# Negative number

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 (they are positive or zero). The non-positive numbers are the real numbers that are not positive (they are negative or zero).

In the context of complex numbers, positive implies real, but for clarity one may say "positive real number".

## Negative numbers

Negative integers can be regarded as an extension of the natural numbers, such that the equation xy = 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.

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.

## Non-negative numbers

A number is non-negative 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.

A real matrix A is called nonnegative if every entry of A is nonnegative.

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.

## Signum 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):

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.

## Complex Signum function

It is possible to define a function csgn(x) on the complex numbers which is 1 for positive numbers, −1 for negative numbers and 0 for zero (sometimes called the complex signum function):

Where the complex inequality should be interpreted as follows

We then have (except for x=0):

## Arithmetic involving signed numbers

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:

5 + (−3) = 5 − 3 = 2
(if you have \$5 and acquire a debt of \$3, then you have a net worth of \$2)
–2 + (−5) = −2 − 5 = −7

(In order to avoid confusion between the concepts of subtraction and negation, often the negative sign is written as a superscript:
2 + 5 = 2 − 5 = 7)

Subtracting a positive number from a smaller positive number yields a negative result:

4 − 6 = −2
(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:

−3 − 6 = −9
(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:

5 − (−2) = 5 + 2 = 7
(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:

−8 − (−3) = −5
(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 Ã— 3 = −6. Multiplication of two negative numbers yields a positive result: −4 Ã— −3 = 12.

One way of understanding this is to regard multiplication by a positive number as repeated addition. Think of 3 x 2 as 3 groups, with 2 in each group. Thus, 3 Ã— 2 = 2 + 2 + 2 = 6 and so naturally −2 Ã— 3 = (−2) + (−2) + (−2) = −6.

Multiplication by a negative number can be regarded as repeated addition as well. For instance, 3 Ã— -2 can be thought of as 3 groups, with -2 in each group. 3 Ã— −2 = (-2) + (−2) + (-2) = −6. Notice that this keeps multiplication commutative: 3 Ã— −2 = −2 Ã— 3 = −6.

Applying the same interpretation of "multiplication by a negative number" for a value that is also negative, we have:
 −4 Ã— −3 =   − (−4) − (−4) − (−4) =  4 + 4 + 4 =  12

However, from a formal viewpoint, multiplication between two negative numbers is directly received by means of the distributivity of multiplication over addition:
 −1 Ã— −1 =  (−1) Ã— (−1) + (−2) + 2 =  (−1) Ã— (−1) + (−1) Ã— 2 + 2 =  (−1) Ã— (−1 + 2) + 2 =  (−1) Ã— 1 + 2 =  (−1) + 2 =  1

### Division

Division is similar to multiplication. If both the dividend and the divisor have different signs, the result is negative:
8 / −2 = −4
−10 / 2 = −5

If both numbers are of the same sign, the result is positive (even if they are both negative):
−12 / −3 = 4

## 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:
(a, b) + (c, d) = (a + c, b + d)
(a, b) Ã— (c, d) = (a Ã— c + b Ã— d, a Ã— d + b Ã— c)

We define an equivalence relation ~ upon these pairs with the following rule:
(a, b) ~ (c, d) if and only if a + d = b + c.
This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set NÂ²/~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense.

We can also define a total order on Z by writing
(a, b) ≤ (c, d) if and only if a + db + c.

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
(a, b) − (c, d) = (a + d, b + c).

## First usage of negative numbers

For a long time, 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 BC50 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. They were able to solve simultaneous equations involving negative numbers. The ancient Indian Bakhshali Manuscript, written around the seventh century CE,[1] carried out calculations with negative numbers, using a "+" as a negative sign. These are the earliest known uses of negative numbers.

In Hellenistic Egypt, Diophantus in the 3rd century CE referred to the equation equivalent to 4x + 20 = 0 (the solution would be negative) in Arithmetica, saying that the equation was absurd, indicating that no concept of negative numbers existed in the ancient Mediterranean.

During the 7th century, negative numbers were in use in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written in 628) discusses the use of negative numbers to produce the general form quadratic formula that remains in use today. He also finds negative solutions to quadratic equations and gives rules regarding operations involving negative numbers and zero, such as "a debt cut off from nothingness becomes a credit, a credit cut off from nothingness becomes a debt." He called positive numbers "fortunes", zero a "cipher", and negative numbers a "debt".23 In the 12th century in India, Bhaskara also gives negative roots for quadratic equations but rejects the negative roots since they were inappropriate in the context of the problem, stating that the negative values "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."

From the 8th century, the Islamic world learnt about negative numbers from Arabic translations of Brahmagupta's works, and by about 1000 AD, Arab mathematicians had realized the use of negative numbers for debt.

Knowledge of negative numbers eventually reached Europe through Latin translations of Arabic and Indian works.

European mathematicians however, 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 non-zero 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”.

The English mathematician Francis Maseres [1] wrote in 1759 that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". He came to the conclusion that negative numbers did not exist.1

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.4

## Footnotes and references

1 Maseres, Francis, 1731–1824. A dissertation on the use of the negative sign in algebra: containing a demonstration of the rules usually given concerning it; and shewing how quadratic and cubic equations may be explained, without the consideration of negative roots. To which is added, as an appendix, Mr. Machin's Quadrature of the Circle, 1758. Quoting from Maseres' work, "If any single quantity is marked either with the sign + or the sign − without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of −5, or the product of −5 into −5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon."
• 2 Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 "In Our Time", on Negative Numbers, 9 March 2006.
• 3 Knowledge Transfer and Perceptions of the Passage of Time, ICEE-2002 Keynote Address by Colin Adamson-Macedo. "Referring again to Brahmagupta's great work, all the necessary rules for algebra, including the 'rule of signs', were stipulated, but in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (= fortunes) is used to represent positive numbers, whereas 'rina' (= debts) were negative". http://www.ineer.org/Events/ICEE2002Info/Adamson-MacedoKeynotePaper.pdf
• 4 Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton University Press, 2006; a history of controversies on negative numbers, mainly from the 1600s until the early 1900s.
1. ^ Hayashi, Takao (2005), "Indian Mathematics", in Flood, Gavin, The Blackwell Companion to Hinduism, Oxford: Basil Blackwell, 616 pages, pp. 360-375, ISBN 9781405132510. Quote:"The dates so far proposed for the Bakhshali work vary from the third to the twelfth centuries AD, but a recently made comparative study has shown many similarities, particularly in the style of exposition and terminology, between Bakhshalī work and Bhāskara I's commentary on the Āryabhatīya. This seems to indicate that both works belong to nearly the same period, although this does not deny the possibility that some of the rules and examples in the Bakhshālī work date from anterior periods."

number is an abstract idea used in counting and measuring. A symbol which represents a number is called a numeral, but in common usage the word number is used for both the idea and the symbol.
• The notation means that a is less than b and
• The notation means that a is greater than b.

0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
In mathematics, a complex number is a number of the form

where a and b are real numbers, and i is the imaginary unit, with the property i Â² = −1.
In mathematics, a natural number can mean either an element of the set (i.e the positive integers or the counting numbers) or an element of the set (i.e. the non-negative integers).
trillion fold).]]

Temperature is a physical property of a system that underlies the common notions of hot and cold; something that is hotter generally has the greater temperature. Temperature is one of the principal parameters of thermodynamics.
Bookkeeping (book-keeping or book keeping) is the recording of all financial transactions undertaken by an individual or organization. The organization may be a business, a charitable organization or even a local sports club.
Debt is that which is owed; usually referencing assets owed, but the term can cover other obligations. In the case of assets, debt is a means of using future purchasing power in the present before a summation has been earned.
Red is any of a number of similar colors evoked by light consisting predominantly of the longest wavelengths of light discernible by the human eye, in the wavelength range of roughly 625–750 nm.
0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
A matrix is totally positive if the determinant of any square submatrix is positive. For example, a Vandermonde matrix whose nodes are positive and increasing is a totally positive matrix.
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A
sign function is a logical function which extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function (after the Latin form of "sign").
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.
Heaviside step function, H, also called unit step function, is a discontinuous function whose value is zero for negative argument and one for positive argument. It seldom matters what value is used for H(0), since is mostly used as a distribution.
derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
sign function is a logical function which extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function (after the Latin form of "sign").
• The notation means that a is less than b and
• The notation means that a is greater than b.

Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:
4 • (2 + 3) = (4 • 2) + (4 • 3).

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.

Specifically, if c times b equals a, written:
where b is not zero, then a