Irrational numbers are numbers having a decimal expansion that neither terminates nor expresses periodicity. For a long time now, mathematicians have struggled with the concept of irrational numbers because of their peculiar nature. Hippassus of Metapontum is believed to have been the first person to discover irrational numbers. He is supposed to have tried using the famous theorem a 2 + b 2 =c 2 to find the length of the diagonal of a unit square (Shahhriari, 2014). While Hippassus is believed to have discovered irrational numbers, the method which he used to do so is not clear. Some people think that he might have used Euclid's proof of the irrationality of numbers, which was written 300 years after the era of Hippassus. However, some scholars argue that Euclid’s proof is far too advanced beyond the knowledge of Hippassus at the time.
Well, what happened is not a burning issue today. One thing that remains surprising is the fact that there was a point in time when the effort to prove the existence of an irrational number was considered moral wrongdoing. Throughout all recorded history, there is evidence of numbers, with the earliest basis for math and numbers coming from the need to measure and count things. The process through which natural, positive, non-zero numbers would emerge while counting was natural and intuitive. Also, the measurement would at some point present a value that could not be divided into a whole number, or dimensions that lay between two whole numbers. The discovery of positive rational numbers was a pretty intuitive process ( Andrei, 2015) .
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While numbers are mostly believed to have originated from practical human needs, some, like the Pythagoreans, developed as a result of a spiritual basis of their religion and philosophy. Pythagorean ethics, physics, spirituality, and cosmology were predicated on the principle of “all is number.” They believed that rational numbers could be used to describe and comprehend all things from the pitches of musical scales, the number of stars up in the sky, and even the qualities of virtue . This theory was the depiction of the Pythagorean celebration of sunrise. The Pythagorean gave the ability of rational numbers to be perceived in almost everything, be it a musical harmony or a sunrise a mystical significance.
The fact that there is an infinite amount of rational numbers is one of the reasons why they form the basis of everything within the universe. It is intuitive and reasonable to say that an infinite amount of rational numbers is enough to describe anything that is in existence. Rational numbers are incomprehensibly dense within the number line. There is little “space” between the numbers, but if you ever needed to describe something existing between two numbers, then a fraction could easily come in handy. It is probable that the Pythagoreans had initially measured the diagonal of a unit square manually. Then they viewed this measurement as an approximation very close to a specific rational number, which must be the actual length of the diagonal. Before the discovery by Hippassus, they could not have suspected the existence of logically real numbers that in principle could not be counted to or measured ( Van Hoof et al., 2018) .
An irrational number was an indication of meaninglessness in the then seemingly orderly world. Therefore, the Pythagoreans believed that numbers had to be something you could count on and that all things had to be counted as rational numbers. The discovery of irrational numbers by Hippassus was an indication of the existence of things that could not be understood or explained through rational numbers. It was a threat not only to Pythagorean philosophy but their mathematics as well. While the number line is considered to be infinitely dense with rational numbers, the existence of irrational numbers is proof of the existence of holes that cannot be described as a ratio of two integers.
References
Andrei, A. (2015). Proving the unaccountability of the number of irrational powers of irrational numbers evaluated as rationales and solutions approximation for x^ x= y and x^ = y. arXiv preprint arXiv: 1510.08022 .
Shahhriari, S. (2014): On the distance from a rational power to the nearest integer. Journal of Number Theory , 117 (1), 222-239.
Van Hoof, J., Degrande, T., Ceulemans, E., Verschaffel, L., & Van Dooren, W. (2018). Towards a mathematically more correct understanding of rational numbers: A longitudinal study with upper elementary school learners. Learning and Individual Differences , 61 , 99-108.
