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A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither positive nor negative. The non-negative numbers are the real numbers that are not negative (positive or zero). The non-positive numbers are the real numbers that are not positive (negative or zero). Number is the current mathematics collaboration of the week! Please help improve it to featured article standard. ...
The feasible regions of linear programming are defined by a set of inequalities. ...
0 (zero), alternatively called naught, nil, nada, ought zilch, zip, nothing or nought, is both a number and a numeral. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In the context of complex numbers, positive implies real, but for clarity one may say "positive real number". Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). ...
Negative numbers Negative integers can be regarded as an extension of the natural numbers, such that the equation x − y = z has a meaningful solution for all values of x and y. The other sets of numbers are then derived as progressively more elaborate extensions and generalizations from the integers. In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...
Negative numbers are useful to describe values on a scale that goes below zero, such as temperature, and also in bookkeeping where they can be used to represent debts. In bookkeeping, debts are often represented by red numbers, or a number in parentheses. Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ...
Accountancy (British English) or accounting (American English) is the process of maintaining, auditing, and processing financial information for business purposes. ...
Debt is that which is owed. ...
Red is a color at the highest frequencies of light discernible by the human eye. ...
Non-negative numbers A number is nonnegative if and only if it is greater than or equal to zero, i.e. positive or zero. Thus the nonnegative integers are all the integers from zero on upwards, and the nonnegative reals are all the real numbers from zero on upwards. 0 (zero), alternatively called naught, nil, nada, ought zilch, zip, nothing or nought, is both a number and a numeral. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
A real matrix A is called nonnegative if every entry of A is nonnegative. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ...
A real matrix A is called totally nonnegative by matrix theorists or totally positive by computer scientists if the determinant of every square submatrix of A is nonnegative. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ...
In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
Sign function It is possible to define a function sgn(x) on the real numbers which is 1 for positive numbers, −1 for negative numbers and 0 for zero (sometimes called the signum function): Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ...
 We then have (except for x=0):  where |x| is the absolute value of x and H(x) is the Heaviside step function. See also derivative. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...
The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...
In mathematics, the derivative is one of the two central concepts of calculus. ...
Arithmetic involving signed numbers
Addition and subtraction For purposes of addition and subtraction, one can think of negative numbers as debts.
Adding a negative number is the same as subtracting the corresponding positive number:
 - (if you have $5 and acquire a debt of $3, then you have a net worth of $2)
 Subtracting a positive number from a smaller positive number yields a negative result:  - (if you have $4 and spend $6 then you have a debt of $2).
Subtracting a positive number from any negative number yields a negative result:  - (if you have a debt of $3 and spend another $6, you have a debt of $9).
Subtracting a negative is equivalent to adding the corresponding positive:  - (if you have a net worth of $5 and you get rid of a debt of $2, then your new net worth is $7).
Also:  - (if you have a debt of $8 and get rid of a debt of $3, then you still have a debt of $5).
Multiplication Multiplication of a negative number by a positive number yields a negative result: (−2) × 3 = −6. The reason is that this multiplication can be understood as repeated addition: (−2) × 3 = (−2) + (−2) + (−2) = −6. Alternatively: if you have a debt of $2, and then your debt is tripled, you end up with a debt of $6. In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...
Multiplication of two negative numbers yields a positive result: (−3) × (−4) = 12. This situation cannot be easily understood as repeated addition, and the analogy to debts doesn't help either. The ultimate reason for this rule is that we want the distributive law to work: In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...
 The left hand side of this equation equals 0 × (−4) = 0. The right hand side is a sum of −12 + (−3) × (−4); for the two to be equal, we need (−3) × (−4) = 12. This can, with a stretch of the imagination, be seen as debts; if you have 3 less debts of $4 than someone who has no debts at all (or you have no debts of your own, and 3 debts of $4 owed to you) your net worth is $12.
Division Division is similar to multiplication. If both the dividend and the divisor have different signs, the result is negative: In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ...
A dividend is the distribution or sharing of parts of profits to a companys shareholders. ...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
  If both numbers are of the same sign, the result is positive (even if they are both negative): 
Formal construction of negative and non-negative integers In a similar manner to rational numbers, we can extend the natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers (a, b). We can extend addition and multiplication to these pairs with the following rules: In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...
  We define an equivalence relation ~ upon these pairs with the following rule: In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
 if and only if  This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N2/~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets...
We can also define a total order on Z by writing In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...
 if and only if  This will lead to an additive zero of the form (a, a), an additive inverse of (a, b) of the form (b, a), a multiplicative unit of the form (a+1, a), and a definition of subtraction 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...
First usage of negative numbers For centuries, negative solutions to problems were considered “false” because they couldn't be found in the real world (in the sense that one cannot have a negative number of, for example, seeds). The abstract concept was recognised as early as 100 BC - 50 BC. The Chinese ”Nine Chapters on the Mathematical Art” (Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. This is the earliest known mention of negative numbers in the East, the first indication in a western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to 4x + 20 = 0 (the solution would be negative) in Arithmetica, saying that the equation gave an absurd result. Centuries: 2nd century BC - 1st century BC - 1st century Decades: 130s BC 120s BC 110s BC - 100s BC - 90s BC 80s BC 70s BC 60s BC 50s BC Years: 105 BC 104 BC 103 BC 102 BC 101 BC - 100 BC - 99 BC 98 BC 97 BC 96 BC 95...
Centuries: 2nd century BC - 1st century BC - 1st century Decades: 100s BC 90s BC 80s BC 70s BC 60s BC - 50s BC - 40s BC 30s BC 20s BC 10s BC 0s BC Years: 55 BC 54 BC 53 BC 52 BC 51 BC 50 BC 49 BC 48 BC 47...
Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ...
In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...
The phrase The East has multiple meanings: Eastern society, referring to a specific worldview U.S. Eastern states, East Coast of the United States This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
// Overview Events 212: Constitutio Antoniniana grants citizenship to all free Roman men 212-216: Baths of Caracalla 230-232: Sassanid dynasty of Persia launches a war to reconquer lost lands in the Roman east 235-284: Crisis of the Third Century shakes Roman Empire 250-538: Kofun era, the first...
Diophantus of Alexandria - Διόφαντος ο ΑλεξανδÏεÏÏ‚ - (circa 200/214 – circa 284/298) was a Greek mathematician. ...
Arithmetica, an ancient text on mathematics written by classical period Greek mathematician Diophantus in the second century AD is a collection of 130 algebra problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. ...
During the 600s, negative numbers were in use in India to represent debts. Diophantus’ previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots." Centuries: 6th century 7th century 8th century Decades: 550s - 560s - 570s - 580s - 590s - 600s - 610s - 620s - 630s - 640s - 650s Years: 599 600 601 602 603 604 605 606 607 608 609 Events: Births: Deaths: 604 - Pope Gregory I the Great Categories: 600s ...
Diophantus of Alexandria - Διόφαντος ο ΑλεξανδÏεÏÏ‚ - (circa 200/214 – circa 284/298) was a Greek mathematician. ...
Brahmagupta (ब्रह्मगुप्त) (598_668) was an Indian mathematician and astronomer. ...
The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic...
Events Khusro II of Persia overthrown Pippin of Landen becomes Mayor of the Palace Brahmagupta writes the Brahmasphutasiddhanta Births Deaths Empress Suiko of Japan Theodelinda, queen of the Lombards Categories: 628 ...
Graph of a quadratic function: y = x2 - x - 2 = (x+1)(x-2) The x-coordinates of the points where the graph crosses the x-axis, x = -1 and x = 2, are the roots of the quadratic equation: x2 - x - 2 = 0 In mathematics, a quadratic equation is a polynomial...
(11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ...
BhÄskara (1114-1185), also called BhÄskara II and BhÄskarÄAchÄrya (Bhaskara the teacher) was an Indian mathematician. ...
European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most nonzero digit. 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” This article is about the continent. ...
(16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ...
Drawing of Leonardo Pisano Leonardo of Pisa or Leonardo Pisano (Pisa, c. ...
Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. ...
// Events August 1 - Arthur of Brittany captured in Mirebeau, north of Poitiers Beginning of the Fourth Crusade. ...
Drawing of Leonardo Pisano Leonardo of Pisa or Leonardo Pisano (Pisa, c. ...
Nicolas Chuquet (born 1445 (some sources say c. ...
(14th century - 15th century - 16th century - other centuries) As a means of recording the passage of time, the 15th century was that century which lasted from 1401 to 1500. ...
In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...
Negative numbers were not well-understood until modern times. 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. (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ...
It has been suggested that Leonhard Euler/EB1911 biography be merged into this article or section. ...
Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ...
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