超实数系统是数系发展的里程碑

   迄今为止，数学文明已有3000年的历史，而数系概念是数学文明的核心。

  从数学发展文明史的视角来看，经历两千多年艰苦探索，才形成了严格定义的实数概念（1999，希尔伯特）。

  1960年，鲁宾逊第一次严格定义了 超实数系统，为微积分现代化奠定了坚实的理论基础。

  由此可见，超实数系统是数系发展的里程碑。一千年之后，人们使用的必定视超实数系统。

   请见本文附件。

袁萌  陈启清  7月21日

附件：

实数的History

Real numbers () include the rational numbers (), which include the integers (), which include the natural numbers ()

Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Sulba 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 since Manava (c. 750–690 BC), who were aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined.[1] Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2.

The Middle Ages brought the acceptance of zero, negative, integral, 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,[2] which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers.[3] The Egyptian mathematician Ab Kmil Shuj 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.[4]

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" ones.

In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that π cannot be rational; Adrien-Marie Legendre (1794) completed the proof,[5] and showed that π is not the square root of a rational number.[6] Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of the Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.

Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Georg Cantor (1873) extended and greatly simplified this proof.[7] Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Adolf Hurwitz[8] and Paul Gordan.[9]

The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was published by Georg Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof.