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The Discovery of the “Unnatural”: Irrational Numbers

Updated: Oct 26, 2023


Written by Arda Kizilkaya


Irrational numbers are real numbers that do not go into the category of rational numbers, and help us with numerous different topics like the growth of technology, creating new hypotheses, and so on.



The First “Discovery” of Irrational Numbers


The root of the Irrational numbers discovery dates back to the 6th century BC when the Pythagoreans and other civilizations sought to think that all things are numbers, they also believed that all magnitudes were commensurable, but civilizations around the world believed that all numbers were of a whole number of units, we can also say that civilizations thought that all numbers were rational. A positive rational number is the exact ratio of two positive integers p/q, where q does not equal 0. A negative rational number was likely to be accepted in China because of the belief in dualism in Chinese national philosophy, it is also likely to be accepted in India (or Bharat) by Brahmagupta, who was born in India in 30 BC. In his treatise Brahmasphutasiddhanta he thought that negative numbers were a sense of “fortune and debt”. Despite all this everybody thought that no number in the universe could be irrational or incommensurable.


But some legends say that around one of them called Hippasus of Metapontum discovered that square root 2 is an incommensurable or an irrational number while some sources say that he found it by questioning what a square edge length is its area is 2, while some other sources say that he found it by measuring a right triangles long edge. Although Hippasus suggested that irrational numbers existed, the people did not believe in him, and according to some sources, they even threw him into the sea. But we still cannot confirm that any of this is true or false because these are legends; they cannot be true historical facts until there is proof that it happened. There are many irrational numbers but some of the most known ones are.



The First Found Irrational Number: √2


It is believed that the first use of irrational numbers was in the Indian Sulbasutras. They found it so they could meet the requirements of their ritual sacrifices in which they had to construct a square fire altar twice the area of a given square altar. Which led to them finding √2. In Sulbasutras there is also a debate that these numbers can't be computed exactly. Thus, the concept of irrational numbers was implicitly accepted by Indian Brahmins. While decimal fractions and decimal place value notation have a long history, decimal fraction approximations of √2 appeared in 200-870 AD, in the Jain School of Mathematics, which is in India. In terms of decimal expansions being the complete opposite of rational numbers, an irrational number never repeats the same sequence, it’s always different. However, it also doesn’t terminate itself. Decimal expansion is the only thing that immediately shows the difference between rational and irrational numbers.



The Golden Irrational Number: Golden Ratio


The golden ratio is the positive number that satisfies the equation x²=x+1. The first known mention of the golden ratio is around 300 BC in Euclid’s famous book “Euclid’s Elements”. A Classical Greek book on mathematics and geometry. But it wasn’t deemed as an irrational number until in the 1200s, Magister Campanus Nouariensis demonstrated the irrationality of the golden ratio. Since then the golden ratio has been used in art, architecture, music, and so on.



Constant of Calculus: Euler’s Number


Euler’s number is an irrational number that is used in calculus, finance, and so on. Although the name suggests that Leonhard Euler created the number, Euler's number was not created by Euler; instead, it was done in 1683 by Jacob Bernoulli, but Euler did prove that Euler’s number is irrational. He also proved that “e” can be represented as an infinite sum of inverse factorials.



References:
  1. Origin of irrational numbers and their approximations. (2021, March 9). MDPI. https://www.mdpi.com/2079-3197/9/3/29

  2. Golden ratio | Examples, definition, & facts. (April 27). Encyclopedia Britannica. https://www.britannica.com/science/golden-ratio

  3. Euler's number (E) explained, and how it is used in finance. (November 14). Investopedia. https://www.investopedia.com/terms/e/eulers-constant.asp





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