section_path: History
section_name: History
paragraph_text: History
  
  Simple fractions were used by the Egyptians around 1000 BC; the Vedic " Shulba Sutras " ("The rules of chords") in c. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava (c. 750–690 BC), who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined.
  
  Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational.
  
  For Greek mathematicians, numbers were only the natural numbers. Real numbers were called "proportions", being the ratios of two lengths, or equivalently being measures of a length in terms of another length, called unit length. Two lengths are "commensurable", if there is a unit in which they are both measured by integers, that is, in modern terminology, if their ratio is a rational number. Eudoxus of Cnidus (c. 390−340 BC) provided a definition of the equality of two irrational proportions in a way that is similar to Dedekind cuts (introduced more than 2,000 years later), except that he did not use any arithmetic operation other than multiplication of a length by a natural number (see Eudoxus of Cnidus). This may be viewed as the first definition of the real numbers.
  
  The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra). Arabic mathematicians merged the concepts of " number " and " magnitude " into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots, and fourth roots). In Europe, such numbers, not commensurable with the numerical unit, were called irrational or surd ("deaf").
  
  In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.
  
  In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" numbers.
  
  In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Lambert (1761) gave a flawed proof that π cannot be rational; Legendre (1794) completed the proof and showed that π is not the square root of a rational number. Liouville (1840) showed that neither e nor e can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Cantor (1873) extended and greatly simplified this proof. Hermite (1873) proved that e is transcendental, and Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), Hilbert (1893), Hurwitz, and Gordan.
  
  The concept that many points existed between rational numbers, such as the square root of 2, was well known to the ancient Greeks. The existence of a continuous number line was considered self-evident, but the nature of this continuity, presently called completeness, was not understood. The rigor developed for geometry did not cross over to the concept of numbers until the 1800s.