82-463: [REDACTED] Look up numerical in Wiktionary, the free dictionary. Numerical may refer to: Number Numerical digit Numerical analysis Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Numerical . If an internal link led you here, you may wish to change
164-473: A + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i , ( a + b i ) ( c + d i ) = ( a c − b d ) + ( a d + b c ) i . {\displaystyle {\begin{aligned}(a+bi)+(c+di)&=(a+c)+(b+d)i,\\[5mu](a+bi)(c+di)&=(ac-bd)+(ad+bc)i.\end{aligned}}} When multiplied by
246-673: A + b i ) = − b + a i , − i ( a + b i ) = b − a i . {\displaystyle i(a+bi)=-b+ai,\quad -i(a+bi)=b-ai.} The powers of i repeat in a cycle expressible with the following pattern, where n is any integer: i 4 n = 1 , i 4 n + 1 = i , i 4 n + 2 = − 1 , i 4 n + 3 = − i . {\displaystyle i^{4n}=1,\quad i^{4n+1}=i,\quad i^{4n+2}=-1,\quad i^{4n+3}=-i.} Thus, under multiplication, i
328-1029: A l l a c y ( − 1 ) ⋅ ( − 1 ) = 1 = 1 (incorrect). {\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}\mathrel {\stackrel {\mathrm {fallacy} }{=}} {\textstyle {\sqrt {(-1)\cdot (-1)}}}={\sqrt {1}}=1\qquad {\text{(incorrect).}}} Generally, the calculation rules x t y ⋅ y t y = x ⋅ y t y {\textstyle {\sqrt {x{\vphantom {ty}}}}\cdot \!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x\cdot y{\vphantom {ty}}}}} and x t y / y t y = x / y {\textstyle {\sqrt {x{\vphantom {ty}}}}{\big /}\!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x/y}}} are guaranteed to be valid only for real, positive values of x and y . When x or y
410-478: A + bi can be represented by: a I + b J = ( a − b b a ) . {\displaystyle aI+bJ={\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.} More generally, any real-valued 2 × 2 matrix with a trace of zero and a determinant of one squares to − I , so could be chosen for J . Larger matrices could also be used; for example, 1 could be represented by
492-524: A Sanskrit word Shunye or shunya to refer to the concept of void . In mathematics texts this word often refers to the number zero. 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 ). There are other uses of zero before Brahmagupta, though the documentation
574-480: A number line , the imaginary axis , which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally. Integer sums of the real unit 1 and the imaginary unit i form a square lattice in the complex plane called the Gaussian integers . The sum, difference, or product of Gaussian integers is also a Gaussian integer: (
656-404: A , b positive and the other negative. The incorrect use of this identity, and the related identity in the case when both a and b are negative even bedeviled Euler . This difficulty eventually led him to the convention of using the special symbol i in place of − 1 {\displaystyle {\sqrt {-1}}} to guard against this mistake. The 18th century saw
738-603: A base 4, base 5 "finger" abacus. By 130 AD, Ptolemy , influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals . Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica ( Almagest ),
820-439: A given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing. 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
902-451: A notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis 's De algebra tractatus . 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. Gauss studied complex numbers of
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#1732765335241984-451: A part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The symbol ∞ {\displaystyle {\text{∞}}}
1066-503: A placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans. The use of 0 as a number should be distinguished from its use as a placeholder numeral in place-value systems . Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting . Indian texts used
1148-414: A representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums is isomorphic to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects. More generally, in the geometric algebra of any higher-dimensional Euclidean space ,
1230-421: A right angle between them. Addition by a complex number corresponds to translation in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every similarity transformation of the plane can be represented by a complex-linear function z ↦ a z + b . {\displaystyle z\mapsto az+b.} In the geometric algebra of
1312-425: A rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz . A modern geometrical version of infinity is given by projective geometry , which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in
1394-679: A system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior Hindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for zero , which was developed by ancient Indian mathematicians around 500 AD. The first known documented use of zero dates to AD 628, and appeared in
1476-449: A unit bivector of any arbitrary planar orientation squares to −1 , so can be taken to represent the imaginary unit i . The imaginary unit was historically written − 1 , {\textstyle {\sqrt {-1}},} and still is in some modern works. However, great care needs to be taken when manipulating formulas involving radicals . The radical sign notation x {\textstyle {\sqrt {x}}}
1558-479: A way to swap true roots and false roots as well. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral. The first use of negative numbers in a European work was by Nicolas Chuquet during the 15th century. He used them as exponents , but referred to them as "absurd numbers". As recently as
1640-430: Is isomorphic to the complex numbers, and the variable x {\displaystyle x} expresses the imaginary unit. The complex numbers can be represented graphically by drawing the real number line as the horizontal axis and the imaginary numbers as the vertical axis of a Cartesian plane called the complex plane . In this representation, the numbers 1 and i are at the same distance from 0 , with
1722-444: Is a subset of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as A more complete list of number sets appears in the following diagram. The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0
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#17327653352411804-837: Is a generator of a cyclic group of order 4, a discrete subgroup of the continuous circle group of the unit complex numbers under multiplication. Written as a special case of Euler's formula for an integer n , i n = exp ( 1 2 π i ) n = exp ( 1 2 n π i ) = cos ( 1 2 n π ) + i sin ( 1 2 n π ) . {\displaystyle i^{n}={\exp }{\bigl (}{\tfrac {1}{2}}\pi i{\bigr )}^{n}={\exp }{\bigl (}{\tfrac {1}{2}}n\pi i{\bigr )}={\cos }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}+{i\sin }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}.} With
1886-425: Is a negative scalar. The quotient of a vector with itself is the scalar 1 = u / u , and when multiplied by any vector leaves it unchanged (the identity transformation ). The quotient of any two perpendicular vectors of the same magnitude, J = u / v , which when multiplied rotates the divisor a quarter turn into the dividend, Jv = u , is a unit bivector which squares to −1 , and can thus be taken as
1968-461: Is a solution to the quadratic equation x + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers , using addition and multiplication . A simple example of the use of i in a complex number is 2 + 3 i . Imaginary numbers are an important mathematical concept; they extend the real number system R {\displaystyle \mathbb {R} } to
2050-579: Is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2 . Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency. The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to Pythagoras , more specifically to
2132-482: Is inherently positive or negative in the sense that real numbers are. A more formal expression of this indistinguishability of + i and − i is that, although the complex field is unique (as an extension of the real numbers) up to isomorphism , it is not unique up to a unique isomorphism. That is, there are two field automorphisms of the complex numbers C {\displaystyle \mathbb {C} } that keep each real number fixed, namely
2214-538: Is largely due to Ernst Kummer , who also invented ideal numbers , which were expressed as geometrical entities by Felix Klein in 1893. 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 . Prime numbers have been studied throughout recorded history. They are positive integers that are divisible only by 1 and themselves. Euclid devoted one book of
2296-588: Is not as complete as it is in the Brāhmasphuṭasiddhānta . Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The paradoxes of Zeno of Elea depend in part on
2378-405: Is often used to represent an infinite quantity. Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity —the general consensus being that only the latter had true value. Galileo Galilei 's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in
2460-434: Is real but negative, these problems can be avoided by writing and manipulating expressions like i 7 {\textstyle i{\sqrt {7}}} , rather than − 7 {\textstyle {\sqrt {-7}}} . For a more thorough discussion, see the articles Square root and Branch point . As a complex number, the imaginary unit follows all of the rules of complex arithmetic . When
2542-500: Is reserved either for the principal square root function, which is defined for only real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results: − 1 = i ⋅ i = − 1 ⋅ − 1 = f
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2624-460: Is said to have an argument of − π 2 , {\displaystyle -{\tfrac {\pi }{2}},} related to the convention of labelling orientations in the Cartesian plane relative to the positive x -axis with positive angles turning anticlockwise in the direction of the positive y -axis. Also, despite the signs written with them, neither + i nor − i
2706-485: Is the Hindu–Arabic numeral system , which allows for the representation of any non-negative integer using a combination of ten fundamental numeric symbols, called digits . In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers ), and for codes (as with ISBNs ). In common usage, a numeral is not clearly distinguished from
2788-434: Is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply
2870-508: Is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite , so there is an uncountably infinite number of transcendental numbers. The earliest known conception of mathematical infinity appears in the Yajur Veda , an ancient Indian script, which at one point states, "If you remove
2952-568: The Brāhmasphuṭasiddhānta , the main work of the Indian mathematician Brahmagupta . He treated 0 as a number and discussed operations involving it, including division . By this time (the 7th century) the concept had clearly reached Cambodia as Khmer numerals , and documentation shows the idea later spreading to China and the Islamic world . Brahmagupta's Brāhmasphuṭasiddhānta
3034-510: The 4 × 4 identity matrix and i could be represented by any of the Dirac matrices for spatial dimensions. Polynomials (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose coefficients are real numbers form a ring , denoted R [ x ] , {\displaystyle \mathbb {R} [x],} an algebraic structure with addition and multiplication and sharing many properties with
3116-697: The Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic , and presented the Euclidean algorithm for finding the greatest common divisor of two numbers. In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras. In 1796, Adrien-Marie Legendre conjectured
3198-402: The Euclidean plane , the geometric product or quotient of two arbitrary vectors is a sum of a scalar (real number) part and a bivector part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector
3280-783: The Pythagorean Hippasus of Metapontum , who produced a (most likely geometrical) proof of the irrationality of the square root of 2 . The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. 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. The 16th century brought final European acceptance of negative integral and fractional numbers. By
3362-726: The complex number system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as rings and fields , and the application of the term "number" is a matter of convention, without fundamental significance. Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks . 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. Nonetheless, tallying systems are considered
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3444-433: The complex plane , which is a special interpretation of a Cartesian plane , i is the point located one unit from the origin along the imaginary axis (which is orthogonal to the real axis ). Being a quadratic polynomial with no multiple root , the defining equation x = −1 has two distinct solutions, which are equally valid and which happen to be additive and multiplicative inverses of each other. Although
3526-462: The natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals ; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system , which is an organized way to represent any number. The most common numeral system
3608-479: The number that it represents. In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers , rational numbers such as one half ( 1 2 ) {\displaystyle \left({\tfrac {1}{2}}\right)} , real numbers such as the square root of 2 ( 2 ) {\displaystyle \left({\sqrt {2}}\right)} and π , and complex numbers which extend
3690-595: The prime number theorem , describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture , which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis , formulated by Bernhard Riemann in 1859. The prime number theorem
3772-461: 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. This was doubly unsettling since they did not even consider negative numbers to be on firm ground at
3854-467: The 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had remained almost dormant since Euclid . In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine , Georg Cantor , and Richard Dedekind
3936-595: The 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless. It is likely that the concept of fractional numbers dates to prehistoric times . The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus . Classical Greek and Indian mathematicians made studies of
4018-611: The 3rd century AD in Greece. Diophantus referred to the equation equivalent to 4 x + 20 = 0 (the solution is negative) in Arithmetica , saying that the equation gave an absurd result. During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta , in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce
4100-603: 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. When division produced 0 as a remainder, nihil , also meaning nothing , was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N,
4182-538: The complex number system C , {\displaystyle \mathbb {C} ,} in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra ). Here, the term "imaginary" is used because there is no real number having a negative square . There are two complex square roots of −1: i and − i , just as there are two complex square roots of every real number other than zero (which has one double square root ). In contexts in which use of
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#17327653352414264-1045: The construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of i with −1 ). Higher integral powers of i are thus i 3 = i 2 i = ( − 1 ) i = − i , i 4 = i 3 i = ( − i ) i = 1 , i 5 = i 4 i = ( 1 ) i = i , {\displaystyle {\begin{alignedat}{3}i^{3}&=i^{2}i&&=(-1)i&&=-i,\\[3mu]i^{4}&=i^{3}i&&=\;\!(-i)i&&=\ \,1,\\[3mu]i^{5}&=i^{4}i&&=\ \,(1)i&&=\ \ i,\end{alignedat}}} and so on, cycling through
4346-416: The development of Greek mathematics , stimulating the investigation of many problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the hypercomplex numbers , which consist of various extensions or modifications of
4428-655: The first kind of abstract numeral system. The first known system with place value was the Mesopotamian base 60 system ( c. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt . Numbers should be distinguished from numerals , the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. Roman numerals,
4510-417: The form a + bi , where a and b are integers (now called Gaussian integers ) or rational numbers. His student, Gotthold Eisenstein , studied the type a + bω , where ω is a complex root of x − 1 = 0 (now called Eisenstein integers ). Other such classes (called cyclotomic fields ) of complex numbers derive from the roots of unity x − 1 = 0 for higher values of k . This generalization
4592-548: The four values 1 , i , −1 , and − i . As with any non-zero real number, i = 1. As a complex number, i can be represented in rectangular form as 0 + 1 i , with a zero real component and a unit imaginary component. In polar form , i can be represented as 1 × e (or just e ), with an absolute value (or magnitude) of 1 and an argument (or angle) of π 2 {\displaystyle {\tfrac {\pi }{2}}} radians . (Adding any integer multiple of 2 π to this angle works as well.) In
4674-744: The general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots". European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci , 1202) and later as losses (in Flos ). René Descartes called them false roots as they cropped up in algebraic polynomials yet he found
4756-579: The idea of a cut (Schnitt) in the system of real numbers , separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker , and Méray. The search for roots of quintic and higher degree equations was an important development, the Abel–Ruffini theorem ( Ruffini 1799, Abel 1824) showed that they could not be solved by radicals (formulas involving only arithmetical operations and roots). Hence it
4838-431: The identity and complex conjugation . For more on this general phenomenon, see Galois group . Using the concepts of matrices and matrix multiplication , complex numbers can be represented in linear algebra. The real unit 1 and imaginary unit i can be represented by any pair of matrices I and J satisfying I = I , IJ = JI = J , and J = − I . Then a complex number a + bi can be represented by
4920-806: The imaginary unit i , any arbitrary complex number in the complex plane is rotated by a quarter turn ( 1 2 π {\displaystyle {\tfrac {1}{2}}\pi } radians or 90° ) anticlockwise . When multiplied by − i , any arbitrary complex number is rotated by a quarter turn clockwise. In polar form: i r e φ i = r e ( φ + π / 2 ) i , − i r e φ i = r e ( φ − π / 2 ) i . {\displaystyle i\,re^{\varphi i}=re^{(\varphi +\pi /2)i},\quad -i\,re^{\varphi i}=re^{(\varphi -\pi /2)i}.} In rectangular form, i (
5002-580: The imaginary unit is repeatedly added or subtracted, the result is some integer times the imaginary unit, an imaginary integer ; any such numbers can be added and the result is also an imaginary integer: a i + b i = ( a + b ) i . {\displaystyle ai+bi=(a+b)i.} Thus, the imaginary unit is the generator of a group under addition, specifically an infinite cyclic group . The imaginary unit can also be multiplied by any arbitrary real number to form an imaginary number . These numbers can be pictured on
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#17327653352415084-575: The letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in electrical engineering and control systems engineering , the imaginary unit is normally denoted by j instead of i , because i is commonly used to denote electric current . Square roots of negative numbers are called imaginary because in early-modern mathematics , only what are now called real numbers , obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so
5166-457: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Numerical&oldid=933023096 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Number A number is a mathematical object used to count, measure, and label. The most basic examples are
5248-610: The matrix aI + bJ , and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic. The most common choice is to represent 1 and i by the 2 × 2 identity matrix I and the matrix J , I = ( 1 0 0 1 ) , J = ( 0 − 1 1 0 ) . {\displaystyle I={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.} Then an arbitrary complex number
5330-509: The properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is often regarded as unlucky , and " a million " may signify "a lot" rather than an exact quantity. Though it is now regarded as pseudoscience , belief in a mystical significance of numbers, known as numerology , permeated ancient and medieval thought. Numerology heavily influenced
5412-408: The property that its square is −1: i 2 = − 1. {\displaystyle i^{2}=-1.} With i defined this way, it follows directly from algebra that i and − i are both square roots of −1. Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number,
5494-403: The real numbers with a square root of −1 (and its combinations with real numbers by adding or subtracting its multiples). Calculations with numbers are done with arithmetical operations, the most familiar being addition , subtraction , multiplication , division , and exponentiation . Their study or usage is called arithmetic , a term which may also refer to number theory , the study of
5576-522: The ring of integers . The polynomial x 2 + 1 {\displaystyle x^{2}+1} has no real-number roots , but the set of all real-coefficient polynomials divisible by x 2 + 1 {\displaystyle x^{2}+1} forms an ideal , and so there is a quotient ring R [ x ] / ⟨ x 2 + 1 ⟩ . {\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .} This quotient ring
5658-457: The set of all natural numbers is N , also written N {\displaystyle \mathbb {N} } , and sometimes N 0 {\displaystyle \mathbb {N} _{0}} or N 1 {\displaystyle \mathbb {N} _{1}} when it is necessary to indicate whether the set should start with 0 or 1, respectively. Imaginary unit The imaginary unit or unit imaginary number ( i )
5740-399: The square root of a negative number was previously considered undefined or nonsensical. The name imaginary is generally credited to René Descartes , and Isaac Newton used the term as early as 1670. The i notation was introduced by Leonhard Euler . A unit is an undivided whole, and unity or the unit number is the number one ( 1 ). The imaginary unit i is defined solely by
5822-567: The theory of rational numbers, as part of the general study of number theory . The best known of these is Euclid's Elements , dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra , which also covers number theory as part of a general study of mathematics. The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it
5904-408: The theory was made by Georg Cantor ; in 1895 he published a book about his new set theory , introducing, among other things, transfinite numbers and formulating the continuum hypothesis . 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 system of hyperreal numbers represents
5986-430: The time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation seemed capriciously inconsistent with the algebraic identity which is valid for positive real numbers a and b , and was also used in complex number calculations with one of
6068-530: The two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled + i (or simply i ) and the other is labelled − i , though it is inherently ambiguous which is which. The only differences between + i and − i arise from this labelling. For example, by convention + i is said to have an argument of + π 2 {\displaystyle +{\tfrac {\pi }{2}}} and − i
6150-578: The uncertain interpretation of 0. (The ancient Greeks even questioned whether 1 was a number.) The late Olmec people of south-central Mexico began to use a symbol for zero, a shell glyph , in the New World, possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar . Maya arithmetic used base 4 and base 5 written as base 20. George I. Sánchez in 1961 reported
6232-420: The work of Abraham de Moivre and Leonhard Euler . De Moivre's formula (1730) states: while Euler's formula of complex analysis (1748) gave us: The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received
6314-472: The writings of Joseph Louis Lagrange . Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus first connected the subject with determinants , resulting, with the subsequent contributions of Heine, Möbius , and Günther, in the theory of Kettenbruchdeterminanten . The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e
6396-456: Was brought about. In 1869, Charles Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. 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). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on
6478-539: Was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted. Numbers can be classified into sets , called number sets or number systems , such as the natural numbers and the real numbers . The main number systems are as follows: N 0 {\displaystyle \mathbb {N} _{0}} or N 1 {\displaystyle \mathbb {N} _{1}} are sometimes used. Each of these number systems
6560-434: Was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory . Simple continued fractions , closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler , and at the opening of the 19th century were brought into prominence through
6642-424: Was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 ( cardinality of the empty set , i.e. 0 elements, where 0 is thus the smallest cardinal number ) in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for
6724-519: Was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol. The abstract concept of negative numbers was recognized as early as 100–50 BC in China. 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. The first reference in a Western work was in
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