Additive inverse

<h2 id="common-examples">Common examples</h2><p>
When working with <a href="/facts/Integer/PUwwrawx">integers</a>, <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, and <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, the additive inverse of any number can be found by multiplying it by <a href="/facts/%25E2%2588%25921/WtI4msbk">−1</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></p>
Simple cases of additive inverses<table><tbody><tr><th>                    n              {\displaystyle n}  </th><th>                    −        n              {\displaystyle -n}  </th></tr><tr><td>                    7              {\displaystyle 7}  </td><td>                    −        7              {\displaystyle -7}  </td></tr><tr><td>                    0.35              {\displaystyle 0.35}  </td><td>                    −        0.35              {\displaystyle -0.35}  </td></tr><tr><td>                                          1            4                                {\displaystyle {\frac {1}{4}}}  </td><td>                    −                              1            4                                {\displaystyle -{\frac {1}{4}}}  </td></tr><tr><td>                    π              {\displaystyle \pi }  </td><td>                    −        π              {\displaystyle -\pi }  </td></tr><tr><td>                    1        +        2        i              {\displaystyle 1+2i}  </td><td>                    −        1        −        2        i              {\displaystyle -1-2i}  </td></tr></tbody></table>
<p>The concept can also be extended to algebraic expressions, which is often used when balancing <a href="/facts/Equation/pxFzLAVR">equations</a>.
</p>
Additive inverses of algebraic expressions<table><tbody><tr><th>                    n              {\displaystyle n}  </th><th>                    −        n              {\displaystyle -n}  </th></tr><tr><td>                    a        −        b              {\displaystyle a-b}  </td><td>                    −        (        a        −        b        )        =        −        a        +        b              {\displaystyle -(a-b)=-a+b}  </td></tr><tr><td>                    2                  x                      2                          +        5              {\displaystyle 2x^{2}+5}  </td><td>                    −        (        2                  x                      2                          +        5        )        =        −        2                  x                      2                          −        5              {\displaystyle -(2x^{2}+5)=-2x^{2}-5}  </td></tr><tr><td>                                          1                          x              +              2                                            {\displaystyle {\frac {1}{x+2}}}  </td><td>                    −                              1                          x              +              2                                            {\displaystyle -{\frac {1}{x+2}}}  </td></tr><tr><td>                                          2                          sin        ⁡                  θ                −                              3                          cos        ⁡                  2          θ                      {\displaystyle {\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta }}  </td><td>                    −        (                              2                          sin        ⁡                  θ                −                              3                          cos        ⁡                  2          θ                )        =        −                              2                          sin        ⁡                  θ                +                              3                          cos        ⁡                  2          θ                      {\displaystyle -({\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta })=-{\sqrt {2}}\sin {\theta }+{\sqrt {3}}\cos {2\theta }}  </td></tr></tbody></table>
<h2 id="relation-to-subtraction">Relation to subtraction</h2>
<p>The additive inverse is closely related to <a href="/facts/Subtraction/dEUrp6k0">subtraction</a>, which can be viewed as an addition using the inverse:
</p>
<i>a</i> − <i>b</i>  =  <i>a</i> + (−<i>b</i>).
<p>Conversely, the additive inverse can be thought of as subtraction from zero:
</p>
−<i>a</i> = 0 − <i>a</i>.
<p>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 class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>
<h2 id="formal-definition">Formal definition</h2>
<p>Given an algebraic structure defined under addition 
  
    
      
        (
        S
        ,
        +
        )
      
    
    {\displaystyle (S,+)}
  
 with an additive identity 
  
    
      
        e
        ∈
        S
      
    
    {\displaystyle e\in S}
  
, an element 
  
    
      
        x
        ∈
        S
      
    
    {\displaystyle x\in S}
  
 has an additive inverse 
  
    
      
        y
      
    
    {\displaystyle y}
  
 if and only if 
  
    
      
        y
        ∈
        S
      
    
    {\displaystyle y\in S}
  
, 
  
    
      
        x
        +
        y
        =
        e
      
    
    {\displaystyle x+y=e}
  
, and 
  
    
      
        y
        +
        x
        =
        e
      
    
    {\displaystyle y+x=e}
  
.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
</p><p>Addition is typically only used to refer to a <a href="/facts/Commutative_property/WaYxJLxd">commutative</a> operation, but it is not necessarily <a href="/facts/Associative_property/EQX8lV6r">associative</a>. When it is associative, so 
  
    
      
        (
        a
        +
        b
        )
        +
        c
        =
        a
        +
        (
        b
        +
        c
        )
      
    
    {\displaystyle (a+b)+c=a+(b+c)}
  
, 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.
</p><p>The definition requires <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closure</a>, that the additive element 
  
    
      
        y
      
    
    {\displaystyle y}
  
 be found in 
  
    
      
        S
      
    
    {\displaystyle S}
  
. This is why despite addition being defined over the natural numbers, it does not an additive inverse for its members. The associated inverses would be <a href="/facts/Negative_number/LzErzCmI">negative numbers</a>, which is why the integers do have an additive inverse.
</p>
<h2 id="further-examples">Further examples</h2>
<ul><li>In a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>, the additive inverse −v (often called the <i><a href="/facts/Opposite_vector/bCLrdksy">opposite vector</a></i> of v) has the same <a href="/facts/Norm_(mathematics)/xIbR4uE1">magnitude</a> as v and but the opposite direction.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li>
<li>In <a href="/facts/Modular_arithmetic/0wtRa5dU">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 class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a></li>
<li>In a <a href="/facts/Boolean_ring/lgko4uqq">Boolean ring</a>, which has elements 
  
    
      
        {
        0
        ,
        1
        }
      
    
    {\displaystyle \{0,1\}}
  
 addition is often defined as the <a href="/facts/Symmetric_difference/W2QST1Qr">symmetric difference</a>. So 
  
    
      
        0
        +
        0
        =
        0
      
    
    {\displaystyle 0+0=0}
  
, 
  
    
      
        0
        +
        1
        =
        1
      
    
    {\displaystyle 0+1=1}
  
, 
  
    
      
        1
        +
        0
        =
        1
      
    
    {\displaystyle 1+0=1}
  
, and 
  
    
      
        1
        +
        1
        =
        0
      
    
    {\displaystyle 1+1=0}
  
. Our additive identity is 0, and both elements are their own additive inverse as 
  
    
      
        0
        +
        0
        =
        0
      
    
    {\displaystyle 0+0=0}
  
 and 
  
    
      
        1
        +
        1
        =
        0
      
    
    {\displaystyle 1+1=0}
  
.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Absolute_value/dwJBpEs5">Absolute value</a> (related through the identity |−<i>x</i>| = |<i>x</i>|).</li>
<li><a href="/facts/Monoid/6AHve7p8">Monoid</a></li>
<li><a href="/facts/Inverse_function/Tg5nnXlu">Inverse function</a></li>
<li><a href="/facts/Involution_(mathematics)/kKxYrUVa">Involution (mathematics)</a></li>
<li><a href="/facts/Multiplicative_inverse/yczdYyCw">Multiplicative inverse</a></li>
<li><a href="/facts/Reflection_(mathematics)/5JHWnrql">Reflection (mathematics)</a></li>
<li><a href="/facts/Reflection_symmetry/eSeR8nbD">Reflection symmetry</a></li>
<li><a href="/facts/Semigroup/rgSdk2kt">Semigroup</a></li></ul>
<h2 id="notes-and-references">Notes and references</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Gallian, Joseph A. (2017). Contemporary abstract algebra (9th ed.). Boston, MA: Cengage Learning. p. 52. ISBN 978-1-305-65796-0. <a href="978-1-305-65796-0" target="_blank">978-1-305-65796-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 169–170. ISBN 978-1-292-02496-7. <a href="978-1-292-02496-7" target="_blank">978-1-292-02496-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Mazur, Izabela (March 26, 2021). "2.5 Properties of Real Numbers -- Introductory Algebra". Retrieved August 4, 2024. <a href="https://pressbooks.bccampus.ca/intermediatedevelopmentalmath/chapter/properties-of-real-numbers/" target="_blank">https://pressbooks.bccampus.ca/intermediatedevelopmentalmath/chapter/properties-of-real-numbers/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"Standards::Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts". learninglab.si.edu. Retrieved 2024-08-04. <a href="https://learninglab.si.edu/standards/CCSS.Math.Content.7.NS.A.1b/340" target="_blank">https://learninglab.si.edu/standards/CCSS.Math.Content.7.NS.A.1b/340</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Weisstein, Eric W. "Negative". MathWorld. Retrieved 2025-01-04. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Kinard, James T.; Kozulin, Alex (2008-06-02). Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom. Cambridge University Press. ISBN 978-1-139-47239-5. <a href="978-1-139-47239-5" target="_blank">978-1-139-47239-5</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Brown, Christopher. "SI242: divisibility". www.usna.edu. Retrieved 2024-08-04. <a href="https://www.usna.edu/Users/cs/wcbrown/courses/F23SI242/lec/l25/lec.html" target="_blank">https://www.usna.edu/Users/cs/wcbrown/courses/F23SI242/lec/l25/lec.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>"2.2.5: Properties of Equality with Decimals". K12 LibreTexts. 2020-07-21. Retrieved 2024-08-04. <a href="https://k12.libretexts.org/Bookshelves/Mathematics/Algebra/02%3A_Linear_Equations/2.02%3A_One-Step_Equations_and_the_Properties_of_Equality/2.2.05%3A_Properties_of_Equality_with_Decimals" target="_blank">https://k12.libretexts.org/Bookshelves/Mathematics/Algebra/02%3A_Linear_Equations/2.02%3A_One-Step_Equations_and_the_Properties_of_Equality/2.2.05%3A_Properties_of_Equality_with_Decimals</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 37–39. ISBN 978-1-292-02496-7. <a href="978-1-292-02496-7" target="_blank">978-1-292-02496-7</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>"2.2.5: Properties of Equality with Decimals". K12 LibreTexts. 2020-07-21. Retrieved 2024-08-04. <a href="https://k12.libretexts.org/Bookshelves/Mathematics/Algebra/02%3A_Linear_Equations/2.02%3A_One-Step_Equations_and_the_Properties_of_Equality/2.2.05%3A_Properties_of_Equality_with_Decimals" target="_blank">https://k12.libretexts.org/Bookshelves/Mathematics/Algebra/02%3A_Linear_Equations/2.02%3A_One-Step_Equations_and_the_Properties_of_Equality/2.2.05%3A_Properties_of_Equality_with_Decimals</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Cajori, Florian (2011). A History of Mathematical Notations: two volume in one. New York: Cosimo Classics. pp. 246–247. ISBN 978-1-61640-571-7. <a href="978-1-61640-571-7" target="_blank">978-1-61640-571-7</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 37–39. ISBN 978-1-292-02496-7. <a href="978-1-292-02496-7" target="_blank">978-1-292-02496-7</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Axler, Sheldon (2024), Axler, Sheldon (ed.), "Vector Spaces", Linear Algebra Done Right, Undergraduate Texts in Mathematics, Cham: Springer International Publishing, pp. 1–26, doi:10.1007/978-3-031-41026-0_1, ISBN 978-3-031-41026-0 <a href="978-3-031-41026-0" target="_blank">978-3-031-41026-0</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Gupta, Prakash C. (2015). Cryptography and network security. Eastern economy edition. Delhi: PHI Learning Private Limited. p. 15. ISBN 978-81-203-5045-8. <a href="978-81-203-5045-8" target="_blank">978-81-203-5045-8</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>Martin, Urusula; Nipkow, Tobias (1989-03-01). "Boolean unification — The story so far". Journal of Symbolic Computation. Unification: Part 1. 7 (3): 275–293. doi:10.1016/S0747-7171(89)80013-6. ISSN 0747-7171. <a href="https://www.sciencedirect.com/science/article/pii/S0747717189800136" target="_blank">https://www.sciencedirect.com/science/article/pii/S0747717189800136</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
</ol>

Additive inverse open-in-new

Additive inverse