Irrational numbers in the context of "E (mathematical constant)"

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⭐ Core Definition: Irrational numbers

In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

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Irrational numbers in the context of Hippasus

Hippasus of Metapontum (/ˈhɪpəsəs/; Ancient Greek: Ἵππασος ὁ Μεταποντῖνος, Híppasos; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this and crediting it to himself instead of Pythagoras, which was the norm in Pythagorean society. The few ancient sources who describe this story, however, either do not mention Hippasus by name (e.g., Pappus) or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.

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