this beautiful and seemingly perfect worldview was about to face a serious challenge.   

Figure 3: Hippasus

One day, a brilliant Pythagorean student named Hippasus was studying the diagonal of a square with sides of length 1 unit. Using the Pythagorean Theorem, he calculated the length of the diagonal as √2. Out of curiosity, Hippasus tried to express √2 as a fraction — a ratio of two whole numbers — since this was the accepted way of describing all numbers in Pythagoreanism.

To his surprise, no such fraction could be found. No matter how hard he tried, he discovered that √2 could never be written as an exact fraction. This led him to a shocking conclusion: √2 is not a rational number — it is irrational. This was the first known discovery of irrational numbers in recorded mathematical history.

When Hippasus presented this discovery to his teacher, Pythagoras, the reaction was far from welcoming. Pythagoras, who had built his entire philosophy around the supremacy of rational numbers, was outraged and deeply disturbed. The very foundation of his worldview was now at risk — if irrational numbers existed, it meant that not all quantities could be explained using the simple ratios of whole numbers.

Rather than embracing this groundbreaking discovery, Pythagoras and his loyal followers viewed it as a threat — a dangerous idea that could shatter the harmony of their mathematical universe. According to historical accounts (though some consider it more legend than fact), Hippasus was severely punished for his discovery and his defiance of Pythagorean teachings. In some versions of the story, he was drowned at sea by his own fellow Pythagoreans, who wanted to suppress the unsettling truth that irrational numbers existed.

''Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty''

-Archimedes

Despite this tragic end, Hippasus' discovery marked a turning point in mathematical history — the realization that numbers, and the very nature of mathematical reality, were far more complex than the Pythagoreans had imagined. Today, irrational numbers like √2 are essential building blocks of mathematics, forming the basis for fields such as algebra, calculus, and number theory.

Despite their seemingly strange nature, irrational numbers play an important role in mathematics and science. They fill the gaps between rational numbers, ensuring that the number line is continuous and complete. Many important mathematical constants, such as π (pi) and e, are irrational, and they appear in fundamental equations across geometry, calculus, and physics. 

In geometry, irrational numbers often emerge when calculating diagonal lengths, circumferences, and areas of curved shapes. In science and engineering, they are essential for accurate measurements, especially in fields like quantum mechanics, relativity, and signal processing. Without irrational numbers, our mathematical understanding of nature would be incomplete, making them indispensable to both pure mathematics and its countless practical applications.

REFRENCES:

1. Wikimedia Commons: Figure representing Pythagoras Theorem, images of the bust of Pythagoras and Hippasus.

2. Books like 'Tell Me Why' which I used to read in my childhood and even continue to read.

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