Negative and non-negative numbers

A negative number is a number that is less than zero, such as −2. A positive number is a number that is greater than zero, such as 2. 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 "x" - "y" = "z" has a meaningful solution "z" 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 credits. In bookkeeping, amounts owing to other people/organizations 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.

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 "–x" be its negative. Let $y~equiv\; -x$.Let $y\; prime$ be another negative of "x". By an axiom of the real number system

:$x\; +\; y\; prime\; =\; 0$,:$x\; +\; y\; =\; 0~$.

And so, $~~x\; +\; yprime\; =\; x\; +\; y$. Using the law of cancellation for addition, it is seen that $yprime\; =\; y$. Therefore $~y~$ is the same number as $y\; prime$ and is the unique negative of "x".

ignum 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 sign function)::

We then have (except for "x"=0)::$sgn(x)\; =\; frac\{x\}$

Division

Division is similar to multiplication. Brahmagupta stated for the first time that negative divided by negative to be positive. Positive divided by negative to be negative. (Reference: Arithmetic and mensuration of Brahmagupta by HT Colebrooke). Brahmagupta's convention has survived to date: if the dividend and divisor have different signs, then the result is negative.

:8 / −2 = −4:−10 / 2 = −5

If dividend and divisor have the same sign, the result is positive, even if both are 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") × ("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").

First use of negative numbers

For a long time, negative solutions to problems were considered "false". The Chinese and Indian works contain the earliest known uses of negative numbers.

In Hellenistic Egypt, Diophantus in the third century A.D. referred to an equation that was equivalent to 4"x" + 20 = 0 (which has a negative solution) in "Arithmetica", saying that the equation was absurd. This indicates that no concept of negative numbers existed in the ancient Mediterranean.

The abstract concept was recognised as early as 100 BC–50 BC. A Han Dynasty (202 BC – 220 AD) Chinese work, "Nine Chapters on the Mathematical Art" ("Jiu-zhang Suanshu"), edited later by Liu Hui in 263, used red counting rods to denote positive coefficients and black rods for negative. [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.] The Chinese were also able to solve simultaneous equations involving negative numbers (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 use of negative numbers was known in early India, and their role in situations like mathematical problems of debt was understood. Bourbaki, page 49] Consistent and correct rules for working with these numbers were formulated.Britannica Concise Encyclopedia (2007). "algebra"] The diffusion of this concept led the Arab intermediaries to pass it to Europe.

The ancient Indian "Bakhshali Manuscript", which Pearce Ian claimed was written some time between 200 B.C. and A.D. 300, [cite web|title=The Bakhshali manuscript|author=Pearce, Ian|publisher=The MacTutor History of Mathematics archive|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript.html|month=May | year=2002|accessdate=2007-07-24] while George Gheverghese Joseph dates it to about 400 AD and Takao Hayashi to no later than the early 7th century, [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.] carried out calculations with negative numbers, using "+" as a negative sign. [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.]

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." [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".] 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. [ 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."]

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, a viewpoint which was shared by John Wallis. It was common practice at that time to ignore any negative results derived from equations, on the assumption that they were meaningless.rf|4|Martinez1 The argument that negative numbers are greater than infinity involved the quotient 1/"x" and considering what happens as "x "approaches and crosses the point "x" = 0 from the positive side.

Notes

References

* Bourbaki, Nicolas (1998). "Elements of the History of Mathematics". Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3540647678.

External links

* [http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maseres.html Maseres' biographical information]
* [http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml BBC Radio 4 series "In Our Time," on "Negative Numbers", March 9, 2006]
* [http://www.free-ed.net/sweethaven/Math/arithmetic/SignedValues01_EE.asp Endless Examples & Exercises: "Operations with Signed Integers"]
* [http://mathforum.org/dr.math/faq/faq.negxneg.html Interesting and insightful conception of negative numbers]

See also

* Signed number representations
* Positive and negative parts
* Integers
* Rational numbers
* Real numbers
* History of zero
* −0