Negative and non-negative numbers

Being negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as − ${\sqrt{2}}$ , −1.414, −1. A positive number (e.g., positive real number, positive rational number, positive integer) is one that is greater than zero, such as ${\sqrt{2}}$ , 1.414, 1. Zero itself is neither positive nor negative. The non-negative numbers are the numbers that are not negative (they are positive or zero). The non-positive numbers are the 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".

1 Negative numbers
2 Non-negative numbers
3 The negative of a number is unique
4 Signum function
- 4.1 Real
- 4.2 Complex
5 Arithmetic involving signed numbers
6 Formal construction of negative and non-negative integers
7 Extensions
8 History
9 See also
10 Notes
11 References
12 External links

Negative numbers

Negative integers can be regarded as an extension of the natural numbers, such that the expression x – y has a well-defined value for all values of x and y. Other number systems, such as the rational 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 credits. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.

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. The set of all non-negative integers forms a commutative monoid under addition. By extending this set to the set of all integers under addition, we obtain an Abelian group.

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.

The negative of a number is unique

The negative of a number is unique, as is shown by the following proof.

Let x be a number and let y be its negative. Suppose y′ is another negative of x. By an axiom of the real number system

$x + y \prime = 0$ ,

$x + y\,\, = 0$ .

And so, x + y′ = x + y. Using the law of cancellation for addition, it is seen that y′ = y. Thus y is equal to any other negative of x. That is, y is the unique negative of x.

Signum function

Real

The function sgn(x), defined on the real numbers and called the signum function or sign function, is 1 for positive numbers, −1 for negative numbers and 0 for zero:

$\sgn(x)=\left\{\begin{matrix} -1 &�: x < 0, \\ \;0 &�: x = 0, \\ \;1 &�: x > 0. \end{matrix}\right.$

We then have (except for x = 0):

$\sgn(x) = \frac{x}{|x|} = \frac{|x|}{x} = \frac{d{|x|}}{d{x}} = 2H(x)-1.$

Here |x| is the absolute value of x and H(x) is the Heaviside step function. See also derivative.

Complex

Contrary to the real numbers, the complex numbers cannot be made into an ordered field, so there is no possible meaning one could give to the terms "positive" and "negative" for complex numbers that would have the usual properties with respect to arithmetic operations. Nevertheless there are two reasonable generalizations of the real signum function to complex numbers.

One definition is the function sgn(x), with values on the unit circle with the number 0 added, given by

$\sgn(x) = \frac{x}{|x|}$ if $x \neq 0,$

and sgn(0) = 0.

There is also a function csgn(x) on the complex numbers (sometimes called the complex sign function), which extends the signum function while still taking values -1, 0 and 1 only, which is given by:

$\operatorname{csgn}(x)=\left\{\begin{matrix} -1: & \operatorname{Re}(x) < 0 \vee (\operatorname{Re}(x) = 0 \land \operatorname{Im}(x) < 0)\\ \;0: & x = 0, \\ \;1: & \operatorname{Re}(x) > 0 \vee (\operatorname{Re}(x) = 0 \land \operatorname{Im}(x) > 0). \end{matrix}\right.$

We then have (except for x = 0), using a similar convention for the square root of a complex number:

$\operatorname{csgn}(x) = \frac{x}{\sqrt{x^2}} = \frac{\sqrt{x^2}}{x} = \frac{d{\sqrt{x^2}}}{d{x}} = 2H(x)-1.$

Arithmetic involving signed numbers

The minus sign "−" is used for both the operation of subtraction and to signify that a number is negative. The ambiguity does not generally cause problems in arithmetic, as the result of adding a negative number to another is the same as subtracting the number. A negative number may be parenthesised with its sign, e.g. an addition is clearer if written 7 + (−5) rather than 7 + −5, and gives the same result as the subtraction 7 − 5.

Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in^[1]

⁻2 + ⁻5 gives ⁻7

Addition and subtraction

For purposes of addition and subtraction, negative numbers are analogous to debts.

Adding a negative number is the same as subtracting the corresponding positive number:

5 + (−3) = 5 − 3 = 2

(5 in hand and a debt of 3 gives the same result as 5 and expenditure of 3, leaving a net value of 2)

(−2) + (−5) = (−2) − 5 = −7

(A debt of 2 and an additional debt of 5 gives the same result as a debt of 2 and expenditure of 5, and leaves a debt of 7)

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

4 − 6 = −2

(4 in hand and expenditure of 6 leaves a debt of 2).

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

(−3) − 6 = −9

(a debt of 3 and expenditure of 6 leaves a debt of 9).

Subtracting a negative is equivalent to adding the corresponding positive:

5 − (−2) = 5 + 2 = 7

(5 in hand and removing a debt of 2 gives the same result as 5 and adding 2, and leaves a value of 7).

(−8) − (−3) = −5

(a debt of 8 and removing a debt of 3 leaves a debt of 5).

Multiplication

Brahmagupta stated in Brahmasputhasiddhanta "positive times positive is positive and negative times negative is positive". Diophantus had earlier stated the rule but only as a route towards getting an eventual positive result. However due to a distrust of negative numbers even as late as the 18th century this rule was challenged by Lazare Carnot. He asked how the square of a smaller number could be larger than the square of a larger number, for example, how could the square of −3 be larger than the square of −2, as −3 is smaller than −2? There is no need for an answer to this subjective question as this fact does not violate the laws of mathematics.

Multiplication of a negative number by a positive number yields a negative result: (−2) × 3 = −6 (and commutativity adds that 3 × (−2) = −6 also). Multiplication by a positive integer is the same as repeated addition. For instance 3 × 2 can be regarded as 3 groups, with 2 in each group. Thus, 3 × 2 = 2 + 2 + 2 = 6 and this can be extended to 3 × (−2) = (−2) + (−2) + (−2) = −6.

Multiplication of two negative numbers yields a positive result: (−4) × (−3) = 12.

Multiplication is seen to be distributive over addition for both positive and negative numbers.

Division

The sign rules for division are the same as for multiplication. Brahmagupta stated that a negative number divided by a negative number is positive. A positive number divided by a negative number is negative. (Reference: Arithmetic and mensuration of Brahmagupta by HT Colebrooke). Brahmagupta's rule has survived to date: if the dividend and divisor have opposite signs, then the result is negative.

8 ÷ (−2) = −4

(−10) ÷ 2 = −5

If dividend and divisor have the same sign, the result is always positive.

(−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) × (c, d) = (a × c + b × d, a × d + b × 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 N²/~, 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 + d ≤ b + 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).

This construction is a special case of the Grothendieck construction.

Extensions

By extension, the terms negative, non-negative, positive and non-positive may be applied to other mathematical objects whose values are real numbers. For example:

A positive matrix is a matrix of real numbers in which all the elements are positive.
A positive function is a function whose range is a subset of the positive numbers.
A positive linear functional is a linear functional on an ordered vector space which takes positive values when applied to positive vectors.

History

Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (Jiu zhang suan-shu), which in its present form dates from the period of the Han Dynasty (202 BC – 220 AD), but may well contain much older material.^[2] The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative.^[3] (This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values). The Chinese were also able to solve simultaneous equations involving negative numbers.

For a long time, negative solutions to problems were considered "false". In Hellenistic Egypt, Diophantus in the third century A.D. referred to an equation that was equivalent to 4x + 20 = 0 (which has a negative solution) in Arithmetica, saying that the equation was absurd.

The use of negative numbers was known in early India, and their role in situations like mathematical problems of debt was understood.^[4] Consistent and correct rules for working with these numbers were formulated.^[5] The diffusion of this concept led the Arab intermediaries to pass it to Europe.^[4]

The ancient Indian Bakhshali Manuscript, which Pearce Ian claimed was written some time between 200 B.C. and A.D. 300,^[6] while George Gheverghese Joseph dates it to about 400 AD and no later than the early 7th century,^[7] carried out calculations with negative numbers, using "+" as a negative sign.^[8]

During the 7th century A.D., negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written in A.D. 628), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today. He also found negative solutions of quadratic equations and gave 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 "debts." ^[9]^[10]

During the 8th century A.D., the Islamic world learned about negative numbers from Arabic translations of Brahmagupta's works, and by A.D. 1000 Arab mathematicians were using negative numbers for debts.

In the 12th century A.D. in India, Bhaskara also gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots."

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

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, A.D. 1202) and later as losses (in Flos).

In the 15th century, Nicolas Chuquet, a Frenchman, used negative numbers as exponents and referred to them as “absurd numbers.”

In A.D. 1759, Francis Maseres, an English mathematician, wrote 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 were nonsensical. ^[11]

In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless.^[12]

Notes

↑ Grant P. Wiggins; Jay McTighe (2005). Understanding by design. ACSD Publications. p. 210. ISBN 1416600353.
↑ Struik, page 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
↑ Temple, Robert. (1986). The Genius of China: 3,000 Years of Science, Discovery, and Invention. With a forward by Joseph Needham. New York: Simon and Schuster, Inc. ISBN 0671620282. Page 141.
↑ ^4.0 ^4.1 Bourbaki, page 49
↑ Britannica Concise Encyclopedia (2007). algebra
↑ Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. http://www-history.mcs.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript.html. Retrieved 2007-07-24.
↑ Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon and Schuster. ISBN 0684837188. Page 65–66.
↑ Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon and Schuster. ISBN 0684837188. Page 65.
↑ Colva M. Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 programme "In Our Time," on 9 March 2006.
↑ 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".
↑ 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."
↑ 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.

References

Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3540647678.
Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.

External links

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