Irrational numbers make up a significant sector of mathematical sciences and find utilization not only in theoretical models but also in engineering, construction, and scientific applications. Although most likely, irrational numbers — such as Pi, the Euler’s number, and Golden Ratio — have always existed, humankind did not immediately guess about their form. The discovery of irrational numbers is historically associated with difficulties in solving practical problems when the sides of figures, lengths of materials, or area of objects for some reason were not integers. For instance, the number of Pi was actively used by ancient Greek and Egyptian mathematicians when they discovered that the length of a rope wrapped around a wheel was about three times the diameter (Hom, 2018). Since this proportion was observed for all circles, this ratio was mathematically justified soon. Archimedes was the first researcher to give an approximate estimate of the value of Pi. To do this, the man used the circles inscribed in the polygons and approximated their area. Next, Euler’s number was discovered by a Swiss mathematician with the same surname: when studying logarithmic patterns, he found a constantly repeating numerical value, which was later called in such a way (Najera, 2019). Moreover, Euclid was another researcher who contributed to the world of irrational numbers because man discovered the Golden Ratio (Mann, 2019). A number approximately equal to 1.618 (or more accurately, (1+√5)/2) was used to construct the right triangle in the author’s works, although it was later even given a divine meaning.

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It is difficult to say when exactly mathematicians have given up: most likely, each philosopher tried to present the value as a decimal fraction and realized that each term, or even a set of terms, never repeats itself, means it is an irrational number. Specific methods could have been long division, such as measuring the length of a circle by diameter or calculating a triangle’s hypotenuse. Indeed, modern computer technologies make it possible to determine with great precision the values of irrational numbers, and, for example, it is even known that any person’s date of birth can be found in the number of Pi (“*Find your Pi day*,” n.d.). This fact makes it impossible to present irrational numbers in the form of fractional expressions since, by definition, these are infinite, non-periodic numbers. Simultaneously, one must admit that, along with the infinity of all numeric sets, there is no strict number of irrational numbers: their amount is not limited.

## References

*Find your Pi day*. (n.d.). Wolfram. Web.

Hom, E. J. (2018). *What is Pi?* Live Science. Web.

Mann, A. (2019). *Phi: The Golden Ratio*. Live Science. Web.

Najera, J. (2020). *The history of Euler’s number (e)*. Medium. Web.