Instruct: Given a question, retrieve passages that answer the question.
Query: Who introduced the concept of complex numbers?
The complex numbers give rise to the fundamental theorem of algebra: every non-constant polynomial equation with complex coefficients has a complex solution. This property is true of the complex numbers, but not the reals. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.
Algebra was also pioneered by Persian Scientist Muhammad ibn Mūsā al-Khwārizmī during this time in his landmark text, Kitab al-Jabr wa-l-Muqabala, from which the term algebra is derived. He is thus considered to be the father of algebra. The terms algorism and algorithm are also derived from the name of al-Khwarizmi, who was responsible for introducing the Arabic numerals and Hindu-Arabic numeral system beyond the Indian Subcontinent.
From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th  century. The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's "Liber Abaci" of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.
Johann Carl Friedrich Gauss (; ; ; (30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the "Princeps mathematicorum" () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.
Stefan of Pise translated into Latin around 1127 an Arab manual of medical theory. The method of algorism for performing arithmetic with Indian-Arabic numerals was developed by the Persian al-Khwarizmi in the 9th century, and introduced in Europe by Leonardo Fibonacci (1170–1250). A translation by Robert of Chester of the "Algebra" by al-Kharizmi is known as early as 1145. Ibn al-Haytham (Alhazen, 980–1037) compiled treatises on optical sciences, which were used as references by Newton and Descartes. Medical sciences were also highly developed in Islam as testified by the Crusaders, who relied on Arab doctors on numerous occasions. Joinville reports he was saved in 1250 by a “Saracen” doctor.
Euclid (; – "Eukleídēs", ; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His "Elements" is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the "Elements", Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.
It was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted in Arabic mathematics by the 9th century. Influential were the books of Muḥammad ibn Mūsā al-Khwārizmī ("On the Calculation with Hindu Numerals", ) and Al-Kindi ("On the Use of the Hindu Numerals", ). The system later spread to medieval Europe by the High Middle Ages.
The work was translated into Arabic around 820 by Al-Khwarizmi, whose "On the Calculation with Hindu Numerals" was in turn influential in the adoption of the Hindu-Arabic numerals in Europe from the 12th century.
Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.
Pickover states, “The Latin translation [of Al-Khuwarizmi's Algebra] introduced the decimal positional number system to Europe” (84). Later, Pickover writes that Fibonacci's Liber Abaci "introduced the Hindu-Arabic numerals and decimal system to Western Europe" (100).
Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers. Although he was preceded by the Babylonians and the Chinese, the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum). The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the "mensa Pythagorica".
During the early medieval era, Roman numerals were used in Europe to represent numbers, but the Arabs represented numbers using the Hindu numeral system, a positional notation that used ten symbols. In about 1202, Fibonacci published his book "Liber Abaci" (Book of Calculation) which introduced the concept of positional notation into Europe. These symbols evolved into the numerals "0", "1", "2" etc. At that time there was dispute regarding the difference between rational numbers and irrational numbers and there was no consistency in the way in which decimal fractions were represented.
Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem faster than anyone else in his class of 100 students. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100. There are many other anecdotes about his precocity while a toddler, and he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opus, "Disquisitiones Arithmeticae", in 1798, at the age of 21—though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists. In particular, the idea of an abstract linear space had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers) without necessarily identifying these elements with "geometric" vectors, such as position and momentum vectors in physical systems. Other objects studied by mathematicians at the turn of the 20th century, in particular spaces of sequences (including series) and spaces of functions, can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors.
Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was the first to calculate the tangents of the kappa curve. He is also notable for being the inaugural holder of the prestigious Lucasian Professorship of Mathematics, a post later held by his student, Isaac Newton who also went on to develop calculus in its modern form.
In 628, the Indian mathematician Brahmagupta wrote "Brāhmasphuṭasiddhānta" which includes, among many other things, a study of equations of the form . In particular he considered what is now called Pell's equation, , and found a method for its solution. In Europe this problem was studied by Brouncker, Euler and Lagrange.
Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appear in the archaeological record. The Babylonians also possessed a place-value system, and used a sexagesimal numeral system, still in use today for measuring angles and time.
Calculus is a genius, who demonstrates himself throughout the series to be an expert in many fields of science, holding three PhDs in nuclear and theoretical physics, and planetary astronomy. He is also an experienced engineer, archaeologist, biologist and chemist. Many of his inventions precede or mirror similar technological developments in the real world (most notably the Moon rocket, but also his failed attempt at creating a colour television set). He seeks to benefit humankind through his inventions, developing a pill that cures alcoholism by making alcohol unpalatable to the patient, and refusing under great duress to yield his talents to producing weapons of mass destruction. Much of Calculus's more dangerous work is criticized by Captain Haddock, although Calculus usually interprets this the other way round: his deafness often leads him to misinterpret Haddock's words, preventing him from hearing his real opinion.
In 1850, William Rankine first used the phrase "the law of the conservation of energy" for the principle.
Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers".
A further source of confusion was that the equation formula_135 seemed to be capriciously inconsistent with the algebraic identity formula_136, which is valid for non-negative real numbers and , and which was also used in complex number calculations with one of , positive and the other negative. The incorrect use of this identity (and the related identity formula_137) in the case when both and are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol in place of to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
Muses also envisioned a mathematical number concept, Musean hypernumbers, that includes hypercomplex number algebras such as complex numbers and split-complex numbers as primitive types. He assigned them levels based on certain arithmetical properties they may possess. While many open questions remain, in particular about defining relations of these levels, Muses pictured a wide range of applicability for this concept. Some of these are based on properties of magic squares, and even related to religious belief. He believed that these hypernumbers were central to issues of consciousness.
Zero-knowledge proofs were first conceived in 1985 by Shafi Goldwasser, Silvio Micali, and Charles Rackoff in their paper "The Knowledge Complexity of Interactive Proof-Systems". This paper introduced the IP hierarchy of interactive proof systems ("see interactive proof system") and conceived the concept of "knowledge complexity", a measurement of the amount of knowledge about the proof transferred from the prover to the verifier. They also gave the first zero-knowledge proof for a concrete problem, that of deciding quadratic nonresidues mod "m" (this more or less means that there isn't any number "x" where formula_86 is "equivalent" to some given number). Together with a paper by László Babai and Shlomo Moran, this landmark paper invented interactive proof systems, for which all five authors won the first Gödel Prize in 1993.
In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. This eventually led to the concept of the extended complex plane.