Examine individual changes

'{{Short description|Integer}} {{about|the number}} {{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 = [[Bengali language|Bengali]] | lang3 symbol = −{{resize|150%|১}} | lang4 = [[Binary numeral system|Binary]] ([[byte]]) | lang4 symbol = {{aligned table|leftright=y | [[Signed magnitude|S&M]]: | 100000001₂ | [[Two's complement|2sC]]: | 11111111₂ }} | lang5 = [[Hexadecimal|Hex]] ([[byte]]) | lang5 symbol = {{aligned table|leftright=y | [[Signed magnitude|S&M]]: | 0x101₁₆ | [[Two's complement|2sC]]: | 0xFF₁₆ }} }} In [[mathematics]], '''−1''' ('''negative one''' or '''minus one''') 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 (−2) and less than&nbsp;[[0 (number)|0]]. == <big>1</big> == :{{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]] generalizing integers and [[real number]]s.<ref name="MultIdRng">{{Cite book |last=Nathanson |first=Melvyn B. |author-link=Melvyn B. Nathanson |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=[[Springer Science+Business Media|Springer]] |location=New York |volume=195 |year=2000 |pages=xviii, 1−514 |doi=10.1007/978-0-387-22738-2_2 |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]] {{mvar|[[Imaginary unit|i]]}} satisfies {{Math|''i''² {{=}} −1}}, and as such can be considered as a [[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 |chapter-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]]. In the algebra of [[quaternion]]s – where the fundamental theorem does not apply – which contains the complex numbers, the equation {{Math|''x''² {{=}} −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 |chapter-url=https://archive.org/details/historicaltopics0000nati/page/38/mode/2up |chapter-url-access=registration |publisher=[[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. |author-link=Ian R. Porteous |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=[[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''⁻¹}} where for every {{Mvar|x}} except 0, {{Math|''f''(''x'')}} represents its [[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]]: :{{Math|''x''⁻¹ {{=}} {{sfrac|1|''x''}}}}. This definition is then applied to negative integers, preserving the exponential law {{Math|''x''^''a''''x''^''b'' {{=}} ''x''^{(''a'' + ''b'')}}} for real numbers {{Mvar|a}} and {{Mvar|b}}. A −1 [[superscript]] in {{Math|''f''⁻¹(''x'')}} takes the [[inverse function]] of {{Math|''f''(''x'')}}, where {{math|(&thinsp;''f''(''x''))⁻¹}} specifically denotes a [[pointwise]] reciprocal.{{efn|1=For example, {{Math|sin⁻¹(''x'')}} is a notation for the [[arcsine]] function. }} Where {{Math|''f''}} is [[Bijection|bijective]] specifying an output [[codomain]] of every {{Math|''y'' ∈ ''Y''&thinsp;}} from every input [[Domain of a function|domain]] {{Math|''x'' ∈ ''X''}}, there will be :{{Math|''f''^&thinsp;−1(&thinsp;''f''(''x'')) {{=}} ''x'',&thinsp;}} and {{Math|&thinsp;''f''^&thinsp;−1(&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''⁻¹}} 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 |doi=10.1007/b102438 |isbn=978-0-7923-9259-0 |oclc=26212117 |s2cid=964280 |zbl=0805.68072 |via=[[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]], 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]]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]] {{math|1=∅ }} or [[Number#Main classification|non-integer]] where the general [[Expression (mathematics)|expression]] describing the [[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]]s as there are [[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]], −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 == {{Portal|Mathematics}} * [[Balanced ternary]] * [[Menelaus's theorem]] == References == {{notelist}} {{reflist}} [[Category:Integers|-1]] [[Category:1 (number)|Negative one]]'