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Encyclopedia > Number
Number systems in mathematics
Basic $mathbb{N}submathbb{Z}submathbb{Q}submathbb{R}submathbb{C}$

Natural numbers $mathbb{N}$
Negative numbers
Integers $mathbb{Z}$
Rational numbers $mathbb{Q}$
Irrational numbers
Real numbers $mathbb{R}$
Imaginary numbers
Complex numbers $mathbb{C}$
Algebraic numbers
Transcendental numbers
Look up number in Wiktionary, the free dictionary. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... A negative number is a number that is less than zero, such as âˆ’3. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...

Complex extensions

Quaternions $mathbb{H}$
Octonions $mathbb{O}$
Sedenions $mathbb{S}$
Cayley-Dickson construction
Split-complex numbers $mathbb{R}^{1,1}$
Bicomplex numbers
Biquaternions
Coquaternions
Tessarines
Hypercomplex numbers
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... A bicomplex number is a number written in the form, a + bi1 + ci2 + dj, where i1, i2 and j are imaginary units. ... In mathematics, a biquaternion (or complex quaternion) is an element of the (unique) quaternion algebra over the complex numbers. ... In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ... The tessarines are a mathematical idea introduced by James Cockle in 1848. ... The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. ...

Other extensions

Musean hypernumbers
Superreal numbers
Hyperreal numbers
Surreal numbers
Dual numbers
Transfinite numbers
Musean Hypernumbers are a concept envisioned by Charles A. MusÃ¨s (1919â€“2000) to form a complete, integrated, connected, and natural number system . MusÃ¨s sketched certain fundamental types of hypernumbers and arranged them in ten levels, each with its own associated arithmetic and geometry. ... The superreal numbers compose a more inclusive category than hyperreal number. ... The system of hyperreal numbers represents 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 calculus by Newton and Leibniz. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... A variety of dualities in mathematics are listed at duality (mathematics). ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ...

Other

Nominal numbers
Serial numbers
Ordinal numbers
Cardinal numbers
Prime numbers
p-adic numbers
Constructible numbers
Computable numbers
Integer sequences
Mathematical constants
Large numbers
π = 3.141592654…
e = 2.718281828…
i (Imaginary unit) i2 = − 1
∞ (infinity) Nominal numbers are numbers used for identification only. ... A serial number is a unique number that is one of a series assigned for identification which varies from its successor or predecessor by a fixed discrete integer value. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ... In mathematics, an integer sequence is a sequence (i. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... This article or section is not written in the formal tone expected of an encyclopedia article. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... The infinity symbol âˆž in several typefaces. ...

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A number is an abstract idea used in counting and measuring. A symbol which represents a number is called a numeral, but in common usage the word number is used for both the idea and the symbol. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (ISBNs). In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers. Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers... Various meters Measurement is an observation that reduces an uncertainty expressed as a quantity. ... A numeral is a symbol or group of symbols that represents a number. ... A telephone number is a sequence of decimal digits that uniquely indicates the network termination point. ... A serial number is a unique number that is one of a series assigned for identification which varies from its successor or predecessor by a fixed discrete integer value. ... The International Standard Book Number, or ISBN (sometimes pronounced is-ben), is a unique identifier for books, intended to be used commercially. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

Certain procedures which input one or more numbers and output a number are called numerical operations. Unary operations input a single number and output a single number. For example, the successor operation adds one to an integer: the successor of 4 is 5. More common are binary operations which input two numbers and output a single number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic. In logic and mathematics, an operation Ï‰ is a function of the form Ï‰ : X1 Ã— â€¦ Ã— Xk â†’ Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In mathematics, a unary operation is an operation with only one operand. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... 3 + 2 = 5 with apples, a popular choice in textbooks This article is about addition in mathematics. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... â€œExponentâ€ redirects here. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...

The branch of mathematics that studies abstract number systems such as groups, rings and fields is called abstract algebra. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...

## Types of numbers GA_googleFillSlot("encyclopedia_square");

Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.) In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a number system is a set of numbers, or number-like objects, together with one or more operations, such as addition or multiplication. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ...

### Natural numbers

The most familiar numbers are the natural numbers or counting numbers: one, two, three, ... . Some people also include zero in the natural numbers; however, others do not. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

In the base ten number system, in almost universal use today, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right. The symbol for the set of all natural numbers is N, also written $mathbb{N}$. The decimal (base ten or occasionally denary) numeral system has ten as its base. ... In mathematics and computer science, a numerical digit is a symbol, e. ... An example of blackboard bold letters. ...

### Integers

Negative numbers are numbers that are less than zero. They are the opposite of positive numbers. For example, if a positive number indicates a bank deposit, then a negative number indicates a withdrawal of the same amount. Negative numbers are usually written by writing a negative sign in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers Z (German Zahl, plural Zahlen), also written $mathbb{Z}$. A negative number is a number that is less than zero, such as &#8722;3. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... The whole numbers are the nonnegative integers (0, 1, 2, 3, ...) The set of all whole numbers is represented by the symbol = {0, 1, 2, 3, ...} Algebraically, the elements of form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). ... The integers are commonly denoted by the above symbol. ... An example of blackboard bold letters. ...

### Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. The fraction m/n or In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... A denominator is a name. ... $m over n ,$

represents m equal parts, where n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is: ${1 over 2} = {2 over 4},$.

If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written −7/1. The symbol for the rational numbers is Q (for quotient), also written $mathbb{Q}$. In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ... In mathematics, a quotient is the end result of a division problem. ... An example of blackboard bold letters. ...

### Real numbers

The real numbers include all of the measuring numbers. Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Following the decimal point, each digit has a place value one-tenth the place value of the digit to its left. Thus Please refer to Real vs. ... For other uses, see Decimal (disambiguation). ... $123.456,$

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six". In, for example, the US and UK, the decimal is represented by a period, in continental Europe by a comma. Zero is often written as 0.0 and negative real numbers are written with a preceding minus sign: Look up period in Wiktionary, the free dictionary. ... The term comma has various uses; comma is the name used for one of the punctuation symbols: , The term comma is also used in music theory for various small intervals that arise as differences between approximately equal intervals. ... The plus (+) and minus (&#8722;) signs are used universally to represent the operations of addition and subtraction, and have been extended to many other meanings, more or less analogous. ... $-123.456,$.

Every rational number is also a real number. To write a fraction as a decimal, divide the numerator by the denominator. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as 1/2 and the real number 0.333... (forever repeating threes) can be written as 1/3. On the other hand, the real number π (pi), the ratio of the circumference of any circle to its diameter, is In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... The circumference is the distance around a closed curve. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... $pi = 3.14159265358979...,$.

Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include $sqrt{2} = 1.41421356237 ...,$

(the square root of 2, that is, the positive number whose square is 2). The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ...

Just as fractions can be written in more than one way, so too can decimals. For example, if we multiply both sides of the equation $1/3 = 0.333...,$

by three, we discover that $1 = 0.999...,$.

Thus 1.0 and 0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so on. In mathematics, the recurring decimal 0. ...

Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. The symbol for the real numbers is R or $mathbb{R}$. A number line, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...

When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.01 meters. If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61. Various meters Measurement is an observation that reduces an uncertainty expressed as a quantity. ... The top portion of this graphic depicts probability densities (for a binomial distribution) that show the relative likelihood that the true percentage is in a particular area given a reported percentage of 50%. The bottom portion of this graphic shows the margin of error, the corresponding zone of 95% confidence. ... Rounding is the process of reducing the number of significant digits in a number. ... In mathematics, truncation is the term used for reducing the number of digits right of the decimal point, by discarding the least significant ones. ... Significant figures (also called significant digits and abbreviated sig figs or sig digs, respectively) is a method of expressing errors in measurements. ... In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ...

In abstract algebra, the real numbers are uniquely characterized by being the only completely ordered field. They are not, however, an algebraically closed field. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ...

### Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from the question of whether a negative number can have a square root. This led to the invention of a new number: the square root of negative one, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... $,a + b i$

where a and b are real numbers. In the expression a + bi, the real number a is called the real part and b is called the imaginary part. If the real part of a complex number is zero, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is zero, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or $mathbb{C}$. â€œSupersetâ€ redirects here. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ...

In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors. Like the real number system, the complex number system is a field and is complete, but unlike the real numbers it is not ordered. That is, there is no meaning in saying that i is greater than 1, nor is there any meaning in saying that that i is less than 1. In technical terms, the complex numbers lack the trichotomy property. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... For other senses of this word, see coefficient (disambiguation). ... In math, see Factorization. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... For the socioeconomic meaning, see social inequality. ...

Complex numbers correspond to points on the complex plane, sometimes called the Argand plane. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, NZQRC. A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X &#8838; Y; Y is a superset of (or includes) X; Y...

### Other types

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields. The superreal numbers compose a more inclusive category than hyperreal number. ... The system of hyperreal numbers represents 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 calculus by Newton and Leibniz. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

The idea behind p-adic numbers is this: While real numbers may have infinitely long expansions to the right of the decimal point, these numbers allow for infinitely long expansions to the left. The number system which results depends on what base is used for the digits: any base is possible, but a system with the best mathematical properties is obtained when the base is a prime number. In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...

For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the collection, while the latter gives its size. For the finite set, the ordinal and cardinal numbers are equivalent, but they differ in the infinite case. In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...

There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, algebraic numbers are the roots of polynomials with rational coefficients. Complex numbers that are not algebraic are called transcendental numbers. In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn&#8722;1 + ··· + an &#8722;1x + an = 0 where n is a positive integer called... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ...

Sets of numbers that are not subsets of the complex numbers include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative. Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as numbers by number theorists. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Sir William Rowan Hamilton (August 4, 1805 â€“ September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... In mathematics, associativity is a property that a binary operation can have. ... In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

## Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral '5' and by the Roman numeral 'V'. Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers. A numeral is a symbol or group of symbols that represents a number. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ...

## History

### History of integers

#### The first numbers

Further information: History of numeral systems

It is speculated that the first known use of numbers dates back to around 30000 BC, bones or other artifacts have been discovered with marks cut into them which are often considered tally marks. The use of these tally marks have been suggested to be anything from counting elapsed time, such as numbers of days, or keeping records of amounts. A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... It has been suggested that this article or section be merged into Unary numeral system. ...

Tallying systems have no concept of place-value (such as in the currently used decimal notation), which limit its representation of large numbers and as such is often considered that this is the first kind of abstract system that would be used, and could be considered a Numeral System.

The first known system with place-value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt.  Originally Ancient Mesopotamian weights and measures came from a collection of city states loosely organized by family, tribe and occupation. ... (35th century BC - 34th century BC - 33rd century BC - other centuries) (5th millennium BC - 4th millennium BC - 3rd millennium BC) Events Stage IIIa2 of the Naqada culture in Egypt (dated in 1998) Significant persons Ur-Nina first king of Lagash in Mesopotamia (c. ... (32nd century BC &#8211; 31st century BC &#8211; 30th century BC &#8211; other centuries) (5th millennium BC &#8211; 4th millennium BC &#8211; 3rd millennium BC) Events 3000 BC &#8211; Menes unifies Upper and Lower Egypt, and a new capital is erected at Memphis. ...

#### History of zero

Further information: History of zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient Indian texts use a Sanskrit word Shunya to refer to the concept of void; in mathematics texts this word would often be used to refer to the number zero. . In a similar vein, Pāṇini (5th century BC) used the null (zero) operator (ie a lambda production) in the Ashtadhyayi, his algebraic grammar for the Sanskrit language. (also see Pingala) For other senses of this word, see zero or 0. ... Positional notation or place-value notation is a numeral system in which each position is related to the next by a constant multiplier called the base (or radix) of that numeral system. ... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... Indian postage stamp depicting (2004), with the implication that he used (à¤ªà¤¾à¤£à¤¿à¤¨à¤¿; IPA ) was an ancient Indian grammarian from Gandhara (traditionally 520â€“460 BC, but estimates range from the 7th to 4th centuries BC). ... The 5th century BC started the first day of 500 BC and ended the last day of 401 BC. // The Parthenon of Athens seen from the hill of the Pnyx to the west. ... The Ashtadhyayi (A&#803;s&#803;t&#257;dhy&#257;y&#299;, meaning eight chapters) is the earliest known grammar of Sanskrit, and one of the first works on descriptive linguistics, generative linguistics, or linguistics altogether. ... In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language â€” that is, of a set of strings. ... Sanskrit ( , for short ) is a classical language of India, a liturgical language of Hinduism, Buddhism, Sikhism, and Jainism, and one of the 23 official languages of India. ... Pingala (à¤ªà¤¿à¤™à¥à¤—à¤² ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ...

Records show that the Ancient Greeks seemed unsure about the status of zero 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 zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned if 1 was a number.) Ancient Greece is a period in Greek history that lasted for around nine hundred years. ... For other uses, see Philosophy (disambiguation). ... Look up Vacuum in Wiktionary, the free dictionary. ... â€œArrow paradoxâ€ redirects here. ... Zeno of Elea (IPA:zÉ›noÊŠ, É›lÉ›É‘Ë)(circa 490 BC? â€“ circa 430 BC?) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. ... This article is about the number one. ...

The late Olmec people of south-central Mexico began to use a true 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, but did not influence Old World numeral systems. Monument 1, one of the four Olmec colossal heads at La Venta. ... The 4th century BC started the first day of 400 BC and ended the last day of 301 BC. It is considered part of the Classical era, epoch, or historical period. ... Centuries: 2nd century BC - 1st century BC - 1st century Decades: 90s BC 80s BC 70s BC 60s BC 50s BC - 40s BC - 30s BC 20s BC 10s BC 0s BC 10s BC Years: 45 BC 44 BC 43 BC 42 BC 41 BC 40 BC 39 BC 38 BC 37... Mayan numerals. ... The Maya calendar is actually a system of distinct calendars and almanacs used by the Maya civilization of pre-Columbian Mesoamerica, and by some modern Maya communities in highland Guatemala. ...

By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (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), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70). For other uses, see number 130. ... This article is about the geographer, mathematician and astronomer Ptolemy. ... For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... â€œByzantineâ€ redirects here. ... The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ... Look up ÎŸ, Î¿ in Wiktionary, the free dictionary. ...

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 zero 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, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol. Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... Events Bernicia settled by the Angles Ethiopia conquers Yemen The Daisan river, a tributary of the Euphrates, floods Edessa and within a couple of hours fills the entire city except for the highest parts. ... Dionysius Exiguus (Dennis the Little, meaning humble) (c. ... Computus (Latin for computation) is the calculation of the date of Easter in the Christian calendar. ... This article is about the Christian festival. ... Bede (IPA: ) (also Saint Bede, the Venerable Bede, or (from Latin) Beda (IPA: )), (ca. ... Events Births Deaths Wihtred, king of Kent Categories: 725 ...

An early documented use of the zero by Brahmagupta (in the Brahmasphutasiddhanta) dates to 628. He treated zero as a number and discussed operations involving it, including division. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world. Brahmagupta (à¤¬à¥à¤°à¤¹à¥à¤®à¤—à¥à¤ªà¥à¤¤) ( ) (589â€“668) was an Indian mathematician and astronomer. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... Events Khusro II of Persia overthrown Pippin of Landen becomes Mayor of the Palace Brahmagupta writes the Brahmasphutasiddhanta Births Deaths Empress Suiko of Japan Theodelinda, queen of the Lombards Categories: 628 ... In mathematics, a division is called a division by zero if the divisor is zero. ... For people named Islam, see Islam (name). ...

#### History of negative numbers

Further information: First usage of negative numbers

The abstract concept of negative numbers was recognised as early as 100 BC - 50 BC. The Chinese Nine Chapters on the Mathematical Art (Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. This is the earliest known mention of negative numbers in the East; the first reference in a western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to 4x + 20 = 0 (the solution would be negative) in Arithmetica, saying that the equation gave an absurd result. A negative number is a number that is less than zero, such as âˆ’3. ... Centuries: 2nd century BC - 1st century BC - 1st century Decades: 130s BC 120s BC 110s BC - 100s BC - 90s BC 80s BC 70s BC 60s BC 50s BC Years: 105 BC 104 BC 103 BC 102 BC 101 BC - 100 BC - 99 BC 98 BC 97 BC 96 BC 95... Centuries: 2nd century BC - 1st century BC - 1st century Decades: 100s BC 90s BC 80s BC 70s BC 60s BC - 50s BC - 40s BC 30s BC 20s BC 10s BC 0s BC Years: 55 BC 54 BC 53 BC 52 BC 51 BC 50 BC 49 BC 48 BC 47... The Nine Chapters on the Mathematical Art (ä¹ç« ç®—è¡“) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later... For other senses of this word, see coefficient (disambiguation). ... The phrase The East has multiple meanings: Eastern society, referring to a specific worldview U.S. Eastern states, East Coast of the United States This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ... // Overview Events 212: Constitutio Antoniniana grants citizenship to all free Roman men 212-216: Baths of Caracalla 230-232: Sassanid dynasty of Persia launches a war to reconquer lost lands in the Roman east 235-284: Crisis of the Third Century shakes Roman Empire 250-538: Kofun era, the first... Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de MÃ©ziriac. ... Arithmetica, an ancient text on mathematics written by classical period Greek mathematician Diophantus in the second century AD is a collection of 130 algebra problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. ...

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 Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce 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." Centuries: 6th century 7th century 8th century Decades: 550s - 560s - 570s - 580s - 590s - 600s - 610s - 620s - 630s - 640s - 650s Years: 600 601 602 603 604 605 606 607 608 609 World population grows to about 208 million. ... Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de MÃ©ziriac. ... Brahmagupta (à¤¬à¥à¤°à¤¹à¥à¤®à¤—à¥à¤ªà¥à¤¤) ( ) (589â€“668) was an Indian mathematician and astronomer. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... Events Khusro II of Persia overthrown Pippin of Landen becomes Mayor of the Palace Brahmagupta writes the Brahmasphutasiddhanta Births Deaths Empress Suiko of Japan Theodelinda, queen of the Lombards Categories: 628 ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara AchÄrya (Bhaskara the teacher), was an Indian mathematician-astronomer. ...

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 debits (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral[citation needed]. The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”. For other uses, see Europe (disambiguation). ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... Drawing of Leonardo Pisano Leonardo of Pisa or Leonardo Pisano (Pisa, c. ... Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. ... // Events August 1 - Arthur of Brittany captured in Mirebeau, north of Poitiers Beginning of the Fourth Crusade. ... Nicolas Chuquet (born 1445 (some sources say c. ... (14th century - 15th century - 16th century - other centuries) As a means of recording the passage of time, the 15th century was that century which lasted from 1401 to 1500. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity[citation needed], and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a cartesian coordinate system. (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... The infinity symbol âˆž in several typefaces. ... René Descartes René Descartes (IPA: , March 31, 1596 &#8211; February 11, 1650), also known as Cartesius, worked as a philosopher and mathematician. ... Fig. ...

### History of rational, irrational, and real numbers

Further information: History of irrational numbers and History of pi

In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... The mathematical constant Ï€ = 3. ...

#### History of rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. Even the Ancient Egyptians wrote math texts describing how to convert general fractions into their special notation. Classical Greek and Indian mathematicians made studies of 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. Prehistory (Greek words &#960;&#961;&#959; = before and &#953;&#963;&#964;&#959;&#961;&#943;&#945; = history) is the period of human history prior to the advent of writing (which marks the beginning of recorded history). ... Map of Ancient Egypt Ancient Egypt was the civilization of the Nile Valley between about 3000 BC and the conquest of Egypt by Alexander the Great in 332 BC. As a civilization based on irrigation it is the quintessential example of an hydraulic empire. ... In arithmetic, a vulgar fraction (or common fraction) consists of one integer divided by a non-zero integer. ... An Egyptian fraction is a sum of distinct unit fractions, i. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 350s BC 340s BC 330s BC 320s BC 310s BC - 300s BC - 290s BC 280s BC 270s BC 260s BC 250s BC Years: 305 BC 304 BC 303 BC 302 BC 301 BC - 300 BC - 299 BC 298 BC... // As per the ÅšvetÄmbara belief, Sthananga Sutra forms part of the first eleven Angas of the Jaina Canon which have survived despite the bad effects of this Hundavasarpini kala. ...

The concept of decimal fractions is closely linked with decimal place value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutras to include calculations of decimal-fraction approximations to pi or the square root of two. Similarly, Babylonian math texts had always used sexagesimal fractions with great frequency. Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and &#8722; (minus... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... The square root of two is the positive real number which, when multiplied by itself, gives a product of two. ...

#### History of irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC. [citation needed] The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to 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 his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning. The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... Centuries: 10th century BC - 9th century BC - 8th century BC Decades: 850s BC 840s BC 830s BC 820s BC 810s BC - 800s BC - 790s BC 780s BC 770s BC 760s BC 750s BC Events and Trends 804 BC - Hadad-nirari IV of Assyria conquers Damascus. ... Centuries: 7th century BC - 6th century BC - 5th century BC Decades: 550s BC - 540s BC - 530s BC - 520s BC - 510s BC - 500s BC - 490s BC - 480s BC - 470s BC - 460s BC - 450s BC Events and Trends 509 BC - Foundation of the Roman Republic 508 BC - Office of pontifex maximus created... Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 and 490 BC) was an Ionian (Greek) philosopher and founder of the religious movement called Pythagoreanism. ... The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. ... Hippasus of Metapontum, born circa 500 B.C. in Magna Graecia, was a Greek philosopher. ... The square root of two is the positive real number which, when multiplied by itself, gives a product of two. ... Pythagoras of Samos (Greek: ; between 580 and 572 BCâ€“between 500 and 490 BC) was an Ionian (Greek) philosopher and founder of the religious movement called Pythagoreanism. ...

The sixteenth century saw the final acceptance by Europeans of negative, integral and fractional numbers. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. But it was not until the nineteenth century that the irrationals were separated into algebraic and transcendental parts, and a scientific study of theory of irrationals was taken once more. It had remained almost dormant since Euclid. The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and the recent endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on 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 (Crelle, 101), and Méray. A negative number is a number that is less than zero, such as âˆ’3. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... For other uses, see Euclid (disambiguation). ... Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Kossak is a surname of several people: Jerzy Kossak, a Polish painter Juliusz Kossak, a Polish painter Wojciech Kossak, a Polish painter Zofia Kossak-Szczucka, Polish author and resistance fighter during the Second World War Categories: Disambiguation ... Heine is a German family name. ... Crelles Journal, or just Crelle, is the common name for the Journal für die reine und angewandte Mathematik founded by August Leopold Crelle. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 â€“ January 6, 1918) was a German mathematician. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Heine is a German family name. ... Salvatore Pincherle (February 11, 1853 â€” July 19, 1936) was an Italian mathematician. ... Paul Tannery (1843â€”1904) was a French mathematician and historian of mathematics. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x â‰¤ a implies that x is in A as well) and B is closed upwards... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ...

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 nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject. In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... The name Kunze can refer to more than one person: John Christopher Kunze ( 1744 â€“ 1807 ), Lutheran theologian Michael Kunze, a German author, lyricist and librettist Michael Kunze, a system architect Category: ... Lemke is a surname, and may refer to Birsel Lemke Jay Lemke Leslie Lemke Mark Lemke Steve Lemke William Lemke Wolf Lemke Categories: | ... The Germanic first name GÃ¼nther, GÃ¼nter, Gunther or Guenther, also Gunthar, refers to various medieval persons, including: Gunther, legendary 5th century king of the Burgundians Blessed Gunther, a Bohemian hermit Gunther of Cologne As a family name, spelled GÃ¼nther: Albert C. L. G. GÃ¼nther (1830â€“1914... Ramus can refer to: Petrus Ramus A portion of a bone, as in the Ramus mandibulÃ¦ This is a disambiguation page: a list of articles associated with the same title. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... August Ferdinand MÃ¶bius. ... The Germanic first name GÃ¼nther, GÃ¼nter, Gunther or Guenther, also Gunthar, refers to various medieval persons, including: Gunther, legendary 5th century king of the Burgundians Blessed Gunther, a Bohemian hermit Gunther of Cologne As a family name, spelled GÃ¼nther: Albert C. L. G. GÃ¼nther (1830â€“1914...

#### Transcendental numbers and reals

The first results concerning transcendental numbers were Lambert's 1761 proof that π cannot be rational, and also that en is irrational if n is rational (unless n = 0). (The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to showed that π is not the square root of a rational number. 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 (formula involving only arithmetical operations and roots). Hence it 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. Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 &#8211; September 25, 1777), was a mathematician, physicist and astronomer. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... For other people with the same name, see John Napier (disambiguation). ... In mathematics, if two variables of bn = x are known, the third can be found. ... Adrien-Marie Legendre (September 18, 1752&#8211;January 10, 1833) was a French mathematician. ... Graph of a polynomial of degree 5, with 4 critical points. ... The Abelâ€“Ruffini theorem (also known as Abels Impossibility Theorem) states that there is no general solution in radicals to polynomial equations of degree five or higher. ... Paolo Ruffini (Valentano, 1765 â€“ Modena, 1822) was an Italian mathematician and philosopher. ... Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ... In mathematics, an nth root of a number a is a number b, such that bn=a. ... In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn&#8722;1 + ··· + an &#8722;1x + an = 0 where n is a positive integer called... Galois at the age of fifteen from the pencil of a classmate. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group theory. ...

Even the set of algebraic numbers was not sufficient and the full set of real number includes transcendental numbers. The existence of which was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally Cantor shows 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. In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... Charles Hermite (pronounced in IPA, ) (December 24, 1822 â€“ January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that &#960; is a transcendental number, i. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 â€“ January 6, 1918) was a German mathematician. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, an uncountable set is a set which is not countable. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics the term countable set is used to describe the size of a set, e. ...

### Infinity

Further information: History of infinity

The earliest known conception of mathematical infinity appears in the Yajur Veda, which at one point states "if you remove 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 circa 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The infinity symbol âˆž in several typefaces. ... The infinity symbol âˆž in several typefaces. ... The Yajur Veda &#2351;&#2332;&#2369;&#2352;&#2381;&#2357;&#2375;&#2342; is one of the four Hindu Vedas; it contains religious texts focussing on liturgy and ritual. ... JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ... The Celtics claim Vienna, Austria. ...

In the West, the traditional notion of mathematical infinity was defined by Aristotle, who distinguished between actual infinity and potential infinity; the general consensus being that only the latter had true value. Galileo's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, the continuum hypothesis. Aristotle (Greek: AristotÃ©lÄ“s) (384 BC â€“ 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Galileo can refer to: Galileo Galilei, astronomer, philosopher, and physicist (1564 - 1642) the Galileo spacecraft, a NASA space probe that visited Jupiter and its moons the Galileo positioning system Life of Galileo, a play by Bertolt Brecht Galileo (1975) - screen adaptation of the play Life of Galileo by Bertolt Brecht... The Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) was Galileos final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years. ... A bijective function. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 â€“ January 6, 1918) was a German mathematician. ... Year 1895 (MDCCCXCV) was a common year starting on Tuesday (link will display full calendar) of the Gregorian calendar (or a common year starting on Sunday of the 12-day-slower Julian calendar). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

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 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. Projective geometry is a non-metrical form of geometry. ... A cube in two-point perspective. ...

### Complex numbers

Further information: History of complex numbers

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 (see Niccolo Fontana Tartaglia, 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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Heros aeolipile Hero (or Heron) of Alexandria (c. ... The 1st century was that century that lasted from 1 to 100 according the Gregorian calendar. ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ... For other meanings, see pyramid (disambiguation). ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Niccolo Fontana Tartaglia. ... Gerolamo Cardano. ...

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation â€œDescartesâ€ redirects here. ... Events February 3 - Tulipmania collapses in Netherlands by government order February 15 - Ferdinand III becomes Holy Roman Emperor December 17 - Shimabara Rebellion erupts in Japan Pierre de Fermat makes a marginal claim to have proof of what would become known as Fermats last theorem. ... $sqrt{-1}^2=sqrt{-1}sqrt{-1}=-1$

seemed to be capriciously inconsistent with the algebraic identity $sqrt{a}sqrt{b}=sqrt{ab}$,

which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity $frac{1}{sqrt{a}}=sqrt{frac{1}{a}}$

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 to guard against this mistake. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula: (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Abraham de Moivre. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ... $(cos theta + isin theta)^{n} = cos n theta + isin n theta ,$

and to Euler (1748) Euler's formula of complex analysis: Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... $cos theta + isin theta = e ^{itheta }. ,$

The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received 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. Caspar Wessel (June 8, 1745 - March 25, 1818) was a Norwegian-Danish mathematician. ... 1799 was a common year starting on Tuesday (see link for calendar). ... Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ...

Also in 1799, 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. The general acceptance of the theory of complex numbers is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known. In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ... Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ...

Gauss studied complex numbers of the form a + bi, where a and b are integral, or rational (and i is one of the two roots of x2 + 1 = 0). His student, Ferdinand Eisenstein, studied the type a + , where ω is a complex root of x3 − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity xk − 1 = 0 for higher values of k. This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation F(x) = 0. Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ... Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ... In mathematics, the nth roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekinds definition of ideals for rings. ... Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... Galois at the age of fifteen from the pencil of a classmate. ...

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 would eventually lead to the concept of the extended complex plane. For the game, see: 1850 (board game) 1850 (MDCCCL) was a common year starting on Tuesday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Sunday  of the 12-day-slower Julian calendar). ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...

### Prime numbers

Prime numbers have been studied throughout recorded history. Euclid devoted one book of 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 mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

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. Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 290s BC 280s BC 270s BC 260s BC 250s BC - 240s BC - 230s BC 220s BC 210s BC 200s BC 190s BC Years: 245 BC 244 BC 243 BC 242 BC 241 BC - 240 BC - 239 BC 238 BC... Eratosthenes (Greek ; 276 BC - 194 BC) was a Greek mathematician, geographer and astronomer. ... In mathematics, the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. ... The Renaissance (French for rebirth, or Rinascimento in Italian), was a cultural movement in Italy (and in Europe in general) that began in the late Middle Ages, and spanned roughly the 14th through the 17th century. ...

In 1796, Adrien-Marie Legendre conjectured 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 was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Year 1796 (MDCCXCVI) was a leap year starting on Friday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Monday of the 11-day slower Julian calendar). ... Adrien-Marie Legendre (September 18, 1752 â€“ January 10, 1833) was a French mathematician. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... In mathematics, Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... Unsolved problems in mathematics: Does every non-trivial zero of the Riemann zeta function have real part Â½? The Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in mathematics. ... Bernhard Riemann. ... Year 1859 (MDCCCLIX) was a common year starting on Saturday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Thursday of the 12-day slower Julian calendar). ... This page is a candidate for speedy deletion. ... Charles-Jean de la VallÃ©e-Poussin (August 14, 1866 - March 2, 1962) was a Belgian mathematician. ... Year 1896 (MDCCCXCVI) was a leap year starting on Wednesday (link will display calendar). ... Results from FactBites:

 What's Special About This Number? (7286 words) is the number of planar partitions of 10. is the number of planar partitions of 11. is the number of planar partitions of 12.
 Number - Wikipedia, the free encyclopedia (3652 words) The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields. This number is denoted by i, a symbol assigned by Leonhard Euler. The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion.
More results at FactBites »

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