A number is an abstract object, tokens of which are symbols used in counting and measuring. For other uses see Abstract In Philosophy it is commonly considered that every object is either abstract or concrete The type versus token distinction separates an abstract concept from the objects which are particular instances of the concept The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece 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 Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as A symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol. In Linguistics, a number name, or numeral, is a symbol or group of symbols or a Word in a Natural language that represents a Number 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). A telephone number or phone number is a sequence of numbers used to call from one Telephone line to another in a Telephone network. A serial number is a unique Number assigned for Identification which varies from its Successor or Predecessor by a fixed discrete Integer 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A negative number is a Number that is less than zero, such as −2 In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted As a result, there is no one encompassing definition of number and the concept of number is open for further development.

Certain procedures which input one or more numbers and output a number are called numerical operations. In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values Unary operations input a single number and output a single number. In Mathematics, a unary operation is an operation with only one Operand, i 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. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. The study of numerical operations is called arithmetic. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone

The branch of mathematics that studies structures of number systems such as groups, rings and fields is called abstract algebra. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules

Contents 1 Types of numbers 1.1 Natural numbers 1.2 Integers 1.3 Rational numbers 1.4 Real numbers 1.5 Complex numbers 1.6 Computable numbers 1.7 Other types 2 Numerals 3 History 3.1 History of integers 3.1.1 The first use of numbers 3.1.2 History of zero 3.1.3 History of negative numbers 3.2 History of rational, irrational, and real numbers 3.2.1 History of rational numbers 3.2.2 History of irrational numbers 3.2.3 Transcendental numbers and reals 3.3 Infinity 3.4 Complex numbers 3.5 Prime numbers 4 References 5 See also 6 External links

Types of numbers

Numbers can be classified into sets, called number systems. In Mathematics, a number system is a set of Numbers (in the broadest sense of the word together with one or more operations such as Addition (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems. Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner )

Natural numbers

The most familiar numbers are the natural numbers or counting numbers: one, two, three, . In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an . . . Some people also include zero in the natural numbers; however, others do not.

In the base ten number system, in almost universal use today for arithmetic operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. In Mathematics and Computer science, a digit is a symbol (a number symbol e 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 .

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets. Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols 3 times.

Integers

Negative numbers are numbers that are less than zero. A negative number is a Number that is less than zero, such as −2 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 (also called a minus sign) in front of the number they are the opposite of. Thus the opposite of 7 is written −7. When the set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers, also called integers, Z (German Zahl, plural Zahlen), also written .

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 integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object Numerator may refer to A numeral used to indicate a count particularly of the equal parts in a fraction For example in 3/4 3 is the numerator The fraction m/n or

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:

.

If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers 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 .

Real numbers

The real numbers include all of the measuring numbers. In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical In Mathematics, the real numbers may be described informally in several different ways 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. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus

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 the US and UK and a number of other countries, the decimal point is represented by a period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. A full stop or period (sometimes stop, full point, decimal point, or dot) is the Punctuation mark commonly placed at the A comma ( ,   is a Punctuation mark It has the same shape as an Apostrophe or single closing Quotation mark in many typefaces but it differs Zero is often written as 0. 0 when necessary to indicate that it is to be treated as a real number rather than as an integer. Negative real numbers are written with a preceding minus sign:

. The plus and minus signs ( + and &minus) are Mathematical symbols used to represent the notions of positive and negative as well as the operations

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. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction 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

. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the

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

(the square root of 2, that is, the positive number whose square is 2). The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2

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

by three, we discover that

.

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.

Every real number is either rational or irrational. Every real number corresponds to a point on the number line. In mathematics a number line is a picture of a straight line in which the Integers are shown as specially-marked points evenly spaced on the 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 .

When a real number represents a measurement, there is always a margin of error. Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as The margin of error is a statistic expressing the amount of random Sampling error in a survey 's results This is often indicated by rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. For lip-rounding in phonetics see Labialisation and Roundedness. In Mathematics, truncation is the term for limiting the number of digits right of the Decimal point, by discarding the least significant ones The remaining digits are called significant digits. The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy 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. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as 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.

In abstract algebra, the real numbers are up to isomorphism uniquely characterized by being the only complete ordered field. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In the mathematical area of Order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations They are not, however, an algebraically closed field. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients

Complex numbers

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This set of numbers arose, historically, from the question of whether a negative number can have a square root. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose 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. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation The complex numbers consist of all numbers of the form

where a and b are real numbers. In the expression a + bi, the real number a is called the real part and bi 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. Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane 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. A Gaussian integer is a Complex number whose real and imaginary part are both Integers The Gaussian integers with ordinary addition and multiplication of complex The symbol for the complex numbers is C or .

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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a coefficient is a Constant multiplicative factor of a certain object In Mathematics, factorization ( also factorisation in British English) or factoring is the decomposition of an object (for Like the real number system, the complex number system is a field and is complete, but unlike the real numbers it is not ordered. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In the mathematical area of Order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation 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. In Mathematics, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not (See also equality

Complex numbers correspond to points on the complex plane, sometimes called the Argand plane. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, N ⊂ Z ⊂ Q ⊂ R ⊂ C.

Computable numbers

Moving to problems of computation, the computable numbers are determined in the set of the real numbers. In Mathematics, Theoretical computer science and Mathematical logic, the computable numbers, also known as the recursive numbers or the The computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes. In Mathematics, a real closed field is a field F in which any of the following equivalent conditions are true There is a Total order

Other types

Hyperreal and hypercomplex numbers are used in non-standard analysis. Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Mathematics, the real numbers may be described informally in several different ways In mathematics the transfer principle is a concept in Abraham Robinson 's Non-standard analysis of the Hyperreal numbers. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields. The superreal numbers are an extension of the Real numbers, similar to the Surreal numbers or Hyperreal numbers but comprising a more inclusive category In Mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

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. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 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 mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. 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.

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. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a coefficient is a Constant multiplicative factor of a certain object Complex numbers that are not algebraic are called transcendental numbers. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation

Sets of numbers that are not subsets of the complex numbers are sometimes called hypercomplex numbers. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic They 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. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Mathematics, associativity is a property that a Binary operation can have Elements of function fields of non-zero characteristic behave in some ways like numbers and are often regarded as numbers by number theorists. In Algebraic geometry, the function field of an Algebraic variety V consists of objects which are interpreted as rational functions on V. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

In addition, various specific kinds of numbers are studied in sets of natural and integer numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

An even number is an integer that is "evenly divisible" by 2, i. e. , divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible". In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without ) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

A perfect number is defined as a positive integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number itself. A negative number is a Number that is less than zero, such as −2 In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. In Mathematics, and specifically in Number theory, a divisor function is an Arithmetical function related to the Divisors of an Integer The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. In mathematics Six is the second smallest Composite number, its proper Divisors being 1, 2 and 3. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. 28 ( twenty-eight) is the Natural number following 27 and preceding 29. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS). Four hundred ninety-six is the Natural number following four hundred ninety-five and preceding four hundred ninety-seven See also 8128 Nicomachus The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences These first four perfect numbers were the only ones known to early Greek mathematics. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century

A figurate number is a number that can be represented as a regular and discrete geometric pattern (e. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations In Mathematics, a polygonal number is a Number that can be arranged as a regular Polygon. Polytopic numbers for r = 2, 3, and 4 are:

• P₂(n) = 1/2 n(n + 1) (triangular numbers)
• P₃(n) = 1/6 n(n + 1)(n + 2) (tetrahedral numbers)
• P₄(n) = 1/24 n(n + 1)(n + 2)(n + 3) (pentatopic numbers)

Numerals

Numbers should be distinguished from numerals, the symbols used to represent numbers. A triangular number is the sum of the n Natural numbers from 1 to n. A tetrahedral number, or triangular pyramidal number, is a Figurate number that represents a Pyramid with a triangular base and three sides called a A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left In Linguistics, a number name, or numeral, is a symbol or group of symbols or a Word in a Natural language that represents a Number The number five can be represented by both the base ten numeral '5' and by the Roman numeral 'V'. Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals. Notations used to represent numbers are discussed in the article numeral systems. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner 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.

History

History of integers

The first use of numbers

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. Tally marks are an implementation of the Unary numeral system. 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.

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. Ancient Mesopotamian units of measurement originated in the loosely organized city-states of Early Dynastic Sumer. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt. Cultures c 3400 BC — Sumerian temple record keepers redesign the Stamp seal in the form of a cylinder Events c 3100 BC — Narmer (Menes unifies Upper and Lower Egypt into one country he rules this new country from Memphis This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. [1]

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. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a 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. Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical [2]. 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. Pāṇini ( IAST: Pāṇini Dēvanāgarī: sa पाणिनि a Patronymic meaning "descendant of {{IAST|Paṇi}} " was an ancient The 5th century BC started the first day of 500 BC and ended the last day of 401 BC. Pāṇini ( IAST: Pāṇini Dēvanāgarī: sa पाणिनि a Patronymic meaning "descendant of {{IAST|Paṇi}} " was an ancient In Formal semantics, Computer science and Linguistics, a formal grammar (also called formation rules) is a precise description of a Formal Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical (also see Pingala)

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. Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language This vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. Zeno of Elea (ˈziːnoʊ əv ˈɛliə Greek: Ζήνων ὁ Ἐλεάτης (ca (The ancient Greeks even questioned if 1 was a number. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity )

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. The Olmec were an ancient Pre-Columbian people living in the Tropical lowlands of south-central Mexico, in what are roughly the modern-day states The United Mexican States ( or commonly Mexico (ˈmɛksɪkoʊ () is a federal constitutional Republic in North America. The 4th century BC started the first day of 400 BC and ended the last day of 301 BC. Year 40 BC was a Common year starting on Friday (link will display the full calendar of the Julian calendar. The Pre-Columbian Maya civilization used a Vigesimal ( base - twenty) Numeral system. The Maya calendar is a system of distinct Calendars and Almanacs used by the Maya civilization of Pre-Columbian Mesoamerica, and by

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. Claudius Ptolemaeus ( Greek: Klaúdios Ptolemaîos; after 83 &ndash ca Hipparchus ( Greek; ca 190 BC &ndash ca 120 BC was a Greek Astronomer, Geographer, and Mathematician of the Hellenistic ʹ the numeral sign redirects here For the accent ´ see Acute accent. 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. ʹ the numeral sign redirects here For the accent ´ see Acute accent. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70). The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early Omicron or Omikron (uppercase Ο, lowercase ο, literally "small o": Όμικρον o mikron, micron meaning 'small' in contrast

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. Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals. Events Dionysius Exiguus proposes a calendar based on the birth of Jesus Christ Dionysius Exiguus ( Dennis the Little or Dennis the Short, meaning humble (c 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). Computus ( Latin for Computation) is the Calculation of the date of Easter in the Christian calendar. Easter ( Greek: Πάσχα Pascha or Pasxa) is the most important religious feast in the Christian Liturgical year. 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. Bede (ˈbiːd (also Saint Bede, the Venerable Bede, or (from Latin Beda (beda (c Events By Place Europe Bede publishes On the reckoning of time ( De temporum ratione) calculating dates

An early documented use of the zero by Brahmagupta (in the Brahmasphutasiddhanta) dates to 628. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including Events By Place Europe Pippin of Landen becomes Mayor of the Palace in Austrasia. He treated zero as a number and discussed operations involving it, including division. In By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world. The Kingdom of Cambodia ( formerly known as Kampuchea (, transliterated: Preăh Réachéanachâkr Kâmpŭchea) is a country in South East China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National For other meanings including people named 'Islam' see Islam (disambiguation.

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. A negative number is a Number that is less than zero, such as −2 Year 50 BC was a year of the pre-Julian calendar. Events By place Rome Consuls Lucius Aemilius Paullus and 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. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and In Mathematics, a coefficient is a Constant multiplicative factor of a certain object 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. The 3rd century is the period from 201 to 300 in accordance with the Julian calendar in the Christian / Common Era. Greece (Ελλάδα transliterated: Elláda, historically, Ellás,) officially the Hellenic Republic (Ελληνική Δημοκρατία 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. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should Arithmetica is an ancient Greek text on Mathematics written by the Mathematician Diophantus in the 3rd century CE.

During the 600s, negative numbers were in use in India to represent debts. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country 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. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including Events By Place Europe Pippin of Landen becomes Mayor of the Palace in Austrasia. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. 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 debits (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). As a means of recording the passage of Time, the 17th Century was that Century which lasted from 1601 - 1700 in the Gregorian calendar Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci Liber Abaci (1202 also spelled as Liber Abbaci) is an historic book on Arithmetic by Leonardo of Pisa known later by his nickname Fibonacci 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. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National The first use of negative numbers in a European work was by Chuquet during the 15th century. Nicolas Chuquet (1445 but some sources say c 1455 &ndash 1488 some sources say c He used them as exponents, but referred to them as “absurd numbers”.

As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity, 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. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system Switzerland (English pronunciation; Schweiz Swiss German: Schwyz or Schwiiz Suisse Svizzera Svizra officially the Swiss Confederation Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

History of rational, irrational, and real numbers

Further information: History of irrational numbers and History of pi

History of rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems Stone Age Paleolithic See also Paleolithic, Recent African Origin, Early Homo sapiens, Early human migrations "Paleolithic" Even the Ancient Egyptians wrote math texts describing how to convert general fractions into their special notation. Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object An Egyptian fraction is the sum of distinct Unit fractions such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16} Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The best known of these is Euclid's Elements, dating to roughly 300 BC. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek Events By place Egypt Pyrrhus, the King of Epirus, is taken as a hostage to Egypt after the Battle of Ipsus Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics. Introduction As per the Śvetāmbara belief Sthananga Sutra forms part of the first eleven Angas of the Jaina Canon which have survived despite the bad effects

The concept of decimal fractions is closely linked with decimal place value notation; the two seem to have developed in tandem. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 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. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 Similarly, Babylonian math texts had always used sexagesimal fractions with great frequency.

History of irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the 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. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced Hippasus of Metapontum (Ίππασος b c 500 BC in Magna Graecia, was a Greek Philosopher. The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √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. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. 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 sixteenth century saw the final acceptance by Europeans of negative, integral and fractional numbers. A negative number is a Number that is less than zero, such as −2 In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object 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. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry 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. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Kossak is the surname of several persons Juliusz Kossak, Polish painter (1824–99 Wojciech Kossak, Polish painter (1857–1942 Heinrich Eduard Heine ( March 15 1821 &ndash October 21, 1881) was a German mathematician. Crelle's Journal, or just Crelle, is the common name for a leading German -language Mathematical journal, the Journal für Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to the year 1872. Heinrich Eduard Heine ( March 15 1821 &ndash October 21, 1881) was a German mathematician. 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). Salvatore Pincherle ( March 11, 1853 &mdash July 10, 1936) was an Italian Mathematician. Paul Tannery (1843–1904 was a French mathematician and historian of mathematics. 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. In Mathematics, a Dedekind cut, named after Richard Dedekind, in a Totally ordered set S is a partition of it into two non-empty In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued

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. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Kunz, Künz, or Kunze is a Surname of Germanic origin and may refer to Alfred Kunz, (1931 - 1998 American murdered Lemke is a surname and may refer to Birsel Lemke Jay Lemke Leslie Lemke Lev Lemke Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

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). Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and (The constant e was first referred to in Napier's 1618 work on logarithms. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line For other people with the same name see John Napier (disambiguation. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce ) Legendre extended this proof to showed that π is not the square root of a rational number. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. 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). In Mathematics, a quintic equation is a Polynomial Equation of degree five The Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no general solution in radicals to Polynomial equations of Paolo Ruffini ( September 22, 1765 – May 9, 1822) was an Italian Mathematician and Philosopher. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation In Mathematics, an n th root of a Number a is a number b such that bn = a. Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory

Even the set of algebraic numbers was not sufficient and the full set of real number includes transcendental numbers. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation The existence of which was first established by Liouville (1844, 1851). Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician. Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Charles Hermite (ʃaʁl ɛʁˈmit ( December 24, 1822 &ndash January 14, 1901) was a French Mathematician who did The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Carl Louis Ferdinand von Lindemann ( April 12, 1852 &ndash March 6 1939) was a German Mathematician, noted for his proof 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. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or

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 (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The Yajurveda ( Sanskrit यजुर्वेदः, a Tatpurusha compound of yajus "sacrificial formula' + veda Infinity was a popular topic of philosophical study among the Jain mathematicians circa 400 BC. Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. Events By place Persian Empire Artaxerxes II King of Persia appoints Tissaphernes to take over all the districts in They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

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. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. In metaphysics, Aristotle distinguished between actual and potential infinities. In metaphysics, Aristotle distinguished between actual and potential infinities. Galileo's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher The Discourses and Mathematical Demonstrations Relating to Two New Sciences ( Discorsi e dimostrazioni matematiche intorno a due nuove scienze, 1638 was Galileo's In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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, transfinite numbers and formulating the continuum hypothesis. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Year 1895 ( MDCCCXCV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar (or a Common year Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite This was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers.

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. Abraham Robinson ( October 6, 1918 &ndash April 11, 1974) was a Mathematician who is most widely known for development of Non-standard 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. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements

A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity," one for each spatial direction. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. 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. Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Hero (or Heron) of Alexandria ( Ήρων ο Αλεξανδρεύς) (c The 1st century was the Century that lasted from 1 to 100 according the Julian calendar. Elements special cases and related concepts Each plane section is a base of the frustum A pyramid is a Building where the upper surfaces are triangular and converge on one point 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). Niccolò Fontana Tartaglia (1499/1500 Brescia, Italy &ndash December 13, 1557, Venice, Italy was a Mathematician 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 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). Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane A further source of confusion was that the equation

seemed to be capriciously inconsistent with the algebraic identity

,

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

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.

The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system "Moivre" redirects here for the French commune see Moivre Marne. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula:

and to Euler (1748) Euler's formula of complex analysis:

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. De Moivre's formula, named after Abraham de Moivre, states that for any Complex number (and in particular for any Real number) x and any This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Caspar Wessel ( June 8, 1745 - March 25, 1818) was a Danish-Norwegian Mathematician. Year 1799 ( MDCCXCIX) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus. John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the

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. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at 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. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation

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). Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German A Gaussian integer is a Complex number whose real and imaginary part are both Integers The Gaussian integers with ordinary addition and multiplication of complex His student, Ferdinand Eisenstein, studied the type a + bω, where ω is a complex root of x³ − 1 = 0. Ferdinand Gotthold Max Eisenstein ( 16 April, 1823 – 11 October, 1852) was a German Mathematician. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity x^k − 1 = 0 for higher values of k. In Number theory, a cyclotomic field is a Number field obtained by adjoining a complex Root of unity to Q, the field of Rational numbers In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) 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 Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group 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.

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 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 In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that

Prime numbers

Prime numbers have been studied throughout recorded history. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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 Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. Events By place Carthage Two of Carthage 's Mercenary commanders — Spendius and Mathos — convince the Eratosthenes of Cyrene ( Greek; 276 BC - 194 BC was a Greek Mathematician, Poet, athlete, Geographer and In Mathematics, the Sieve of Eratosthenes is a simple ancient Algorithm for finding all Prime numbers up to a specified integer But most further development of the theory of primes in Europe dates to the Renaissance and later eras. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere

In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Year 1796 ( MDCCXCVI) was a Leap year starting on Friday (link will display the full calendar of the Gregorian calendar (or a Leap year Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. 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. Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis, formulated by Bernhard Riemann in 1859. 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 Year 1859 ( MDCCCLIX) was a Common year starting on Saturday (link will display the full calendar of the Gregorian calendar (or a Common The prime number theorem was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Jacques Salomon Hadamard ( December 8, 1865 – October 17, 1963) was a French Mathematician best known for his proof of Charles-Jean Étienne Gustave Nicolas Baron de la Vallée Poussin ( August 14[[ 866]] - March 2[[ 962]] was a Belgian Mathematician. Year 1896 ( MDCCCXCVI) was a Leap year starting on Wednesday (link will display the full calendar of the Gregorian Calendar (or a Leap year The conjectures of Goldbach and Riemann yet remain to be proved or refuted.

References

• Erich Friedman, What's special about this number?
• Steven Galovich, Introduction to Mathematical Structures, Harcourt Brace Javanovich, 23 January 1989, ISBN 0-15-543468-3. In Mathematics, a structure on a set, or more generally a type, consists of additional Mathematical objects that in some manner attach to the Events 393 - Roman Emperor Theodosius I proclaims his nine year old son Honorius co-emperor Year 1989 ( MCMLXXXIX) was a Common year starting on Sunday (link displays 1989 Gregorian calendar)
• Paul Halmos, Naive Set Theory, Springer, 1974, ISBN 0-387-90092-6. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician
• Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972. Morris Kline ( May 1, 1908 – June 10, 1992) was a Professor of Mathematics, a writer on the history, philosophy
• Whitehead and Russell, Principia Mathematica to *56, Cambridge University Press, 1910. The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell
• What's a Number? at cut-the-knot

See also

External links

• Mesopotamian and Germanic numbers
• BBC Radio 4, In Our Time: Negative Numbers
• '4000 Years of Numbers', lecture by Robin Wilson, 07/11/07, Gresham College (available for download as MP3 or MP4, and as a text file). Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. The system of Hebrew numerals is a quasi-decimal alphabetic Numeral system using the letters of the Hebrew alphabet. The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century In Mathematics, the parity of an object states whether it is even or odd In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. This is a list of articles about Numbers ( not about Numerals. The following tables list the names and symbols for the numbers 0 through 10 in various Languages and scripts of the world A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. A mythical number is a Number used and accepted as deriving from scientific investigation and/or careful selection but whose origin is unknown and whose basis is A negative number is a Number that is less than zero, such as −2 This list compares various sizes of positive Numbers including counts of things Dimensionless quantity and probabilities. A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Within a set of positive numbers a number is small if it is close to zero Subitizing, coined in 1949 by EL Kaufman et al refers to the rapid accurate and confident judgments of Number performed for small numbers of items Number sign is a name for the symbol #; it is the preferred Unicode name for the Code point associated with that Glyph. The Numero sign (U+2116 or Number sign is used in many languages to indicate ordinal numeration especially in names and titles for example instead of writing the long " IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems Die Grundlagen der Arithmetik (The Foundations of Arithmetic is a book by Gottlob Frege, published in 1884 in which he investigates the philosophical foundations Gresham College is an unusual institution of higher learning off Holborn in central London.

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