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{{Infobox number
| number = -1
| cardinal = &minus;1, minus one, {{nowrap|negative one}}
| ordinal = &minus;1st (negative first)
| lang1 = [[Hindu–Arabic numeral system|Arabic]]
| lang1 symbol = &minus;{{resize|150%|١}}
| lang2 = [[Chinese numerals#Negative numbers|Chinese numeral]]
| lang2 symbol = 负一，负弌，负壹
| lang3 = [[Bengali language|Bengali]]
| lang3 symbol = &minus;{{resize|150%|১}}
| lang4 = [[Binary numeral system|Binary]] ([[byte]])
| lang4 symbol = {{table
| row1 = [[Signed magnitude|S&M]]: {{!!}} style="text-align:right" {{!}} 100000001<sub>2</sub>
| row2 = [[Two's complement|2sC]]: {{!!}} style="text-align:right" {{!}} 11111111<sub>2</sub>
}}
| lang5 = [[Hexadecimal|Hex]] ([[byte]])
| lang5 symbol = {{table
| row1 = [[Signed magnitude|S&M]]: {{!!}} style="text-align:right" {{!}} 0x101<sub>16</sub>
| row2 = [[Two's complement|2sC]]: {{!!}} style="text-align:right" {{!}} 0xFF<sub>16</sub>
}}
}}
In [[mathematics]], '''&minus;1''' is the [[additive inverse]] of [[1 (number)|1]], that is, the number that when [[addition|added]] to 1 gives the additive identity element, 0. It is the [[negative number|negative]] [[integer]] greater than negative two (&minus;2) and less than&nbsp;[[0 (number)|0]].
 
Negative one bears relation to [[Euler's identity]] since <math>e^{i \pi} = -1.\!</math>
 
In [[computer science]], '''&minus;1''' is a common initial value for integers and is also used to show that a variable contains no useful information.
 
'''Negative one''' has some similar but slightly different properties to positive one.<ref>[http://books.google.com/books?id=Xrh89dLWZqEC Mathematical analysis and applications]
By Jayant V. Deshpande, ISBN 1-84265-189-7</ref>
 
==Algebraic properties==
Multiplying a number by −1 is equivalent to changing the sign on the number. This can be proved using the [[distributive law]] and the axiom that 1 is the multiplicative identity: for x [[real number|real]], we have
 
:<math>x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0 \cdot x=0</math>
 
where we used the fact that any real x times 0 equals 0, implied by [[cancellation property|cancellation]] from the equation
 
:<math>0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x \, </math>
 
[[Image:ImaginaryUnit5.svg|thumb|right|0, 1, −1, '''''[[imaginary unit|i]]''''', and −'''''i''''' in the [[complex plane|complex]] or [[cartesian plane]]]]
 
In other words,
 
:<math>x+(-1)\cdot x=0 \, </math>
 
so (&minus;1)&nbsp;·&nbsp;''x'' is the arithmetic inverse of ''x'', or&nbsp;&minus;''x''.
 
===Square of −1===
The [[square (algebra)|square]] of &minus;1, i.e. &minus;1 multiplied by &minus;1, equals 1. As a consequence, a product of two negative real numbers is positive.
 
For an algebraic proof of this result, start with the equation
 
:<math>0 =-1\cdot 0 =-1\cdot [1+(-1)]</math>
 
The first equality follows from the above result. The second follows from the definition of &minus;1 as additive inverse of 1: it is precisely that number that when added to 1 gives 0. Now, using the distributive law, we see that
 
:<math>0 =-1\cdot [1+(-1)]=-1\cdot1+(-1)\cdot(-1)=-1+(-1)\cdot(-1)</math>
 
The second equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies
 
:<math>(-1) \cdot (-1) = 1</math>
 
The above arguments hold in any [[ring (mathematics)|ring]], a concept of [[abstract algebra]] generalizing integers and real numbers.
 
===Square roots of −1===
The [[complex number]] ''[[Imaginary unit|i]]'' satisfies ''i''<sup>2</sup> = −1, and as such can be considered as a [[square root]] of −1. The only other complex number ''x'' satisfying the equation ''x''<sup>2</sup> = −1 is −''i''.<ref>{{cite web|url=http://mathforum.org/library/drmath/view/58251.html |title=Ask Dr. Math |publisher=Math Forum |date= |accessdate=2012-10-14}}</ref> In the algebra of [[quaternion]]s, containing the complex plane, the equation ''x''<sup>2</sup> = −1 has an [[Quaternion#Square roots of −1|infinity of solutions]].
 
==Exponentiation to negative integers==
[[Exponentiation]] of a non-zero real number can be extended to [[Exponentiation#Negative integer exponents|negative integers]]. We make the definition that ''x''<sup>&minus;1</sup> = 1/''x'', meaning that we define raising a number to the power &minus;1 to have the same effect as taking its [[reciprocal (mathematics)|reciprocal]]. This definition then extended to negative integers preserves the exponential law ''x''<sup>''a''</sup>''x''<sup>''b''</sup> = ''x''<sup>(''a'' + ''b'')</sup> for a,b real numbers.
 
Exponentiation to negative integers can be extended to invertible elements of a ring, by defining  ''x''<sup>&minus;1</sup> as the multiplicative inverse of x.
 
−1 that appears next to functions or matrices does not mean raising them to the power −1 but their [[inverse functions]] or [[inverse matrices]]. For example, ''f''<sup>−1</sup>(''x'') is the inverse of ''f''(''x''), or sin<sup>−1</sup>(''x'') is a notation of [[arcsine]] function.
 
==Computer representation==
{{main|Signed number representations}}
Most computer systems represent negative integers using [[two's complement]]. In such systems, −1 is represented using a bit pattern of all ones. For example, an 8-bit signed integer using two's complement would represent −1 as the bitstring "11111111", or "FF" in [[hexadecimal]] (base 16). If interpreted as an unsigned integer, the same bitstring of ''n'' ones represents 2<sup>''n''</sup>&nbsp;−&nbsp;1, the largest possible value that ''n'' bits can hold. For example, the 8-bit string "11111111" above represents 2<sup>8</sup>&nbsp;−&nbsp;1&nbsp;=&nbsp;255.
 
==References==
{{Portal|Mathematics}}
{{Reflist}}
 
[[Category:Integers|-8]]
[[Category:One|Negative one]]

Latest revision as of 22:54, 28 November 2014

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