|
We then have (except for ''x''=0):
:<math>\operatorname{csgn}(x) = \frac{x}{\sqrt{x^2}} = \frac{\sqrt{x^2}}{x} = \frac{d{\sqrt{x^2}}}{d{x}} = 2H(x)-1. </math>
==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:
: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:
:<sup>−</sup>2 + <sup>−</sup>5 = <sup>−</sup>2 − 5 = <sup>−</sup>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 × 3 = −6. Multiplication of two negative numbers yields a positive result: −4 × −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 × 2 = 2 + 2 + 2 = 6 and so naturally −2 × 3 = (−2) + (−2) + (−2) = −6.
Multiplication by a negative number can be regarded as repeated addition as well. For instance, 3 × -2 can be thought of as 3 groups, with -2 in each group. 3 × −2 = (-2) + (−2) + (-2) = −6. Notice that this keeps multiplication [[commutative]]: 3 × −2 = −2 × 3 = −6.
Applying the same interpretation of "multiplication by a negative number" for a value that is also negative, we have:
{| align="center"
|-
|−4 × −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:
{| align="center"
|-
|−1 × −1
|-
|
|= (−1) × (−1) + (−2) + 2
|-
|
|= (−1) × (−1) + (−1) × 2 + 2
|-
|
|= (−1) × (−1 + 2) + 2
|-
|
|= (−1) × 1 + 2
|-
|
|= (−1) + 2
|-
|
|= 1
|}
===Division===
[[Division (mathematics)|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 number]]s, we can extend the [[natural number]]s '''N''' to the [[integer]]s '''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]] '''N'''²/~, 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 usage of negative numbers ==<!-- This section is linked from [[Number]] -->
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 BC]] – [[50 BC]]. The [[Chinese mathematics|Chinese]] "''[[Nine Chapters on the Mathematical Art]]''" (''Jiu-zhang Suanshu'') contains methods for finding the areas of figures; red [[counting rods]] were used to denote positive [[coefficient]]s, black for negative. They were able to solve simultaneous equations involving negative numbers. The [[History of India|ancient Indian]] ''[[Indian mathematics#Bakhshali Manuscript|Bakhshali Manuscript]]'', written around the [[7th century AD|seventh century CE]],<ref>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."</ref> 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|3rd century CE]] referred to the equation equivalent to 4''x'' + 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 [[History of the Mediterranean region|ancient Mediterranean]].
During the [[7th century]], negative numbers were in use in [[India]] to represent debts. The [[Indian mathematics|Indian mathematician]] [[Brahmagupta]], in [[Brahmasphutasiddhanta|Brahma-Sphuta-Siddhanta]] (written in [[628]]) discusses the use of negative numbers to produce the general form [[Quadratic equation#Quadratic formula|quadratic formula]] that remains in use today. He also finds negative solutions to [[quadratic equation]]s and gives rules regarding operations involving negative numbers and [[0 (number)|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".{{rf|2|Dougal1}}{{rf|3|zeros1}} 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 [[Caliph|Islamic world]] learnt about negative numbers from [[Arabic]] translations of [[Brahmagupta]]'s works, and by about [[1000|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.
[[Europe]]an mathematicians however, for the most part, resisted the concept of negative numbers until the [[17th century]], although [[Leonardo of Pisa#Important publications|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 ''[[Leonardo of Pisa|Flos]]''). At the same time, the [[China|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 [http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Maseres.html] 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.{{rf|1|Maseres1}}
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 shared by [[John Wallis]] — and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.{{rf|4|Martinez1}} The argument that negative numbers are greater than infinity involves the quotient <math>\frac{1}{x}</math> and considering what happens as x approaches and then crosses the point x = 0 from the positive side.
== See also ==
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