View source for −1

{{Short description|Integer}} 
{{Infobox number
| number = −1
| divisor = 1
| cardinal = −1, '''minus one''', {{nowrap|negative one}}
| ordinal = −1st (negative first)
| lang1 = [[Hindu–Arabic numeral system|Arabic]]
| lang1 symbol = −{{resize|150%|١}}
| lang2 = [[Chinese numerals#Negative numbers|Chinese numeral]]
| lang2 symbol = 负一，负弌，负壹
| lang3 = [[Social:Bengali language|Bengali]]
| lang3 symbol = −{{resize|150%|১}}
| lang4 = Binary ([[Byte|byte]])
| lang4 symbol = {{aligned table|leftright=y
| S&M: | 100000001<sub>2</sub>
| [[Two's complement|2sC]]: | 11111111<sub>2</sub>
}}
| lang5 = [[Hexadecimal|Hex]] ([[Byte|byte]])
| lang5 symbol = {{aligned table|leftright=y
| S&M: | 0x101<sub>16</sub>
| [[Two's complement|2sC]]: | 0xFF<sub>16</sub>
}}
}}
In [[Mathematics|mathematics]], '''−1''' ('''negative one''' or '''minus one''') is the [[Additive inverse|additive inverse]] of 1, that is, the number that when [[Addition|added]] to 1 gives the [[Additive identity|additive identity]] element, 0. It is the [[Negative number|negative]] [[Integer|integer]] greater than negative two (−2) and less than&nbsp;0.

== Algebraic properties ==

===Multiplication===
{{further|Additive inverse}}

Multiplying a number by −1 is equivalent to changing the [[Sign (mathematics)|sign]] of the number – that is, for any {{Mvar|x}} we have {{Math|(−1)&thinsp;⋅&thinsp;''x'' {{=}} −''x''}}. This can be proved using the distributive law and the axiom that 1 is the multiplicative identity:

:{{Math|''x'' + (−1)&thinsp;⋅&thinsp;''x'' {{=}} 1&thinsp;⋅&thinsp;''x'' + (−1)&thinsp;⋅&thinsp;''x'' {{=}} (1 + (−1))&thinsp;⋅&thinsp;''x'' {{=}} 0&thinsp;⋅&thinsp;''x'' {{=}} 0}}.

Here we have used the fact that any number {{Mvar|x}} times 0 equals 0, which follows by [[Cancellation property|cancellation]] from the equation

:{{Math|0&thinsp;⋅&thinsp;''x'' {{=}} (0 + 0)&thinsp;⋅&thinsp;''x'' {{=}} 0&thinsp;⋅&thinsp;''x'' + 0&thinsp;⋅&thinsp;''x''}}.

[[File:ImaginaryUnit5.svg|thumb|right|0, 1, −1, {{mvar|[[Imaginary unit|i]]}}, and −{{Mvar|i}} in the [[Complex plane|complex]] or cartesian plane]]

In other words,

:{{Math|''x'' + (−1)&thinsp;⋅&thinsp;''x'' {{=}} 0}},

so {{Math|(−1)&thinsp;⋅&thinsp;''x''}} is the additive inverse of {{Mvar|x}}, i.e. {{Math|(−1)&thinsp;⋅&thinsp;''x'' {{=}} −''x''}}, as was to be shown.

=== Square of −1 ===
The [[Square (algebra)|square]] of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of two negative numbers is positive.

For an algebraic proof of this result, start with the equation

:{{Math|0 {{=}} −1&thinsp;⋅ 0 {{=}} −1&thinsp;⋅ [1&thinsp;+ (−1)]}}.

The first equality follows from the above result, and the second follows from the definition of −1 as additive inverse of 1: it is precisely that number which when added to 1 gives 0. Now, using the distributive law, it can be seen that

:{{Math|0 {{=}} −1 ⋅ [1&thinsp;+ (−1)] {{=}} −1&thinsp;⋅ 1 + (−1) ⋅ (−1) {{=}} −1&thinsp;+ (−1) ⋅ (−1)}}.

The third 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) ⋅ (−1) {{=}} 1}}.

The above arguments hold in any [[Ring (mathematics)|ring]], a concept of [[Abstract algebra|abstract algebra]] generalizing integers and [[Real number|real number]]s.<ref name="MultIdRng">{{Cite book |last=Nathanson |first=Melvyn B.  |title=Elementary Methods in Number Theory |chapter=Chapter 2: Congruences |chapter-url=https://link.springer.com/chapter/10.1007/978-0-387-22738-2_2 |series=[[Graduate Texts in Mathematics]] |publisher=[[Company:Springer Science+Business Media|Springer]] |location=New York |volume=195 |year=2000 |pages=xviii, 1−514 |isbn=978-0-387-98912-9 |oclc=42061097 |mr=1732941 }}</ref>{{rp|p.48}}

=== Square roots of −1 ===
Although there are no [[Real number|real]] square roots of −1, the [[Complex number|complex number]] {{mvar|[[Imaginary unit|i]]}} satisfies {{Math|''i''<sup>2</sup> {{=}} −1}}, and as such can be considered as a [[Square root|square root]] of −1.<ref name="imaginary">{{Cite book |last=Bauer |first=Cameron |year=2007 |chapter=Chapter 13: Complex Numbers |title=Algebra for Athletes |edition=2nd |publisher=Nova Science Publishers |location=Hauppauge |page=273 |url=https://books.google.com/books?id=GmB1cSGHbZcC&pg=PA273 |isbn=978-1-60021-925-2 |oclc=957126114 }}</ref> The only other complex number whose square is −1 is −{{Mvar|i}} because there are exactly two square roots of any non‐zero complex number, which follows from the [[Fundamental theorem of algebra|fundamental theorem of algebra]]. In the algebra of [[Quaternion|quaternion]]s – where the fundamental theorem does not apply – which contains the complex numbers, the equation {{Math|''x''<sup>2</sup> {{=}} −1}} has [[Quaternion#Square roots of −1|infinitely many solutions]].<ref>{{Cite book |last=Perlis |first=Sam |chapter=Capsule 77: Quaternions |title=Historical Topics in Algebra |url=https://archive.org/details/historicaltopics0000nati/page/38/mode/2up |url-access=registration |publisher=[[Organization:National Council of Teachers of Mathematics|National Council of Teachers of Mathematics]] |location=Reston, VA |series=Historical Topics for the Mathematical Classroom |volume=31 |year=1971 |page=39 |isbn=9780873530583 |oclc=195566 }}</ref><ref>{{Cite book |last=Porteous |first=Ian R. |chapter=Chapter 8: Quaternions |url=https://www.maths.ed.ac.uk/~v1ranick/papers/porteous3.pdf |title=Clifford Algebras and the Classical Groups |series=Cambridge Studies in Advanced Mathematics |publisher={{wipe|Cambridge University Press}} |location=Cambridge |volume=50 |pages=60 |year=1995 |doi=10.1017/CBO9780511470912.009 |isbn=9780521551779 |oclc=32348823 |mr=1369094 }}</ref>

== Inverse and invertible elements ==
[[File:Geogebra f(x)=1÷x 20211118.svg|350px|thumb|The reciprocal function {{Math|''f''(''x'') {{=}} ''x''<sup>−1</sup>}} where for every {{Mvar|x}} except 0, {{Math|''f''(''x'')}} represents its [[Multiplicative inverse|multiplicative inverse]] ]]

[[Exponentiation]] of a non‐zero real number can be [[Exponentiation#Negative exponents|extended to negative integers]], where raising a number to the power −1 has the same effect as taking its [[Multiplicative inverse|multiplicative inverse]]: 
:{{Math|''x''<sup>−1</sup> {{=}} {{sfrac|1|''x''}}}}. 

This definition is then applied to negative integers, preserving the exponential law {{Math|''x''<sup>''a''</sup>''x''<sup>''b''</sup> {{=}} ''x''<sup>(''a'' + ''b'')</sup>}} for real numbers {{Mvar|a}} and {{Mvar|b}}.

A −1 superscript in {{Math|''f''<sup> −1</sup>(''x'')}} takes the [[Inverse function|inverse function]] of {{Math|''f''(''x'')}}, where {{math|(&thinsp;''f''(''x''))<sup>−1</sup>}} specifically denotes a [[Pointwise|pointwise]] reciprocal.{{efn|1=For example, {{Math|sin<sup>−1</sup>(''x'')}} is a notation for the arcsine function. }} Where {{Math|''f''}} is [[Bijection|bijective]] specifying an output [[Codomain|codomain]] of every {{Math|''y'' ∈ ''Y''&thinsp;}} from every input [[Domain of a function|domain]] {{Math|''x'' ∈ ''X''}}, there will be 

:{{Math|''f''<sup>&thinsp;−1</sup>(&thinsp;''f''(''x'')) {{=}} ''x'',&thinsp;}} and {{Math|&thinsp;''f''<sup>&thinsp;−1</sup>(&thinsp;''f''(''y'')) {{=}} ''y''}}.

When a subset of the codomain is specified inside the function {{Math|''f''}}, its inverse will yield an inverse image, or preimage, of that subset under the function.

=== Rings ===

Exponentiation to negative integers can be further extended to [[Inverse element|invertible elements]] of a ring by defining {{Math|''x''<sup>−1</sup>}} as the multiplicative inverse of {{Mvar|x}}; in this context, these elements are considered [[Unit (ring theory)|units]].<ref name="MultIdRng" />{{rp|p.49}}

In a [[Polynomial#Polynomial ring|polynomial domain]] {{Math|''F''&thinsp;<nowiki>[</nowiki>''x''<nowiki>]</nowiki>}} over any [[Field (mathematics)#Constructing fields|field]] {{Math|''F''}}, the polynomial {{Mvar|x}} has no inverse. If it did have an inverse {{Math|''q''(''x'')}}, then there would be<ref>{{Cite book |last1=Czapor |first1=Stephen R. |last2=Geddes |first2=Keith O. |last3=Labahn |first3=George |chapter=Chapter 2: Algebra of Polynomials, Rational Functions, and Power Series |title=Algorithms for Computer Algebra |url=https://link.springer.com/book/10.1007/b102438 |publisher=Kluwer Academic Publishers |location=Boston |edition=1st |year=1992 |pages=41,42 |isbn=978-0-7923-9259-0 |oclc=26212117 |s2cid=964280 |zbl=0805.68072 |via=[[Company:Springer Science+Business Media|Springer]] }}</ref>

:{{Math|''x'' ''q''(''x'') {{=}} 1 ⇒ ''deg''&thinsp;(''x'') + ''deg''&thinsp;(''q''(''x'')) {{=}} ''deg''&thinsp;(1)}}

:{{Math|{{nbsp|16}}{{hair space}}⇒ 1 + ''deg''&thinsp;(''q''(''x'')) {{=}} 0}}

:{{Math|{{nbsp|16}}{{hair space}}⇒ ''deg''&thinsp;(''q''(''x'')) {{=}} −1 }}

which is not possible, and therefore, {{Math|''F''&thinsp;<nowiki>[</nowiki>''x''<nowiki>]</nowiki>}} is not a field. More specifically, because the polynomial is not [[Polynomial ring#Terminology|continuous]], it is not a unit in {{Math|''F''}}.

==Uses==
===Sequences===
[[Integer sequence]]s commonly use −1 to represent an [[Uncountable set|uncountable set]], in place of "[[Infinity|<span style="font-size:115%; vertical-align:-5%;">{{math|∞}}</span>]]" as a value resulting from a given [[Sequence|index]].<ref name="IntSeq">See searches with "−1 if no such number exists" or "−1 if the number is infinite" in the [[On-Line Encyclopedia of Integer Sequences|OEIS]] for an assortment of relevant sequences.</ref>

As an example, the number of regular convex [[Polytope|polytope]]s in {{math|1=''n'' }}-dimensional space is,
:{{math|1={1, 1, −1, 5, 6, 3, 3, ...} }} for {{math|1=''n'' = {0, 1, 2, ...} }} {{OEIS|A060296 }}.

−1 can also be used as a [[Null (mathematics)|null value]], from an index that yields an [[Empty set|empty set]] {{math|1=∅ }} or [[Number#Main classification|non-integer]] where the general [[Expression (mathematics)|expression]] describing the [[Sequence|sequence]] is not [[Satisfiability|satisfied]], or met.<ref name="IntSeq" />

For instance, the smallest {{math|1=''k'' > 1 }} such that in the interval {{math|1=1...''k'' }} there are as many integers that have exactly twice {{math|1=''n'' }} [[Divisor|divisor]]s as there are [[Prime number|prime number]]s is, 
:{{math|1= {2, 27, −1, 665, −1, 57675, −1, 57230, −1} }} for {{math|1=''n'' = {1, 2, ..., 9} }} {{OEIS|A356136 }}.

A non-integer or empty element is often represented by [[0#Mathematics|0]] as well.

===Computing===
In [[Software development|software development]], −1 is a common initial value for integers and is also used to show that [[Sentinel value|a variable contains no useful information]].{{Citation needed|date=November 2023}}

== See also ==

* [[Balanced ternary]]
* [[Menelaus's theorem]]

== References ==
{{notelist}}
{{reflist}}

[[Category:Integers|−1]]
[[Category:1 (number)|Negative one]]

{{Sourceattribution|−1}}