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Additive inverse

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</style><div role="note" class="hatnote navigation-not-searchable">"Opposite number" redirects here. For other uses, see <a href="Analog_(disambiguation)" class="mw-redirect mw-disambig" title="Analog (disambiguation)">analog</a> and <a href="Counterpart_(disambiguation)" class="mw-redirect mw-disambig" title="Counterpart (disambiguation)">counterpart</a>.</div>

In mathematics, the additive inverse of an <a href="Element_(mathematics)" title="Element (mathematics)">element</a> x, denoted −x,<a href="#cite_note-1">[1]</a> is the element that when <a href="Addition" title="Addition">added</a> to x, yields the <a href="Additive_identity" title="Additive identity">additive identity</a>.<a href="#cite_note-2">[2]</a> This additive identity is often the number <a href="0" title="0">0 (zero)</a>, but it can also refer to a more generalized <a href="Zero_element" title="Zero element">zero element</a>.

In <a href="Elementary_mathematics" title="Elementary mathematics">elementary mathematics</a>, the additive inverse is often referred to as the opposite number,<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> or its negative.<a href="#cite_note-5">[5]</a> The <a href="Unary_operation" title="Unary operation">unary operation</a> of arithmetic negation<a href="#cite_note-6">[6]</a> is closely related to <a href="Subtraction" title="Subtraction">subtraction</a><a href="#cite_note-7">[7]</a> and is important in <a href="Elementary_algebra" title="Elementary algebra">solving algebraic equations</a>.<a href="#cite_note-:0-8">[8]</a> Not all <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> where addition is defined have an additive inverse, such as the <a href="Natural_number" title="Natural number">natural numbers</a>.<a href="#cite_note-:1-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Common_examples">Common examples</h2></div>

When working with <a href="Integer" title="Integer">integers</a>, <a href="Rational_number" title="Rational number">rational numbers</a>, <a href="Real_number" title="Real number">real numbers</a>, and <a href="Complex_number" title="Complex number">complex numbers</a>, the additive inverse of any number can be found by multiplying it by <a href="%E2%88%921" title="−1">−1</a>.<a href="#cite_note-:0-8">[8]</a>

<caption>Simple cases of additive inverses

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -n}</annotation>

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</math><img src="./4f00139753ecf4fe00a10a17bd5620b70a61b29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle -n}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 7}</annotation>

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</math><img src="./ee716ec61382a6b795092c0edd859d12e64cbba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 7}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -7}</annotation>

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</math><img src="./9caacddc9e190990809d63ba721cb15b76c1ac7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -7}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 0.35}</annotation>

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</math><img src="./4805f66ca278a8e44c4a5c6cd1b537c5a7373e1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.134ex; height:2.176ex;" alt="{\displaystyle 0.35}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -0.35}</annotation>

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</math><img src="./7ce90b59d5f80cc7b46add2d96894135d0d31f51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.942ex; height:2.343ex;" alt="{\displaystyle -0.35}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}}</annotation>

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</math><img src="./a2dfb63ee75ec084f2abb25d248bc151a2687508.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{4}}}</annotation>

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</math><img src="./204c4acf21e018ddc2a68336f6332fb511c49862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.807ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{4}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

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</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -\pi }</annotation>

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</math><img src="./f2359073fe90a84a705e02f0c1e63b32df850a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.14ex; height:2.176ex;" alt="{\displaystyle -\pi }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 1+2i}</annotation>

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</math><img src="./ba42d278356f23343a50cb5af6111fc1b4e13a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.968ex; height:2.343ex;" alt="{\displaystyle 1+2i}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -1-2i}</annotation>

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</math><img src="./ed3be81d50acd21c0e1edfbe662fa3fdce7600ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.776ex; height:2.343ex;" alt="{\displaystyle -1-2i}" loading="lazy">

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The concept can also be extended to algebraic expressions, which is often used when balancing <a href="Equation" title="Equation">equations</a>.

<caption>Additive inverses of algebraic expressions

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -n}</annotation>

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</math><img src="./4f00139753ecf4fe00a10a17bd5620b70a61b29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle -n}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle a-b}</annotation>

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</math><img src="./1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -(a-b)=-a+b}</annotation>

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</math><img src="./98d6a9fa4e960684684bb98dfac722dab6790444.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.66ex; height:2.843ex;" alt="{\displaystyle -(a-b)=-a+b}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle 2x^{2}+5}</annotation>

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</math><img src="./745de08a585c5fe06082b3c9d3e787c931456536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle 2x^{2}+5}" loading="lazy">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle -(2x^{2}+5)=-2x^{2}-5}</annotation>

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</math><img src="./4948dbcf16d91f9f65257631ef35fb1967e84488.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.622ex; height:3.176ex;" alt="{\displaystyle -(2x^{2}+5)=-2x^{2}-5}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x+2}}}</annotation>

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</math><img src="./9a9efa14039f8405b9f7dee979a60aef51b54ed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.169ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{x+2}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{x+2}}}</annotation>

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</math><img src="./2fbe11aa765f0401a4836dc3c814a740e6205166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.977ex; height:5.343ex;" alt="{\displaystyle -{\frac {1}{x+2}}}" loading="lazy">

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<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta }}</annotation>

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</math><img src="./acf13ccc2ff03200de292c05ffcdb9bc230be127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.896ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta }}" loading="lazy">

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<msqrt>

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<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle -({\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta })=-{\sqrt {2}}\sin {\theta }+{\sqrt {3}}\cos {2\theta }}</annotation>

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</math><img src="./85e75006590830dfda1cdbca62b97ef57a9ce02c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.315ex; height:3.176ex;" alt="{\displaystyle -({\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta })=-{\sqrt {2}}\sin {\theta }+{\sqrt {3}}\cos {2\theta }}" loading="lazy">

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<div class="mw-heading mw-heading2"><h2 id="Relation_to_subtraction">Relation to subtraction</h2></div>

The additive inverse is closely related to <a href="Subtraction" title="Subtraction">subtraction</a>, which can be viewed as an addition using the inverse:

Conversely, the additive inverse can be thought of as subtraction from zero:

This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

Given an algebraic structure defined under addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,+)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (S,+)}</annotation>

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</math><img src="./757e19e5bf3fd2aa1fcc68e2ff18d9805acc2f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.151ex; height:2.843ex;" alt="{\displaystyle (S,+)}" loading="lazy"> with an additive identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\in S}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle e\in S}</annotation>

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</math><img src="./f71122665efb66eb1109282c5a2c4e8e66c8e921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle e\in S}" loading="lazy">, an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in S}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x\in S}</annotation>

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</math><img src="./51186ba8afb2067573a9082d55dd383df1ea9214.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle x\in S}" loading="lazy"> has an additive inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in S}">

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<annotation encoding="application/x-tex">{\displaystyle y\in S}</annotation>

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</math><img src="./9d26e188ccec145558c09a23bb1d25149ea67973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.495ex; height:2.509ex;" alt="{\displaystyle y\in S}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y=e}">

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<annotation encoding="application/x-tex">{\displaystyle x+y=e}</annotation>

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</math><img src="./075a4fd6a9e2340d278fd43e185c715d1d25dece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.508ex; height:2.343ex;" alt="{\displaystyle x+y=e}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y+x=e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y+x=e}</annotation>

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</math><img src="./9133436a38113f80843b61177ad5ce5387baa28e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.508ex; height:2.343ex;" alt="{\displaystyle y+x=e}" loading="lazy">.<a href="#cite_note-:1-9">[9]</a>

Addition is typically only used to refer to a <a href="Commutative_property" title="Commutative property">commutative</a> operation, but for some systems of numbers, such as <a href="Floating_point" class="mw-redirect" title="Floating point">floating point</a>, it might not be <a href="Associative_property" title="Associative property">associative</a>.<a href="#cite_note-11">[11]</a> When it is associative, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)+c=a+(b+c)}">

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<annotation encoding="application/x-tex">{\displaystyle (a+b)+c=a+(b+c)}</annotation>

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</math><img src="./46b7b8d31d5845966e6abdbb030c73f343c17d4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle (a+b)+c=a+(b+c)}" loading="lazy">, the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist.

The definition requires <a href="Closure_(mathematics)" title="Closure (mathematics)">closure</a>, that the additive element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> be found in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. However, despite being able to add the natural numbers together, the set of natural numbers does not include the additive inverse values. This is because the additive inverse of a natural number (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3}">

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<annotation encoding="application/x-tex">{\displaystyle -3}</annotation>

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</math><img src="./8e8ffb51bb3837cabdd202845d15aedc67a4ac88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -3}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}">

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<annotation encoding="application/x-tex">{\displaystyle 3}</annotation>

</semantics>

</math><img src="./991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy">) is not a natural number; it is an <a href="Integer" title="Integer">integer</a>. Therefore, the natural numbers in set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> do have additive inverses and their associated inverses are <a href="Negative_number" title="Negative number">negative numbers</a>.

<div class="mw-heading mw-heading2"><h2 id="Further_examples">Further examples</h2></div>

<ul><li>In a <a href="Vector_space" title="Vector space">vector space</a>, the additive inverse −v (often called the <a href="Opposite_vector" class="mw-redirect" title="Opposite vector">opposite vector</a> of v) has the same <a href="Norm_(mathematics)" title="Norm (mathematics)">magnitude</a> as v and but the opposite direction.<a href="#cite_note-12">[12]</a></li>

<li>In <a href="Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11).<a href="#cite_note-13">[13]</a></li>

<li>In a <a href="Boolean_ring" title="Boolean ring">Boolean ring</a>, which has elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{0,1\}}</annotation>

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</math><img src="./28de5781698336d21c9c560fb1cbb3fb406923eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{0,1\}}" loading="lazy"> addition is often defined as the <a href="Symmetric_difference" title="Symmetric difference">symmetric difference</a>. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+0=0}">

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<annotation encoding="application/x-tex">{\displaystyle 0+0=0}</annotation>

</semantics>

</math><img src="./fe19ab3f16789e36aa5d09f0f5752037966a184b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 0+0=0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+1=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0+1=1}</annotation>

</semantics>

</math><img src="./3a5a3b1ffec99ff23f991886a85cc7c63572c026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 0+1=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+0=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+0=1}</annotation>

</semantics>

</math><img src="./dcf199855ac1237e8ec0f3ac2c8512985127c966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+0=1}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+1=0}</annotation>

</semantics>

</math><img src="./8a94a15bb0333c813ac2ed0d697e5e57092cbbc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=0}" loading="lazy">. Our additive identity is 0, and both elements are their own additive inverse as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+0=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0+0=0}</annotation>

</semantics>

</math><img src="./fe19ab3f16789e36aa5d09f0f5752037966a184b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 0+0=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+1=0}</annotation>

</semantics>

</math><img src="./8a94a15bb0333c813ac2ed0d697e5e57092cbbc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 1+1=0}" loading="lazy">.<a href="#cite_note-14">[14]</a></li></ul>

<ul><li><a href="Absolute_value" title="Absolute value">Absolute value</a> (related through the identity |−x| = |x|).</li>

<li><a href="Monoid" title="Monoid">Monoid</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse function</a></li>

<li><a href="Involution_(mathematics)" title="Involution (mathematics)">Involution (mathematics)</a></li>

<li><a href="Multiplicative_inverse" title="Multiplicative inverse">Multiplicative inverse</a></li>

<li><a href="Reflection_(mathematics)" title="Reflection (mathematics)">Reflection (mathematics)</a></li>

<li><a href="Reflection_symmetry" title="Reflection symmetry">Reflection symmetry</a></li>

<li><a href="Semigroup" title="Semigroup">Semigroup</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2></div>

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