Quotient

<h2 id="notation">Notation</h2>
<p class="note">Main article: <a href="/facts/Division_(mathematics)/FqLKNkmq">Division (mathematics) § Notation</a></p>
<p>The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line.  The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole.
</p><p>
  
    
      
        
          
            
              1
              2
            
          
        
        
        
          
            
              
              
                
                ←
                
                  dividend  or  numerator
                
              
            
            
              
              
                
                ←
                
                  divisor  or  denominator
                
              
            
          
        
        
          
            }
          
        
        ←
        
          quotient
        
      
    
    {\displaystyle {\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend  or  numerator}}\\&\leftarrow {\text{divisor  or  denominator}}\end{aligned}}{\Biggr \}}\leftarrow {\text{quotient}}}

</p>
<h2 id="integer-part-definition">Integer part definition</h2>
<p>The quotient is also less commonly defined as the greatest <a href="/facts/Natural_number/u65og8JE">whole number</a> of times a divisor may be subtracted from a dividend—before making the <a href="/facts/Remainder/wCmbYeeh">remainder</a> negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative:
</p>
20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0,
<p>while
</p>
20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0.
<p>In this sense, a quotient is the <a href="/facts/Integer_part/ssehyMMf">integer part</a> of the ratio of two numbers.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>
<h2 id="quotient-of-two-integers">Quotient of two integers</h2>
<p class="note">Main article: <a href="/facts/Rational_number/VJQmyY05">Rational number</a></p>
<p>A <a href="/facts/Rational_number/VJQmyY05">rational number</a> can be defined as the quotient of two <a href="/facts/Integer/PUwwrawx">integers</a> (as long as the denominator is non-zero).
</p><p>A more detailed definition goes as follows:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
</p>
A real number <i>r</i> is rational, if and only if it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational.
<p>Or more formally: 
</p>
Given a real number <i>r</i>, <i>r</i> is rational if and only if there exists integers <i>a</i> and <i>b</i> such that 
  
    
      
        r
        =
        
          
            
              a
              b
            
          
        
      
    
    {\displaystyle r={\tfrac {a}{b}}}
  
 and  
  
    
      
        b
        ≠
        0
      
    
    {\displaystyle b\neq 0}
  
.
<p>The existence of <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a>—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>
</p>
<h2 id="more-general-quotients">More general quotients</h2>
<p>Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> with an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> defined on it, a "<a href="/facts/Quotient_set/1d2sh0TB">quotient set</a>" may be created which contains those equivalence classes as elements. A <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> may be formed by breaking a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> into a number of similar <a href="/facts/Cosets/zFgQUTLY">cosets</a>, while a <a href="/facts/Quotient_space_(linear_algebra)/WmQ3lqdI">quotient space</a> may be formed in a similar process by breaking a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> into a number of similar <a href="/facts/Linear_subspace/2wtKTgHL">linear subspaces</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Product_(mathematics)/aXY11nGS">Product (mathematics)</a></li>
<li><a href="/facts/Quotient_category/y3CATQTt">Quotient category</a></li>
<li><a href="/facts/Quotient_graph/EpJlV5UV">Quotient graph</a></li>
<li><a href="/facts/Division_(mathematics)/FqLKNkmq">Integer division</a></li>
<li><a href="/facts/Quotient_module/IJFG2yqv">Quotient module</a></li>
<li><a href="/facts/Quotient_object/lEcCIhrP">Quotient object</a></li>
<li><a href="/facts/Quotient_of_a_formal_language/86AcQ1Ab">Quotient of a formal language</a>, also left and right quotient</li>
<li><a href="/facts/Quotient_ring/X2WkXyEw">Quotient ring</a></li>
<li><a href="/facts/Quotient_set/1d2sh0TB">Quotient set</a></li>
<li><a href="/facts/Quotient_space_(topology)/muE0qQVC">Quotient space (topology)</a></li>
<li><a href="/facts/Quotient_type/vp6aKpy5">Quotient type</a></li>
<li><a href="/facts/Quotition_and_partition/72AQmEp3">Quotition and partition</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li> Media related to Quotients at Wikimedia Commons</li></ul>

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

<ol>
<li id="fn:1"><p>"Quotient". Dictionary.com. <a href="http://dictionary.reference.com/browse/quotient" target="_blank">http://dictionary.reference.com/browse/quotient</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Weisstein, Eric W. "Integer Division". mathworld.wolfram.com. Retrieved 2020-08-27. <a href="https://mathworld.wolfram.com/IntegerDivision.html#:~:text=Integer%20division%20is%20division%20in,and%20is%20the%20floor%20function." target="_blank">https://mathworld.wolfram.com/IntegerDivision.html#:~:text=Integer%20division%20is%20division%20in,and%20is%20the%20floor%20function.</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23. <a href="https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en" target="_blank">https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>James, R. C. (1992-07-31). Mathematics Dictionary. Springer Science & Business Media. ISBN 978-0-412-99041-0. <a href="978-0-412-99041-0" target="_blank">978-0-412-99041-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"IEC 60050 - Details for IEV number 102-01-22: "quotient"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13. <a href="https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-22" target="_blank">https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-22</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>"ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23. <a href="https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en" target="_blank">https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>"IEC 60050 - Details for IEV number 102-01-23: "ratio"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13. <a href="https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-23" target="_blank">https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-01-23</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>"IEC 60050 - Details for IEV number 112-03-18: "rate"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13. <a href="https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=112-03-18" target="_blank">https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=112-03-18</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Thompson, A.; Taylor, B. N. (March 4, 2020). "NIST Guide to the SI, Chapter 7: Rules and Style Conventions for Expressing Values of Quantities". Special Publication 811 | The NIST Guide for the use of the International System of Units. National Institute of Standards and Technology. Retrieved October 25, 2021. <a href="https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values" target="_blank">https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-7-rules-and-style-conventions-expressing-values</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>"ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23. <a href="https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en" target="_blank">https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-2:v1:en</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Weisstein, Eric W. "Quotient". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Epp, Susanna S. (2011-01-01). Discrete mathematics with applications. Brooks/Cole. p. 163. ISBN 9780495391326. OCLC 970542319. <a href="9780495391326" target="_blank">9780495391326</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>"Irrationality of the square root of 2". www.math.utah.edu. Retrieved 2020-08-27. <a href="https://www.math.utah.edu/~pa/math/q1.html" target="_blank">https://www.math.utah.edu/~pa/math/q1.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
</ol>

Quotient open-in-new

Quotient