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Additive map

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</style><div role="note" class="hatnote navigation-not-searchable">For additive functions in number theory, see <a href="Additive_function" title="Additive function">Additive function</a>. For additive functions on the reals, see <a href="Cauchy's_functional_equation" title="Cauchy's functional equation">Cauchy's functional equation</a>.</div>

In <a href="Abstract_algebra" title="Abstract algebra">algebra</a>, an additive map, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy">-linear map or additive function is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> that preserves the addition operation:<a href="#cite_note-1">[1]</a>

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

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for every pair of elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <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"> in the <a href="Domain_of_a_function" title="Domain of a function">domain</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

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

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</math><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"> For example, any <a href="Linear_map" title="Linear map">linear map</a> is additive. When the domain is the <a href="Real_number" title="Real number">real numbers</a>, this is <a href="Cauchy's_functional_equation" title="Cauchy's functional equation">Cauchy's functional equation</a>. For a specific case of this definition, see <a href="Additive_polynomial" title="Additive polynomial">additive polynomial</a>.

More formally, an additive map is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-<a href="Module_homomorphism" title="Module homomorphism">module homomorphism</a>. Since an <a href="Abelian_group" title="Abelian group">abelian group</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-<a href="Module_(mathematics)" title="Module (mathematics)">module</a>, it may be defined as a <a href="Group_homomorphism" title="Group homomorphism">group homomorphism</a> between abelian groups.

A map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\times W\to X}">

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<annotation encoding="application/x-tex">{\displaystyle V\times W\to X}</annotation>

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</math><img src="./9e5f94fbf1682c7c3ffb91557ec93cf3aa8bcd1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.657ex; height:2.176ex;" alt="{\displaystyle V\times W\to X}" loading="lazy"> that is additive in each of two arguments separately is called a bi-additive map or a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-bilinear map.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

Typical examples include maps between <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="Vector_space" title="Vector space">vector spaces</a>, or <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> that preserve the <a href="Additive_group" title="Additive group">additive group</a>. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> are additive maps, then the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}">

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

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</math><img src="./4d94a24abd865f6f9fd67a7df7e531cae1c769b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}" loading="lazy"> (defined <a href="Pointwise" title="Pointwise">pointwise</a>) is additive.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Definition of scalar multiplication by an integer

Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is an additive group with identity element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.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 0}" loading="lazy"> and that the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x.}">

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

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</math><img src="./d98781bc4ba61e6bfaa7a8bfd7704f2671865444.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.785ex; height:2.176ex;" alt="{\displaystyle -x.}" loading="lazy"> For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} ,}">

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

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</math><img src="./32f0d0d4c10d0cdc803b261e36c2a7c0f3c122e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.433ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {Z} ,}" loading="lazy"> let:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0,\\&&&x&&+\cdots +&&x&&~~~~{\text{(}}n&&{\text{ summands) }}&&~{\text{ when }}n>0,\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{(}}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0,\\\end{alignedat}}\right.}">

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<mtext> summands) </mtext>

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<mtext> summands) </mtext>

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<annotation encoding="application/x-tex">{\displaystyle nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0,\\&&&x&&+\cdots +&&x&&~~~~{\text{(}}n&&{\text{ summands) }}&&~{\text{ when }}n>0,\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{(}}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0,\\\end{alignedat}}\right.}</annotation>

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Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)x=-x}">

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

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</math><img src="./30f858b063f4aac63865a4cec03d4d1ce09b7f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.346ex; height:2.843ex;" alt="{\displaystyle (-1)x=-x}" loading="lazy"> and it can be shown that for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle m,n\in \mathbb {Z} }</annotation>

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</math><img src="./f932c786e686277b8f060aae74134ce9fc3e3348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.86ex; height:2.509ex;" alt="{\displaystyle m,n\in \mathbb {Z} }" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

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

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</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m+n)x=mx+nx}">

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<annotation encoding="application/x-tex">{\displaystyle (m+n)x=mx+nx}</annotation>

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</math><img src="./fc3af07b6efc316942020c9d3f41b63013419ad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.448ex; height:2.843ex;" alt="{\displaystyle (m+n)x=mx+nx}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(nx)=(-n)x=n(-x).}">

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

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</math><img src="./9ca878151b3393b8a2033389a651b1153de50e6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.869ex; height:2.843ex;" alt="{\displaystyle -(nx)=(-n)x=n(-x).}" loading="lazy">

This definition of scalar multiplication makes the cyclic subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} x}">

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

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</math><img src="./71ce3c350a3fab3228411d22f2d2c9e892310b76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.88ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} x}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> into a <a href="Module_(mathematics)" title="Module (mathematics)">left <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-module</a>; if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is commutative, then it also makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> into a left <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-module.

Homogeneity over the integers

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

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</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is an additive map between additive groups then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0}">

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

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</math><img src="./8d308c32c9894b88115262081194321ae7d9bbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}" loading="lazy"> and for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X,}">

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

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</math><img src="./e3ebdb0a09f0721ccdd0b779e0a21caf386be82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.797ex; height:2.509ex;" alt="{\displaystyle x\in X,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)=-f(x)}">

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

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</math><img src="./b022ffe516cf5bc26a68fd954753aa2bddf508f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle f(-x)=-f(x)}" loading="lazy"> (where negation denotes the additive inverse) and<a href="#cite_note-3">[proof 1]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(nx)=nf(x)\quad {\text{ for all }}n\in \mathbb {Z} .}</annotation>

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Consequently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x-y)=f(x)-f(y)}">

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

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</math><img src="./1923d4019d3d710301e6b9ab2da07bfacdb97cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.013ex; height:2.843ex;" alt="{\displaystyle f(x-y)=f(x)-f(y)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}">

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

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</math><img src="./6d72f66ab332ed430aa9b34ff18c9723c4fea2a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}" loading="lazy"> (where by definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y:=x+(-y)}">

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

</semantics>

</math><img src="./0e5dbf4e384d0d631c9ca9e6b5df3943fbe40d62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.014ex; height:2.843ex;" alt="{\displaystyle x-y:=x+(-y)}" loading="lazy">).

In other words, every additive map is <a href="Homogeneous_function" title="Homogeneous function">homogeneous over the integers</a>. Consequently, every additive map between <a href="Abelian_group" title="Abelian group">abelian groups</a> is a <a href="Module_homomorphism" title="Module homomorphism">homomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-modules</a>.

Homomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">-modules

If the additive <a href="Abelian_group" title="Abelian group">abelian groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are also a <a href="Unital_module" class="mw-redirect" title="Unital module">unital</a> <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> over the rationals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> (such as real or complex <a href="Vector_space" title="Vector space">vector spaces</a>) then an additive map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> satisfies:<a href="#cite_note-4">[proof 2]</a>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(qx)=qf(x)\quad {\text{ for all }}q\in \mathbb {Q} {\text{ and }}x\in X.}</annotation>

</semantics>

</math>

In other words, every additive map is <a href="Homogeneous_over_the_rational_numbers" class="mw-redirect" title="Homogeneous over the rational numbers">homogeneous over the rational numbers</a>. Consequently, every additive maps between unital <a href="Module_(mathematics)" title="Module (mathematics)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">-modules</a> is a <a href="Module_homomorphism" title="Module homomorphism">homomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy">-modules</a>.

Despite being homogeneous over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./91185244fbdded6ea99a5e9e6603299128b10928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} ,}" loading="lazy"> as described in the article on <a href="Cauchy's_functional_equation" title="Cauchy's functional equation">Cauchy's functional equation</a>, even when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Y=\mathbb {R} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=Y=\mathbb {R} ,}</annotation>

</semantics>

</math><img src="./62b3ea61474ac7c5cea150e562b4e520e2f1f3e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.275ex; height:2.509ex;" alt="{\displaystyle X=Y=\mathbb {R} ,}" loading="lazy"> it is nevertheless still possible for the additive function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"> to not be <a href="Real_homogeneity" class="mw-redirect" title="Real homogeneity">homogeneous over the real numbers</a>; said differently, there exist additive maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"> that are not of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=s_{0}x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=s_{0}x}</annotation>

</semantics>

</math><img src="./19bd246acb5f4f540bdbe08e97040d5115916d5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.99ex; height:2.843ex;" alt="{\displaystyle f(x)=s_{0}x}" loading="lazy"> for some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}\in \mathbb {R} .}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{0}\in \mathbb {R} .}</annotation>

</semantics>

</math><img src="./2c25abc53c9a31a894c1d60d796a9a7dfee1c8da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.31ex; height:2.509ex;" alt="{\displaystyle s_{0}\in \mathbb {R} .}" loading="lazy">

In particular, there exist additive maps that are not <a href="Linear_map" title="Linear map">linear maps</a>.

<ul><li><a href="Antilinear_map" title="Antilinear map">Antilinear map</a> – Conjugate homogeneous additive map</li></ul>

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

/* start https://en.wikipedia.org/ */

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/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFLeslie_Hogben2013" class="citation cs2"><a href="Leslie_Hogben" title="Leslie Hogben">Leslie Hogben</a> (2013), Handbook of Linear Algebra (3 ed.), CRC Press, pp. 30–8, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781498785600</bdi></cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFN._Bourbaki1989" class="citation cs2">N. Bourbaki (1989), Algebra Chapters 1–3, Springer, p. 243</cite>

</li>

</ol></div></div>

</div>

Proofs

</mstyle>

</mrow>

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

</semantics>

</math><img src="./341a536345afdb0876c8ccf79c5a5822fa29ac78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.042ex; height:2.843ex;" alt="{\displaystyle f(0)=f(0+0)=f(0)+f(0)}" loading="lazy"> so adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -f(0)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4dea37b1aa2d9b9d7791bf91fa2f28b153a75766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.058ex; height:2.843ex;" alt="{\displaystyle -f(0)}" loading="lazy"> to both sides proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)=0.}</annotation>

</semantics>

</math><img src="./cc3f237e89ebcbd24f17125497c63b4d3749dbf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.158ex; height:2.843ex;" alt="{\displaystyle f(0)=0.}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=f(0)=f(x+(-x))=f(x)+f(-x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./55a8f5c1883f701a2f67447a2f876aad4c1fa39f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.397ex; height:2.843ex;" alt="{\displaystyle 0=f(0)=f(x+(-x))=f(x)+f(-x)}" loading="lazy"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)=-f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b022ffe516cf5bc26a68fd954753aa2bddf508f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle f(-x)=-f(x)}" loading="lazy"> where by definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)f(x):=-f(x).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1)f(x):=-f(x).}</annotation>

</semantics>

</math><img src="./2f23d230746326d37bdb28f0a3c06d8d04cee8e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.815ex; height:2.843ex;" alt="{\displaystyle (-1)f(x):=-f(x).}" loading="lazy"> Induction shows that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> is positive then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(nx)=nf(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(nx)=nf(x)}</annotation>

</semantics>

</math><img src="./fe1988372667eb7701fdcdc77faab40926315db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.723ex; height:2.843ex;" alt="{\displaystyle f(nx)=nf(x)}" loading="lazy"> and that the additive inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nf(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9e6e535422c012b5d4b7720ccc2e9a0c265bad7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.812ex; height:2.843ex;" alt="{\displaystyle nf(x)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(-f(x)),}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b55c90038fe71a8ca438cd16197781e9197f7b5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.077ex; height:2.843ex;" alt="{\displaystyle n(-f(x)),}" loading="lazy"> which implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f((-n)x)=f(n(-x))=nf(-x)=n(-f(x))=-(nf(x))=(-n)f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3602793929f87cc7706608c61dfdd3424fd841be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.261ex; height:2.843ex;" alt="{\displaystyle f((-n)x)=f(n(-x))=nf(-x)=n(-f(x))=-(nf(x))=(-n)f(x)}" loading="lazy"> (this shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(nx)=nf(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(nx)=nf(x)}</annotation>

</semantics>

</math><img src="./fe1988372667eb7701fdcdc77faab40926315db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.723ex; height:2.843ex;" alt="{\displaystyle f(nx)=nf(x)}" loading="lazy"> holds for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ad9f68fb8426c4411972241aac6c359e7812eaba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n<0}" loading="lazy">). <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\frac {m}{n}}\in \mathbb {Q} }">

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q={\frac {m}{n}}\in \mathbb {Q} }</annotation>

</semantics>

</math><img src="./0bfb1324b8854629d46e2dcda17be7c1aabea5e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.693ex; height:4.676ex;" alt="{\displaystyle q={\frac {m}{n}}\in \mathbb {Q} }" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./f932c786e686277b8f060aae74134ce9fc3e3348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.86ex; height:2.509ex;" alt="{\displaystyle m,n\in \mathbb {Z} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./57ab13b00b6f8bb19aea0dcfcc78f18a2b6cbffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.302ex; height:2.176ex;" alt="{\displaystyle n>0.}" loading="lazy"> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y:={\frac {1}{n}}x.}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y:={\frac {1}{n}}x.}</annotation>

</semantics>

</math><img src="./cf8a418e79b9c9715257b1f4e0c96f37f1482176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.108ex; height:5.176ex;" alt="{\displaystyle y:={\frac {1}{n}}x.}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ny=n\left({\frac {1}{n}}x\right)=\left(n{\frac {1}{n}}\right)x=(1)x=x,}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ny=n\left({\frac {1}{n}}x\right)=\left(n{\frac {1}{n}}\right)x=(1)x=x,}</annotation>

</semantics>

</math><img src="./893b08cee01a8913dd11a289964ca0959edb91a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.749ex; height:6.176ex;" alt="{\displaystyle ny=n\left({\frac {1}{n}}x\right)=\left(n{\frac {1}{n}}\right)x=(1)x=x,}" loading="lazy"> which implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=f(ny)=nf(y)=nf\left({\frac {1}{n}}x\right)}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=f(ny)=nf(y)=nf\left({\frac {1}{n}}x\right)}</annotation>

</semantics>

</math><img src="./99fce7f69b235d3e448e5056d592ea702f076752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.031ex; height:6.176ex;" alt="{\displaystyle f(x)=f(ny)=nf(y)=nf\left({\frac {1}{n}}x\right)}" loading="lazy"> so that multiplying both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cf0aefecf48d43fdedd71e318ae6129bd67be252.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.231ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{n}}}" loading="lazy"> proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left({\frac {1}{n}}x\right)={\frac {1}{n}}f(x).}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left({\frac {1}{n}}x\right)={\frac {1}{n}}f(x).}</annotation>

</semantics>

</math><img src="./beb21bce7cb8a80b3cb13ed69f7065c2d4aa423e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.041ex; height:6.176ex;" alt="{\displaystyle f\left({\frac {1}{n}}x\right)={\frac {1}{n}}f(x).}" loading="lazy"> Consequently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(qx)=f\left({\frac {m}{n}}x\right)=mf\left({\frac {1}{n}}x\right)=m\left({\frac {1}{n}}f(x)\right)=qf(x).}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(qx)=f\left({\frac {m}{n}}x\right)=mf\left({\frac {1}{n}}x\right)=m\left({\frac {1}{n}}f(x)\right)=qf(x).}</annotation>

</semantics>

</math><img src="./8589a5583d2f1f0f0cc0b764171bc62c99961e35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.847ex; height:6.176ex;" alt="{\displaystyle f(qx)=f\left({\frac {m}{n}}x\right)=mf\left({\frac {1}{n}}x\right)=m\left({\frac {1}{n}}f(x)\right)=qf(x).}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8733090f2d787d03101c3e16dc3f6404f0e7dd4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }" loading="lazy">

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFRoger_C._LyndonPaul_E._Schupp2001" class="citation cs2"><a href="Roger_C._Lyndon" class="mw-redirect" title="Roger C. Lyndon">Roger C. Lyndon</a>; <a href="Paul_E._Schupp" class="mw-redirect" title="Paul E. Schupp">Paul E. Schupp</a> (2001), Combinatorial Group Theory, Springer</cite></li></ul></div><div><div class="zim-footer">

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