This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (July 2009)

Mathematics portal

1,000,000,000 (one billion, short scale; one thousand million or milliard, long scale) is the natural number following 999,999,999 and preceding 1,000,000,001.

In scientific notation, it is written as 10⁹.

A billion traditionally equated to one million, million, (1,000,000,000,000) though the short scale (American English) billion of merely one thousand million (1,000,000,000) has been adopted in finance and increasingly in other fields. The unambiguous term thousand million has the same meaning in both scales.

In South Asian English, it is known as 100 crore or 1 arab.

The term milliard can also be used to refer to 1,000,000,000; this name very often appears in other languages.

This terminology is rarely used in English outside of British English. The SI prefix giga indicates 1,000,000,000 times the base unit.

See Orders of magnitude (numbers) for larger numbers; and long and short scales.

Selected 10-digit numbers (1,000,000,000–9,999,999,999)[link]

Visualisation of powers of ten from one to 1 billion

• 1023456789 – smallest pandigital number in base 10
• 1026753849 – smallest pandigital square that includes 0
• 1073676287 – 15th Carol number
• 1073741824 = 2³⁰
• 1073807359 – 14th Kynea number
• 1129760415 – 23rd Motzkin number
• 1134903170 – 45th Fibonacci number
• 1162261467 = 3¹⁹
• 1220703125 = 5¹³
• 1234567890 – pandigital number with the digits in order
• 1311738121 – 25th Pell number
• 1382958545 – 15th Bell number
• 1406818759 – 30th Wedderburn–Etherington number
• 1836311903 – 46th Fibonacci number
• 1882341361 – The least prime whose reversal is both square (40391²) and triangular (triangular of 57121).
• 1977326743 = 7¹¹
• 2147483647 – 8th Mersenne prime and the largest signed 32-bit integer
• 2147483648 = 2³¹
• 2214502422 – 6th primary pseudoperfect number
• 2357947691 = 11⁹
• 2971215073 – 11th Fibonacci prime (47th Fibonacci number)
• 3166815962 – 26th Pell number
• 3192727797 – 24th Motzkin number
• 3323236238 – 31st Wedderburn–Etherington number
• 3405691582 – hexadecimal CAFEBABE; used as a placeholder in programming.
• 3405697037 – hexadecimal CAFED00D; used as a placeholder in programming.
• 3486784401 = 3²⁰
• 4294836223 – 16th Carol number
• 4294967291 – Largest prime 32-bit unsigned integer
• 4294967295 – Maximum 32-bit unsigned integer (hexadecimal FFFFFFFF)
• 4294967296 = 2³²
• 4294967297 – the first composite Fermat number
• 4295098367 – 15th Kynea number
• 4807526976 – 48th Fibonacci number
• 5784634181 – 13th alternating factorial
• 6103515625 = 5¹⁴
• 6210001000 – only self-descriptive number in base 10
• 6227020800 = 13!
• 6983776800 – 15th colossally abundant number
• 7645370045 – 27th Pell number
• 7778742049 – 49th Fibonacci number
• 7862958391 – 32nd Wedderburn–Etherington number
• 8589869056 – 6th perfect number
• 8589934592 = 2³³
• 9043402501 – 25th Motzkin number
• 9814072356 – largest square pandigital number, largest pandigital pure power
• 9876543210 – largest pandigital number without redundant digits

Sense of scale[link]

The facts below give a sense of how large 1,000,000,000 (10⁹) is in the context of time according to current scientific evidence:

• 10⁹ seconds is 114 days short of 32 calendar years (≈ 31.7 years).
• About 10⁹ minutes ago, the Roman Empire was flourishing and Christianity was emerging. (10⁹ minutes is roughly 1,900 years.) A billion minutes ago, Trajan was emperor in Rome^[1].
• About 10⁹ hours ago, modern human beings and their ancestors were living in the Stone Age (more precisely, the Middle Paleolithic). (10⁹ hours is roughly 114,000 years.)
• About 10⁹ days ago, Australopithecus, an ape-like creature related to an ancestor of modern humans, roamed the African savannas. (10⁹ days is roughly 2.7 million years.)
• About 10⁹ months ago, dinosaurs walked the Earth during the late Cretaceous. (10⁹ months is roughly 82 million years.)
• About 10⁹ years ago, the first multicellular eukaryotes appeared on Earth.
• The universe is currently thought to be about 13.7 × 10⁹ years old.

In terms of distance:

• 10⁹ inches is 15,783 miles (25,400 km), more than halfway around the world and sufficient to reach any point on the globe from any other point.
• 10⁹ metres is almost three times the distance from the Earth to the Moon.
• 10⁹ kilometres is over six times the distance from the Earth to the Sun.

In terms of finance:

• The possession of assets with total value of 10⁹ United States dollars would place a person among the world's wealthiest individuals.
• As visualized in a work by artist Michael Marcovici, this amount of money in stacks of hundred-dollar bills can fit on twelve wrapped pallets.^[2]

In terms of area:

• A billion square inches would be a square about one half mile on a side.
• A piece of finely woven bed sheet cloth that contained a billion holes would measure about 500 square feet (46 m²), large enough to cover a moderate sized apartment.

In terms of volume:

• There are a billion cubic millimeters in a cubic meter.
• A billion grains of table salt or granulated sugar would occupy a volume of about two and a half cubic feet.
• A billion cubic inches would be a volume comparable to a large commercial building slightly larger than a typical supermarket.

In terms of natural landscape; a small mountain, slightly larger than Stone Mountain Georgia, United States, would weigh (have a mass of) a billion tons.

In terms of count:

A is a cube; B consists of 1000 cubes of type A. C consists of 1000 Bs; and D 1000 Cs. Thus there are 1 million As in C; and 1,000,000,000 As in D.

References[link]

A number is a mathematical object used to count and measure. 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.

Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs).

In common use, the word number can mean the abstract object, the symbol, or the word for the number.

Classification of numbers[link]

Different types of numbers are used in many cases. 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.)

Natural numbers[link]

The most familiar numbers are the natural numbers or counting numbers: one, two, three, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.^{[citation needed]} Today, different mathematicians use the term to describe both sets, including zero or not. The mathematical symbol for the set of all natural numbers is N, also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} .

In the base ten numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 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.

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics,^[1] natural numbers can be represented by classes of equivalent sets. 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 (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.

Integers[link]

The negative of a positive integer is defined as a number that produces zero when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (which includes zero), the result is defined as the set of integer numbers, also called integers, Z also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z} . Here the letter Z comes from German Zahl, meaning "number".

The set of integers forms a ring with operations addition and multiplication.^[2]

Rational numbers[link]

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. Fractions are written as two numbers, the numerator and the denominator, with a dividing bar between them. In the fraction written ^m⁄_n or

Failed to parse (Missing texvc executable; please see math/README to configure.): m \over n \,

m represents 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): {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 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q} .

Real numbers[link]

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. 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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 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. Zero is often written as 0.0 when it must be treated as a real number rather than an integer. In the US and UK a number between −1 and 1 is always written with a leading zero to emphasize the decimal. Negative real numbers are written with a preceding minus sign:

Failed to parse (Missing texvc executable; please see math/README to configure.): -123.456.\,

Every rational number is also a real number. 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, otherwise written 0.3) 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3.14159265358979\dots.\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} = 1.41421356237 \dots\,

(the square root of 2, that is, the positive number whose square is 2).

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. 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, also written as Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} .

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 0.001 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.

In abstract algebra, it can be shown that any complete ordered field is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field.

Complex numbers[link]

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from trying to find closed formulas for the roots of cubic and quartic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \,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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C} .

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 i is less than 1. In technical terms, the complex numbers lack the trichotomy property.

Complex numbers correspond to points on the complex plane, sometimes called the Argand plane.

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} .

Computable numbers[link]

Moving to problems of computation, the computable numbers are determined in the set of the real numbers. 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. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms. 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.

Other types[link]

Algebraic numbers are those that can be expressed as the solution to a polynomial equation with integer coefficients. The complement of the algebraic numbers are the transcendental numbers.

Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R.

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties.

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.

A relation number is defined as the class of relations consisting of all those relations that are similar to one member of the class.^[3]

Sets of numbers that are not subsets of the complex numbers are sometimes called hypercomplex numbers. 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. Elements of function fields of non-zero characteristic behave in some ways like numbers and are often regarded as numbers by number theorists.

Specific uses[link]

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.

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".) 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.

A perfect number is a positive integer that is the sum of its proper positive divisors—the sum of the positive divisors not including the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS). These first four perfect numbers were the only ones known to early Greek mathematics.

A figurate number is a number that can be represented as a regular and discrete geometric pattern (e.g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number. 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[link]

Numbers should be distinguished from numerals, the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system.^{[citation needed]} Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by both the base ten numeral '5', by the Roman numeral 'Ⅴ' and ciphered letters. 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.

History[link]

First use of numbers[link]

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks.^[4] These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract 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.^[5]

[edit] Zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient texts used zero. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting entries. Indian texts used a Sanskrit word Shunye to refer to the concept of void. In mathematics texts this word often refers to the number zero.^[6]

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 whether 1 was a number.)

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. Mayan arithmetic used base 4 and base 5 written as base 20. Sanchez in 1961 reported a base 4, base 5 'finger' abacus.

By 130 AD, 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).

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.

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 (the 7th century) the concept had clearly reached Cambodia as Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world.

[edit] 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 (Chinese: Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.^[7] 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 4x + 20 = 0

(the solution is negative) in Arithmetica, saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus’ previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in 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."

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers either by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral.^[8] 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”.

As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a Cartesian coordinate system.

[edit] Rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. The best known of these is Euclid's Elements, dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.

The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it 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 (base 60) fractions with great frequency.

[edit] Irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800–500 BC.^[9] 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 he could not accept irrational numbers, so he sentenced Hippasus to death by drowning.

The sixteenth century brought final European acceptance of negative integral and fractional numbers. By the seventeenth century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the nineteenth century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook scientific study of irrationals. It had remained almost dormant since Euclid. 1872 brought publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. In 1869, Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on 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.

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.

[edit] 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 show 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.

The existence of transcendental numbers^[10] 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.

[edit] Infinity and infinitesimals

The earliest known conception of mathematical infinity appears in the Yajur Veda, an ancient Indian script, 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 c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity—the general consensus being that only the latter had true value. Galileo'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, transfinite numbers and formulating the continuum hypothesis. 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. 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 infinitesimal calculus by Newton and Leibniz.

A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity," one for each spatial direction. Each family of parallel lines in 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.

[edit] 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 such as Niccolo Fontana Tartaglia and Gerolamo Cardano. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1

seemed capriciously inconsistent with the algebraic identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{a}\sqrt{b}=\sqrt{ab},

which is valid for positive real numbers a and b, and 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{-1}

to guard against this mistake.

The 18th century saw the work of Abraham de Moivre and Leonhard Euler. de Moivre's formula (1730) states:

Failed to parse (Missing texvc executable; please see math/README to configure.): (\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos \theta + i\sin \theta = e ^{i\theta }. \,

The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received 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.

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 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.

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, Gotthold Eisenstein, studied the type a + bω, where ω is a complex root of x³ − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity x^k − 1 = 0 for higher values of k. This generalization is largely due to Ernst 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.

In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. This eventually led to the concept of the extended complex plane.

[edit] 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 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.

In 1796, Adrien-Marie Legendre conjectured 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. Goldbach and Riemann's conjectures remain unproven and unrefuted.

Word alternatives[link]

Some numbers traditionally have alternative words to express them, including the following:

• Nothing: 0
• Single: 1
• Pair, couple, brace: 2
• Trio: 3
• Half-Dozen: 6
• Decade: 10
• Dozen: 12
• Baker's dozen: 13
• Score: 20
• Half-century: 50
• Century: 100
• Gross: 144
• Ream: 480 (old measure) 500 (new measure)
• Millenium: 1000
• Great gross: 1728
• "n-figure", as in digit, generally for larger-number ranges, also written without a hyphen; often used in financial discussion. For example

• "five-figure": 10,000 to 99,999 (five digits); ten-thousands
• "six-figure": 100,000 to 999,999 (six digits); hundred-thousands
• "seven-figure": 1,000,000 to 9,999,999 (seven digits); millions

See also[link]

Wikimedia Commons has media related to: Numbers

Notes[link]

References[link]

• Tobias Dantzig, Number, the language of science; a critical survey written for the cultured non-mathematician, New York, The Macmillan company, 1930.
• 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.
• Paul Halmos, Naive Set Theory, Springer, 1974, ISBN 0-387-90092-6.
• Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge University Press, 1910.
• George I. Sanchez, Arithmetic in Maya,Austin-Texas, 1961.

External links[link]

Look up number in Wiktionary, the free dictionary.

Wikiversity has learning materials about Primary mathematics:Numbers

• Nechaev, V.I. (2001), "Number", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=Number 
• 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).
• http://planetmath.org/encyclopedia/MayanMath2.html
• "What's the World's Favorite Number?". 2011-06-22. http://www.npr.org/blogs/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1. Retrieved 2011-09-17. ; "Cuddling With 9, Smooching With 8, Winking At 7". 2011-08-11. http://www.npr.org/templates/transcript/transcript.php?storyId=139797360. Retrieved 2011-09-17. 

Bruno Mars
Bruno Mars performing in Las Vegas, Nevada on September 19, 2010
Background information
Birth name	Peter Gene Hernandez
Born	(1985-10-08) October 8, 1985 (age 26)[1] Honolulu, Hawaii, U.S.
Genres	Reggae fusion, pop, soul, R&B
Occupations	Singer-songwriter, record producer, model, dancer
Instruments	Vocals, drums, guitar, keyboard, bass, piano, harmonica
Years active	2005–present
Labels	Atlantic, Elektra
Associated acts	The Smeezingtons
Website	www.brunomars.com

Peter Gene Hernandez (born October 8, 1985), better known by his stage name Bruno Mars, is an American singer-songwriter and record producer. Raised in Honolulu, Hawaii by a family of musicians, Mars began making music at a young age. He performed in various musical venues in his hometown throughout his childhood. He graduated from high school and then moved to Los Angeles, California to pursue a musical career. Mars produced songs for other artists, joining production team The Smeezingtons.

Mars had an unsuccessful stint with Motown Records, but then signed with Atlantic Records in 2009. He became recognized as a solo artist after lending his vocals and co-writing the hooks for the songs "Nothin' on You" by B.o.B, and "Billionaire" by Travie McCoy. He also co-wrote the hits "Right Round" by Flo Rida featuring Ke$ha, and "Wavin' Flag" by K'naan. In October 2010, he released his debut album, Doo-Wops & Hooligans. Anchored by the worldwide number-one singles "Just the Way You Are" and "Grenade", the album peaked at number three on the Billboard 200.^[2] Mars was nominated for seven Grammys at the 53rd Grammy Awards, winning Best Male Pop Vocal Performance for "Just the Way You Are".

Mars' music is noted for displaying a wide variety of styles and influences, and contains elements of many musical genres. He has worked with an assortment of artists from various genres. Mars acknowledges the influences that his collaborations have had on his own music. As a child, he was highly influenced by artists such as Little Richard,^[3] Elvis Presley and Michael Jackson and would often impersonate these artists from a young age. Mars also incorporates reggae and Motown inspired sounds into his work. Jon Caramanica of The New York Times referred to Mars as "one of the most versatile and accessible singers in pop."^[4] Worldwide, Mars became the best-selling digital artist in 2011. He had three singles in the top ten, including first place with "Just The Way You Are," which sold 12.5 million copies.^[5] Mars won Best International Male Solo Artist at the BRIT Awards in 2012.^[6] Bruno Mars has sold 5 million albums and 36 million singles worldwide.^[7]

Early life[link]

Bruno Mars was born Peter Gene Hernandez on October 8th, 1985, and raised in the Waikiki neighborhood of Honolulu, Hawaii, by parents Pete Hernandez, of Puerto Rican descent, and Bernadette "Bernie," a Filipina.^{[8][9][10][11]} His mother emigrated to Hawaii from the Philippines as a child and his father moved to the island from Brooklyn, New York.^[12] Mars' parents met while performing in a show, where his mother was a hula dancer and his father played percussion.^[12] At the age of two, he was nicknamed "Bruno" by his father, because of his resemblance to legendary professional wrestler Bruno Sammartino.^[13]

Mars was one of six children and came from a musical family who exposed him to a diverse mix of reggae, rock, hip hop, and R&B.^[14][15] In addition to being a dancer, his mother was a singer and his father used his musical ability to perform Little Richard rock n roll music.^[3] Mars' uncle was an Elvis impersonator. By the age of three his uncle had Mars up on the stage performing. He also performed songs by artists such as Michael Jackson, The Isley Brothers, and The Temptations.^[10] At age four, Mars began performing five days a week with his family's band, The Love Notes, in which he became known on the island for his impersonation of Presley.^[16]

In 1990, Mars was featured in MidWeek as "Little Elvis", going on to have a cameo in the film Honeymoon in Vegas in 1992.^[10][17] He later reflected on the influence Presley had on his music: "I watch the best. I'm a big fan of Elvis. I'm a big fan of 1950s Elvis when he would go on stage and scare people because he was a force and girls would go nuts! You can say the same thing for Prince or The Police. It's just guys who know that people are here to see a show, so I watch those guys and I love studying them because I'm a fan."^[18] In 2010, he also acknowledged his Hawaiian roots and musical family as an influence, explaining, "Growing up in Hawaii made me the man I am. I used to do a lot of shows in Hawaii with my father's band. Everybody in my family sings, everyone plays instruments...I've just been surrounded by it."^[19] In 2003, shortly after graduating from President Theodore Roosevelt High School at the age of seventeen, Mars moved to Los Angeles, California, to pursue a musical career.^[10][17] He adopted his stage name from the nickname his father gave him, adding "Mars" at the end because "I felt like I didn't have [any] pizzazz, and a lot of girls say I’m out of this world, so I was like I guess I'm from Mars."^[20]

Career[link]

[edit] 2004–10: Career beginnings and It's Better If You Don't Understand

Bruno Mars performing in Houston, Texas in November 2010.

Shortly after moving to Los Angeles, Mars signed to Motown Records in 2004, in a deal that "went nowhere".^[21] However, Mars' experience with Motown proved to be beneficial to his career when he met songwriter and producer Philip Lawrence, who was also signed to the label. Mars, Lawrence, and engineer Ari Levine began writing songs together, and formed the production team The Smeezingtons.^[21] In 2006, Lawrence introduced Mars to his future manager at Atlantic Records, Aaron Bay-Schuck.^[22] After hearing him play a couple of songs on the guitar, Bay-Schuck wanted to sign him immediately, but it took about three years for Atlantic records to finally sign Mars to the label.^[22] In the meantime, Bay-Schuck hired Mars and the Smeezingtons to write and produce songs for the label's artists.^[22] According to Bay-Schuck in an interview with HitQuarters, Mars had stated that although his ultimate goal was to be a solo artist, he was willing to write and produce for other artists, both to help improve his songwriting and help him discover the type of artist he ultimately wanted to be.^[22] Bay-Schuck credits this as a period of "self-discovery" that contributed significantly to Mars' later success.^[22]

Before becoming a successful solo artist, Hernandez was an acknowledged music producer, writing songs for Alexandra Burke, Travie McCoy, Adam Levine, Brandy, Sean Kingston, and Flo Rida.^[14][23] He also co-wrote the Sugababes' hit song "Get Sexy" and provided backing vocals on their album Sweet 7.^[24][25] His first musical appearance as a singer was in Far East Movement's second studio album Animal, featured on the track "3D".^[26] He was also featured on pastor and hip hop artist Jaeson Ma's debut single "Love" in August 2009.^[27][28] He reached prominence as a solo artist after being featured on and co-writing B.o.B's "Nothin' on You" and Travie McCoy's "Billionaire"; both songs peaked within the top ten of many charts worldwide.^{[29][30][31][32]} He said of them, "I think those songs weren't meant to be full-sung songs. If I'd sung all of "Nothin' on You", it might've sounded like some '90s R&B." Following this success, Mars released his debut extended play (EP), titled It's Better If You Don't Understand, on May 11, 2010.^[33] The EP peaked at the 99th position on the Billboard 200 and produced one single, "The Other Side", featuring singers Cee Lo Green and B.o.B.^[34][35] Mars collaborated with Green once more in August 2010 by co-writing his single "Fuck You!". He performed a medley of "Nothin' on You" and "Airplanes" with B.o.B and Hayley Williams at the 2010 MTV Video Music Awards on September 12, 2010.^[36]

[edit] 2010–present: Doo-Wops & Hooligans

Bruno Mars' debut album, Doo-Wops & Hooligans, was released digitally on October 4, and saw its physical release on October 5, 2010.^[37][38] The lead single, "Just the Way You Are", was released on July 19, 2010,^[39] and has reached the top of the Billboard Hot 100 as well as several other charts worldwide.^[40][41] The music video was released on September 8, 2010.^[42] The second single, "Grenade", was released September 28, 2010, and has also seen successful international chart performance.^[43][44] In the United States, Doo-Wops & Hooligans debuted at number three on the Billboard 200 for the week of October 13, 2010, selling 55,000 copies.^[2] The album also received generally positive reviews from critics. Entertainment Weekly's Leah Greenblatt praised Mars for his "instant-access melodies" and "sly snatches of dance-floor swagger", but noted weaknesses in songs deviant from his conventional pop and soul genres.^[45] He opened for Maroon 5 on the fall leg of the Hands All Over Tour starting October 6, 2010 and co-headlined with McCoy on a European tour starting October 18, 2010.^[38]

On September 19, 2010, Mars was arrested in Las Vegas for possession of cocaine.^[46] When talking to a police officer, Mars reportedly stated that what he did was "foolish" and that "he has never used drugs before".^[47][48] Mars pled guilty to felony drug possession, and in return for his plea, he was told that the charges would be erased from his criminal record as long as he stayed out of trouble for a year, paid a $2,000 fine, did 200 hours of community service and completed a drug counseling course.^[49]

On February 13, 2011, Mars won his first Grammy Award for Best Male Pop Vocal Performance, having received six nominations; Best Rap Song and Best Rap/Sung Collaboration for "Nothin' on You", Record of the Year for "Nothin' on You" and "Fuck You", Song of the Year for "Fuck You", and Producer of the Year, Non-Classical at the 53rd Grammy Awards.

On September 16, 2011, Bad Meets Evil released their single, "Lighters", which featured Mars in the song. The song was met with mixed reviews from critics and many criticized Mars for doing this type of song.^[50] On September 22, 2011, it was announced on Mars' website that his new song "It Will Rain" will appear on The Twilight Saga: Breaking Dawn – Part 1 Original Motion Picture Soundtrack which was released on November 8.^[51] On October 30, 2011, Mars gave an exclusive performance of "Runaway Baby" on the results show of The X Factor UK, the same day that Mars received six nominations for a Grammy Award; Album of the Year and Best Pop Vocal Album for Doo-Wops & Hooligans, Record of the Year, Song of the Year and Best Pop Solo Performance for "Grenade", and Producer of the Year, Non-Classical at the 54th Grammy Awards. During the ceremony, Mars performed "Runaway Baby" but he did not win any awards as Adele won in all the categories he was nominated in and also lost to Paul Epworth for Producer of the Year.

Mars has confirmed on Twitter that his work on his second album has begun.^[52] On March 22, 2012, it was announced that Mars had signed a worldwide publishing deal with BMG Chrysalis US.^[53]

Musical style[link]

Mars performing on November 24, 2010. Mars plays several instruments including piano, guitar, bass and congas.^[54]

Mars' music has been noted for displaying a wide variety of styles and influences, including pop,^[55] rock,^[55] reggae,^[55] R&B,^[56] soul,^[45] and hip hop.^[57] His co-producer Philip Lawrence says, "What people don't know is there's a darker underbelly to Bruno Mars." Mars himself says, "I blame that on me singing to girls back in high school."^[58] Jon Caramanica of The New York Times called him "one of the most versatile and accessible singers in pop, with a light, soul-influenced voice that’s an easy fit in a range of styles, a universal donor."^[4] Mars was initially influenced by R&B artists such as Keith Sweat, Jodeci and R. Kelly, as well as 1950s rock 'n' roll and Motown.^[59] In high school, he began listening to classic rock groups such as The Police, Led Zeppelin, and the Beatles.^[59] All of these genres of music have influenced Mars' musical style; he observed that "It’s not easy to [create] songs with that mixture of rock and soul and hip-hop, and there’s only a handful of them."^[59] Mars also stated that he is a fan of Alicia Keys, Jessie J, Jack White and The Saturdays.^[60]

In addition, Mars claims that his work with other artists influenced his musical style, saying that "'Nothin' on You' had a Motown vibe, 'Billionaire' was a reggae acoustic guitar-driven song, though one of my favorites is the Cee-Lo [Green] song ['Fuck You']. I don't think anyone else could've sung that song. And there's 'Just the Way You Are.' If you know my story, you know I love all different genres of music."^[61] He cites doo-wop as a major influence on his music, referring to the genre as "just straightforward love songs – so charming and simple and romantic."^[61] In addition, Mars states that growing up in Hawaii influenced his music, giving the songs a reggae sound. He explains that "In Hawaii some of the biggest radio stations are reggae. The local bands are heavily influenced by Bob Marley. That music brings people together. It's not urban music or pop music. It's just songs. That's what makes it cross over so well. The song comes first."^[12] Lyrically, many of Mars' songs have been described as "feel-good", carefree, and optimistic,^[62] however, darker subjects are addressed in songs such as "Grenade", "Liquor Store Blues", and "Talking to the Moon", which detail failed relationships and self-destructive behavior.^[45][63]

Discography[link]

• Doo-Wops & Hooligans (2010)
• TBA (2012)

Tours[link]

• Doo-Wops & Hooligans tour (2010–2011)

Awards and nominations[link]

References[link]

External links[link]

Wikimedia Commons has media related to: Bruno Mars

• Official website
• Bruno Mars on Twitter
• Bruno Mars at the Internet Movie Database