Number
[7] Though it is now regarded as pseudoscience, belief in a mystical significance of numbers, known as numerology, permeated ancient and medieval thought.
[9] Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.
[11] These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.
A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers.
The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets.
[14][better source needed] The key to the effectiveness of the system was the symbol for zero, which was developed by ancient Indian mathematicians around 500 AD.
By this time (the 7th century), the concept had clearly reached Cambodia in the form of Khmer numerals,[15] and documentation shows the idea later spreading to China and the Islamic world.
[16] In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an algebraic grammar for the Sanskrit language (also see Pingala).
In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter Omicron (otherwise meaning 70).
Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol.
The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.
Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today.
As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.
Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.
For example, it is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2.
[citation needed] Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency.
He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news.
[22][better source needed] The 16th century brought final European acceptance of negative integral and fractional numbers.
In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine,[23] Georg Cantor,[24] and Richard Dedekind[25] was brought about.
Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by Paul Tannery (1894).
Simple continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler,[27] and at the opening of the 19th century were brought into prominence through the writings of Joseph Louis Lagrange.
Ramus[28] first connected the subject with determinants, resulting, with the subsequent contributions of Heine,[29] Möbius, and Günther,[30] in the theory of Kettenbruchdeterminanten.
In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis.
The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid.
They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as Niccolò Fontana Tartaglia and Gerolamo Cardano.
It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.
Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion.
In the same year, Gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm.
In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
The treatment of negative real numbers is according to the general rules of arithmetic and their denotation is simply prefixing the corresponding positive numeral by a minus sign, e.g. −123.456.
