Natural number

In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.

A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.

Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.

Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a

HEK 293 cells

Human embryonic kidney 293 cells often referred to as HEK 293, HEK-293, 293 cells, or less as HEK cells, are a specific cell line derived from human embryonic kidney cells grown in tissue culture. HEK 293 cells have been used in cell biology research for many years, because of their reliable growth and propensity for transfection, they are used by the biotechnology industry to produce therapeutic proteins and viruses for gene therapy. HEK 293 cells were generated in 1973 by transfection of cultures of normal human embryonic kidney cells with sheared adenovirus 5 DNA in Alex van der Eb's laboratory in Leiden, the Netherlands; the cells were obtained from a single healthy aborted fetus under Dutch law. The cells were cultured by van der Eb, they were published in 1977. They are called HEK since they originated in human embryonic kidney cultures, while the number 293 came from Graham's habit of numbering his experiments. Graham performed the transfection a total of eight times, obtaining just one clone of cells that were cultured for several months.

After adapting to tissue culture, cells from this clone developed into the stable HEK 293 line. Subsequent analysis has shown that the transformation was brought about by inserting ~4.5 kilobases from the left arm of the viral genome, which became incorporated into human chromosome 19. For many years it was assumed that HEK 293 cells were generated by transformation of either a fibroblastic, endothelial or epithelial cell, all of which are abundant in kidneys. However, the original adenovirus transformation was inefficient, suggesting that the cell that produced the HEK 293 line may have been unusual in some fashion. Graham and coworkers provided evidence that HEK 293 cells and other human cell lines generated by adenovirus transformation of human embryonic kidney cells have many properties of immature neurons, suggesting that the adenovirus preferentially transformed a neuronal lineage cell in the original kidney culture. A comprehensive study of the genomes and transcriptomes of HEK 293 and five derivative cell lines compared the HEK 293 transcriptome with that of human kidney, adrenal and central nervous tissue.

The HEK 293 pattern most resembled that of adrenal cells, which have many neuronal properties. Given the location of the adrenal gland, a few adrenal cells could plausibly have appeared in an embryonic kidney derived culture, could be preferentially transformed by adenovirus. Adenoviruses transform neuronal lineage cells much more efficiently than typical human kidney epithelial cells. An embryonic adrenal precursor cell therefore seems the most origin cell of the HEK 293 line; as a consequence, HEK 293 cells should not be used as an in vitro model of typical kidney cells. HEK 293 cells have a complex karyotype, exhibiting two or more copies of each chromosome and with a modal chromosome number of 64, they are described as hypotriploid, containing less than three times the number of chromosomes of a haploid human gamete. Chromosomal abnormalities include a total of three copies of the X chromosome and four copies of chromosome 17 and chromosome 22; the presence of multiple X chromosomes and the lack of any trace of Y chromosome derived sequence suggest that the source fetus was female.

HEK 293 cells are straightforward to transfect. They have been used as hosts for gene expression; these experiments involve transfecting in a gene of interest, analyzing the expressed protein. The widespread use of this cell line is due to its transfectability by the various techniques, including calcium phosphate method, achieving efficiencies approaching 100%. Examples of such experiments include: Effects of a drug on sodium channels Inducible RNA interference system Isoform-selective protein kinase C agonist Interaction between two proteins Nuclear export signal in a protein A more specific use of HEK 293 cells is in the propagation of adenoviral vectors. Viruses offer an efficient means of delivering genes into cells, which they evolved to do, are thus of great use as experimental tools. However, as pathogens, they present a risk to the experimenter; this danger can be avoided by the use of viruses which lack key genes, which are thus unable to replicate after entering a cell. In order to propagate such viral vectors, a cell line that expresses the missing genes is required.

Since HEK 293 cells express a number of adenoviral genes, they can be used to propagate adenoviral vectors in which these genes are deleted, such as AdEasy. An important variant of this cell line is the 293T cell line, it contains the SV40 Large T-antigen that allows for episomal replication of transfected plasmids containing the SV40 origin of replication. This allows for amplification of transfected plasmids and extended temporal expression of desired gene products. Note that any modified cell line can be used for this sort of work. HEK 293, HEK 293T, cells are used for the production of various retroviral vectors. Various retroviral packaging cell lines are based on these cells. Depending on various conditions, the gene expression of HEK 293 cells may vary; the following proteins of interest are found in untreated HEK 293 cells: Corticotrophin releasing factor type 1 receptor Sphingosine-1-phosphate receptors EDG1

United States

The United States of America known as the United States or America, is a country composed of 50 states, a federal district, five major self-governing territories, various possessions. At 3.8 million square miles, the United States is the world's third or fourth largest country by total area and is smaller than the entire continent of Europe's 3.9 million square miles. With a population of over 327 million people, the U. S. is the third most populous country. The capital is Washington, D. C. and the largest city by population is New York City. Forty-eight states and the capital's federal district are contiguous in North America between Canada and Mexico; the State of Alaska is in the northwest corner of North America, bordered by Canada to the east and across the Bering Strait from Russia to the west. The State of Hawaii is an archipelago in the mid-Pacific Ocean; the U. S. territories are scattered about the Pacific Ocean and the Caribbean Sea, stretching across nine official time zones. The diverse geography and wildlife of the United States make it one of the world's 17 megadiverse countries.

Paleo-Indians migrated from Siberia to the North American mainland at least 12,000 years ago. European colonization began in the 16th century; the United States emerged from the thirteen British colonies established along the East Coast. Numerous disputes between Great Britain and the colonies following the French and Indian War led to the American Revolution, which began in 1775, the subsequent Declaration of Independence in 1776; the war ended in 1783 with the United States becoming the first country to gain independence from a European power. The current constitution was adopted in 1788, with the first ten amendments, collectively named the Bill of Rights, being ratified in 1791 to guarantee many fundamental civil liberties; the United States embarked on a vigorous expansion across North America throughout the 19th century, acquiring new territories, displacing Native American tribes, admitting new states until it spanned the continent by 1848. During the second half of the 19th century, the Civil War led to the abolition of slavery.

By the end of the century, the United States had extended into the Pacific Ocean, its economy, driven in large part by the Industrial Revolution, began to soar. The Spanish–American War and World War I confirmed the country's status as a global military power; the United States emerged from World War II as a global superpower, the first country to develop nuclear weapons, the only country to use them in warfare, a permanent member of the United Nations Security Council. Sweeping civil rights legislation, notably the Civil Rights Act of 1964, the Voting Rights Act of 1965 and the Fair Housing Act of 1968, outlawed discrimination based on race or color. During the Cold War, the United States and the Soviet Union competed in the Space Race, culminating with the 1969 U. S. Moon landing; the end of the Cold War and the collapse of the Soviet Union in 1991 left the United States as the world's sole superpower. The United States is the world's oldest surviving federation, it is a representative democracy.

The United States is a founding member of the United Nations, World Bank, International Monetary Fund, Organization of American States, other international organizations. The United States is a developed country, with the world's largest economy by nominal GDP and second-largest economy by PPP, accounting for a quarter of global GDP; the U. S. economy is post-industrial, characterized by the dominance of services and knowledge-based activities, although the manufacturing sector remains the second-largest in the world. The United States is the world's largest importer and the second largest exporter of goods, by value. Although its population is only 4.3% of the world total, the U. S. holds 31% of the total wealth in the world, the largest share of global wealth concentrated in a single country. Despite wide income and wealth disparities, the United States continues to rank high in measures of socioeconomic performance, including average wage, human development, per capita GDP, worker productivity.

The United States is the foremost military power in the world, making up a third of global military spending, is a leading political and scientific force internationally. In 1507, the German cartographer Martin Waldseemüller produced a world map on which he named the lands of the Western Hemisphere America in honor of the Italian explorer and cartographer Amerigo Vespucci; the first documentary evidence of the phrase "United States of America" is from a letter dated January 2, 1776, written by Stephen Moylan, Esq. to George Washington's aide-de-camp and Muster-Master General of the Continental Army, Lt. Col. Joseph Reed. Moylan expressed his wish to go "with full and ample powers from the United States of America to Spain" to seek assistance in the revolutionary war effort; the first known publication of the phrase "United States of America" was in an anonymous essay in The Virginia Gazette newspaper in Williamsburg, Virginia, on April 6, 1776. The second draft of the Articles of Confederation, prepared by John Dickinson and completed by June 17, 1776, at the latest, declared "The name of this Confederation shall be the'United States of America'".

The final version of the Articles sent to the states for ratification in late 1777 contains the sentence "The Stile of this Confederacy shall be'The United States of America'". In June 1776, Thomas Jefferson wrote the phrase "UNITED STATES OF AMERICA" in all capitalized letters in the headline of his "original Rough draught" of the Declaration of Independence; this draft of the document did not surface unti

Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: "0" and "1". The base-2 numeral system is a positional notation with a radix of 2; each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by all modern computers and computer-based devices; the modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt and India. Leibniz was inspired by the Chinese I Ching; the scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions. Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, 1/64.

Early forms of this system can be found in documents from the Fifth Dynasty of Egypt 2400 BC, its developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt 1200 BC. The method used for ancient Egyptian multiplication is closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value is either doubled or has the first number added back into it; this method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC. The I Ching dates from the 9th century BC in China; the binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang.eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.

Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. The Indian scholar Pingala developed a binary system for describing prosody, he used binary numbers in the form of long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter; the binary representations in Pingala's system increases towards the right, not to the left like in the binary numbers of the modern, Western positional notation. The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Asia. Sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.

In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or ‘Ars generalis’ based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence. In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could be encoded as scarcely visible variations in the font in any random text. For the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only. John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results.

The first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700. Leibniz studied binary numbering in 1679. Leibniz's system uses 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 0 0 0 1 numerical value 20 0 0 1 0 numerical value 21 0 1 0 0 numerical value 22 1 0 0 0 numerical value 23Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus; as a Sinophile, Leibniz was aware of

100 (number)

100 or one hundred is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120. 100 is the square of 10. The standard SI prefix for a hundred is "hecto-". 100 is the basis of percentages. 100 is the sum of the first nine prime numbers, as well as the sum of some pairs of prime numbers e.g. 3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53. 100 is the sum of the cubes of the first four integers. This is related by Nicomachus's theorem to the fact that 100 equals the square of the sum of the first four integers: 100 = 102 = 2.26 + 62 = 100, thus 100 is a Leyland number.100 is an 18-gonal number. It is divisible by 25, the number of primes below it, it can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. It can be expressed as a sum of some of its divisors.

100 is a Harshad number in base 10, in base 4, in that base it is a self-descriptive number. There are 100 prime numbers whose digits are in ascending order. 100 is the smallest number. One hundred is the atomic number of fermium, an actinide and the first of the heavy metals that cannot be created through neutron bombardment. On the Celsius scale, 100 degrees is the boiling temperature of pure water at sea level; the Kármán line lies at an altitude of 100 kilometres above the Earth's sea level and is used to define the boundary between Earth's atmosphere and outer space. There are 100 blasts of the Shofar heard in the service of the Jewish New Year. A religious Jew is expected to utter at least 100 blessings daily. In the Hindu book of the Mahabharata, the king Dhritarashtra had 100 sons known as the Kauravas; the United States Senate has 100 Senators. Most of the world's currencies are divided into 100 subunits; the 100 Euro banknotes feature a picture of a Rococo gateway on the obverse and a Baroque bridge on the reverse.

The U. S. hundred-dollar bill has Benjamin Franklin's portrait. S. bill in print. American savings bonds of $100 have Thomas Jefferson's portrait, while American $100 treasury bonds have Andrew Jackson's portrait. One hundred is also: The number of years in a century; the number of pounds in an American short hundredweight. In Greece, India and Nepal, 100 is the police telephone number. In Belgium, 100 is the firefighter telephone number. In United Kingdom, 100 is the operator telephone number; the HTTP status code indicating that the client should continue with its request. The 100 The age at which a person becomes a centenarian; the number of yards in an American football field. The number of runs required for a cricket batsman to score a significant milestone; the number of points required for a snooker player to score a century break, a significant milestone. The record number of points scored in one NBA game by a single player, set by Wilt Chamberlain of the Philadelphia Warriors on March 2, 1962.

1 vs. 100 AFI's 100 Years... Hundred Hundred Hundred Days Hundred Years' War List of highways numbered 100 Top 100 Greatest 100 Wells, D; the Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group.: 133 Chisholm, Hugh, ed.. "Hundred". Encyclopædia Britannica. Cambridge University Press. On the Number 100

Factorization

In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, is a factorization of the polynomial x2 – 4. Factorization is not considered meaningful within number systems possessing division, such as the real or complex numbers, since any x can be trivially written as × whenever y is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. Factorization was first considered by ancient Greek mathematicians in the case of integers, they proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors.

Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact, exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root-finding algorithms; the case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing factorizations within the ring of polynomials with rational number coefficients.

A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals. Factorization may refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal entries equal to one, an upper triangular matrix U, a permutation matrix P. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one.

For computing the factorization of an integer n, one needs an algorithm for finding a divisor q of n or deciding that n is prime. When such a divisor is found, the repeated application of this algorithm to the factors q and n / q gives the complete factorization of n. For finding a divisor q of n, if any, it suffices to test all values of q such that 1 < q and q2 ≤ n. In fact, if r is a divisor of n such that r2 > n q = n / r is a divisor of n such that q2 ≤ n. If one tests the values of q in increasing order, the first divisor, found is a prime number, the cofactor r = n / q cannot have any divisor smaller than q. For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of r, not smaller than q and not greater than √r. There is no need to test all values of q for applying the method. In principle, it suffices to test only prime divisors; this needs to have a table of prime numbers that may be generated for example with the sieve of Eratosthenes.

As the method of factorization does the same work as the sieve of Eratosthenes, it is more efficient to test for a divisor only those numbers for which it is not clear whether they are prime or not. One may proceed by testing 2, 3, 5, the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3. This method is inefficient for larger integers. For example, Pierre de Fermat was unable to discover that the 6th Fermat number 1 + 2 2 5 = 1 + 2 32 = 4 294 967 297 is not a prime number. In fact, applying the above method would require more than 10000 divisions, for a number that has 10 decimal digits. There are more efficient factoring algorithms; however they remain inefficient, as, with the present state of the art, one cannot factorize with the more powerful computers, a number of 500 decimal digits, the product of two randomly chosen prime numbers. This insures the security of the RSA cryptosystem, used for secure internet communication. For fa

Negative number

In mathematics, a negative number is a real number, less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level negative represents below sea level. If positive represents a deposit, negative represents a withdrawal, they are used to represent the magnitude of a loss or deficiency. A debt, owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature; the laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic.

For example, − = 3 because the opposite of an opposite is the original value. Negative numbers are written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, is pronounced "minus three" or "negative three". To help tell the difference between a subtraction operation and a negative number the negative sign is placed higher than the minus sign. Conversely, a number, greater than zero is called positive; the positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign; every real number other than zero is either negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers are referred to as integers. In bookkeeping, amounts owed are represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty, but may well contain much older material.

Liu Hui established rules for subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of negative numbers around the middle of the 19th century. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd; some mathematicians like Leibniz agreed that negative numbers were false, but still used them in calculations. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: 0 − 3 = −3. In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers.

For example, 5 − 8 = −3since 8 − 5 = 3. The relationship between negative numbers, positive numbers, zero is expressed in the form of a number line: Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less, thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example though 8 is greater than 5, written 8 > 5negative 8 is considered to be less than negative 5: −8 < −5. It follows that any negative number is less than any positive number, so −8 < 5 and −5 < 8. In the context of negative numbers, a number, greater than zero is referred to as positive, thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number, either positive or zero, while nonpositive is used to refer to a number, either negative or zero.

Zero is a neutral number. Goal difference in association football and hockey. Plus-minus differential in ice hockey: the difference in total goals scored for the team and against the team when a particular player is on the ice is the player’s +/− rating. Players can have a negative rating. Run differential in baseball: the run differential is negative if the team allows more runs than they scored. British football clubs are deducted points if they enter administration, thus have a negative points total until they have earned at least that many points that season. Lap times in Formula 1 may be given as the difference compared to a previous lap, will be positive if slower and negative if faster. In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorde