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Undefined (mathematics)
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<h2 id="other-shades-of-meaning">Other shades of meaning</h2> <p>In some mathematical contexts, undefined can refer to a <a href="/facts/Primitive_notion/PrQue1HG">primitive notion</a> which is not defined in terms of simpler concepts.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> For example, in <a href="/facts/Euclid%2527s_Elements/nPKvCIxs">Elements</a>, <a href="/facts/Euclid/dHpVyL4R">Euclid</a> defines a point merely as "that of which there is no part", and a line merely as "length without breadth".<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Although these terms are not further defined, Euclid uses them to construct more complex geometric concepts.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>Contrast also the term <i><a href="/facts/Undefined_behavior/awUm2vGd">undefined behavior</a></i> in computer science, in which the term indicates that a function may produce or return <i>any</i> result, which may or may not be correct. </p> <h2 id="common-examples-of-undefined-expressions">Common examples of undefined expressions</h2> <p>Many fields of mathematics refer to various kinds of expressions as undefined. Therefore, the following examples of undefined expressions are not exhaustive. </p> <h3>Division by zero</h3> <p>In <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a>, and therefore <a href="/facts/Algebra/nMdqczjo">algebra</a>, <a href="/facts/Division_by_zero/wXK7t2Ge">division by zero</a> is undefined.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Use of a division by zero in an arithmetical calculation or proof, can produce absurd or meaningless results. </p><p>Assuming that division by zero exists, can produce <a href="/facts/Consistency/IHkdnA9M">inconsistent</a> logical results, such as the following fallacious "proof" that one is equal to two<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>: </p> Incorrect "proof" that 1 = 2 {\displaystyle 1=2} <table><tbody><tr><td> x {\displaystyle x} </td><td> = y {\displaystyle =y} </td><td>Define x {\displaystyle x} as equal to y {\displaystyle y} </td></tr><tr><td> x 2 {\displaystyle x^{2}} </td><td> = x y {\displaystyle =xy} </td><td>Multiply both sides of equation by x {\displaystyle x} </td></tr><tr><td> x 2 − y 2 {\displaystyle x^{2}-y^{2}} </td><td> = x y − y 2 {\displaystyle =xy-y^{2}} </td><td>Subtract y 2 {\displaystyle y^{2}} from both sides</td></tr><tr><td> ( x + y ) ( x − y ) {\displaystyle (x+y)(x-y)} </td><td> = y ( x − y ) {\displaystyle =y(x-y)} </td><td>Factor both sides of equation</td></tr><tr><td> x + y {\displaystyle x+y} </td><td> = y {\displaystyle =y} </td><td>Divide both sides of equation by x − y {\displaystyle x-y} </td></tr><tr><td> 2 y {\displaystyle 2y} </td><td> = y {\displaystyle =y} </td><td>Replace x {\displaystyle x} with y {\displaystyle y} , because we know that x = y {\displaystyle x=y} </td></tr><tr><td> 2 {\displaystyle 2} </td><td> = 1 {\displaystyle =1} </td><td>Divide both sides by y {\displaystyle y} </td></tr></tbody></table> <p>The above "proof" is not meaningful. Since we know that x = y {\displaystyle x=y} , if we divide both sides of the equation by x − y {\displaystyle x-y} , we divide both sides of the equation by zero. This operation is undefined in arithmetic, and therefore deductions based on division by zero can be contradictory. </p><p>If we assume that a non-zero answer n {\displaystyle n} exists, when some number k ∣ k ≠ 0 {\displaystyle k\mid k\neq 0} is divided by zero, then that would imply that k = n × 0 {\displaystyle k=n\times 0} . But there is no number, which when multiplied by zero, produces a number that is not zero. Therefore, our assumption is incorrect.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h3>Zero to the power of zero</h3> <p>Depending on the particular context, mathematicians may refer to <a href="/facts/Zero_to_the_power_of_zero/pxbWkuFg">zero to the power of zero</a> as undefined,<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> indefinite,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> or equal to 1.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Controversy exists as to which definitions are mathematically rigorous, and under what conditions.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h3>The square root of a negative number</h3> <p>When restricted to the field of real numbers, the square root of a negative number is undefined, as no real number exists which, when squared, equals a negative number. Mathematicians, including <a href="/facts/Gerolamo_Cardano/djTqLl2V">Gerolamo Cardano</a>, <a href="/facts/John_Wallis/hlMcED3G">John Wallis</a>, <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a>, and <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>, explored formal definitions for the square roots of negative numbers, giving rise to the field of <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h3>In trigonometry</h3> <p>In trigonometry, for all n ∈ Z {\displaystyle n\in \mathbb {Z} } , the functions tan θ {\displaystyle \tan \theta } and sec θ {\displaystyle \sec \theta } are undefined for θ = π ( n − 1 2 ) {\textstyle \theta =\pi \left(n-{\frac {1}{2}}\right)} , while the functions cot θ {\displaystyle \cot \theta } and csc θ {\displaystyle \csc \theta } are undefined for all θ = π n {\displaystyle \theta =\pi n} . This is a consequence of the <a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">identities</a> of these functions, which would imply a <a href="/facts/Division_by_zero/wXK7t2Ge">division by zero</a> at those points.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p><p>Also, arcsin k {\displaystyle \arcsin k} and arccos k {\displaystyle \arccos k} are both undefined when k > 1 {\displaystyle k>1} or k < − 1 {\displaystyle k<-1} , because the range of the sin {\displaystyle \sin } and cos {\displaystyle \cos } functions is between − 1 {\displaystyle -1} and 1 {\displaystyle 1} inclusive. </p> <h3>In complex analysis</h3> <p>In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, a point z {\displaystyle z} on the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> where a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> is undefined, is called a <a href="/facts/Mathematical_singularity/Y5sMQY4I">singularity</a>. Some different types of singularities include: </p> <ul><li><a href="/facts/Removable_singularity/6MPFDNha">Removable singularities</a> - in which the function can be extended holomorphically to z {\displaystyle z} </li> <li><a href="/facts/Pole_(complex_analysis)/gnA3U8VP">Poles</a> - in which the function can be extended <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphically</a> to z {\displaystyle z} </li> <li><a href="/facts/Essential_singularity/KMb1R292">Essential singularities</a> - in which no meromorphic extension to z {\displaystyle z} can exist</li></ul> <h2 id="related-terms">Related terms</h2> <h3>Indeterminate</h3> <p>The term <i>undefined</i> should be contrasted with the term <i><a href="/facts/Indeterminate_form/5Y3JWKjS">indeterminate</a></i>. In the first case, undefined generally indicates that a value or property can have <i>no</i> meaningful definition. In the second case, indeterminate generally indicates that a value or property can have <i>many</i> meaningful definitions. Additionally, it seems to be generally accepted that undefined values <i>may not</i> be safely used within a particular formal system, whereas indeterminate values <i>might</i> be, depending on the relevant rules of the particular formal system.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Analytic_function/JhL3z8FP">Analytic function</a> - a function locally given by a convergent power series, which may be useful for dealing with otherwise undefined values</li> <li><a href="/facts/L%2527H%25C3%25B4pital%2527s_rule/pCXK75Io">L'Hôpital's rule</a> - a method in calculus for evaluating indeterminate forms</li> <li><a href="/facts/Indeterminate_form/5Y3JWKjS">Indeterminate form</a> - a mathematical expression for which many assignments exist</li> <li><a href="/facts/NaN/B43LVnTc">NaN</a> - the <a href="/facts/IEEE_754/dD3e7zAl">IEEE-754</a> expression indicating that the result of a calculation is not a number</li> <li><a href="/facts/Primitive_notion/PrQue1HG">Primitive notion</a> - a concept that is not defined in terms of previously-defined concepts</li> <li><a href="/facts/Singularity_(mathematics)/Y5sMQY4I">Singularity</a> - a point at which a mathematical function ceases to be well-behaved</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Smart, James R. (1988). <i>Modern Geometries</i> (3rd ed.). Brooks/Cole. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-534-08310-2.</li> <li>Lo Bello, Anthony (2013). <i>Origins of Mathematical Words</i>. Johns Hopkins University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4214-1098-2.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.youtube.com/watch?v=lHdlHTsXbZg">Undefined and indeterminate - Functions and their graphs - Algebra II - Khan Academy</a> on <a href="/facts/YouTube_video_(identifier)/db276jst">YouTube</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"What exactly does undefined mean?". Mathematics Stack Exchange. Retrieved 2024-12-02. <a href="https://math.stackexchange.com/questions/4524071/what-exactly-does-undefined-mean" target="_blank">https://math.stackexchange.com/questions/4524071/what-exactly-does-undefined-mean</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Horvath, Joan; Cameron, Rich (2022). Make: Calculus: build models to learn, visualize, and explore. Mathematics/Calculus (1st ed.). Santa Rosa, CA: Make Community, LLC. ISBN 978-1-68045-739-1. <a href="978-1-68045-739-1" target="_blank">978-1-68045-739-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Martínez, Alberto A. (2018). Negative Math: How Mathematical Rules Can Be Positively Bent. Princeton, NJ: Princeton University Press. ISBN 978-0-691-13391-1. <a href="978-0-691-13391-1" target="_blank">978-0-691-13391-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Definition:Undefined Term - ProofWiki". proofwiki.org. Retrieved 2024-12-03. <a href="https://proofwiki.org/wiki/Definition:Undefined_Term" target="_blank">https://proofwiki.org/wiki/Definition:Undefined_Term</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Euclides (2008). Fitzpatrick, Richard (ed.). Euclid's elements of geometry: the Greek text of J.L. Heiberg (1883 - 1885): from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883-1885. Translated by Fitzpatrick, Richard (2nd ed.). Lulu.com. p. 6. ISBN 978-0-615-17984-1. <a href="978-0-615-17984-1" target="_blank">978-0-615-17984-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Waismann, Friedrich (1951). Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics. Translated by Benac, Theodore J. New York: Frederick Ungar Publishing Co. p. 73. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Euler, Leonard (1770). Elements of Algebra (4th ed.). London: Longman, Rees, Orme, & Co. p. 28. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Sultan, Alan; Artzt, Alice F. (2011). The mathematics that every secondary school math teacher needs to know. Studies in mathematical thinking and learning. New York: Routledge. p. 6. ISBN 978-0-415-99413-2. <a href="978-0-415-99413-2" target="_blank">978-0-415-99413-2</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Euler, Leonard (1770). Elements of Algebra (4th ed.). London: Longman, Rees, Orme, & Co. p. 28. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Hafstrom, John Edward (1961). Basic concepts in modern mathematics. Dover books on mathematics. Mineola, New York: Dover Publications, Inc (published 2013). p. 19. ISBN 978-0-486-49729-7. {{cite book}}: ISBN / Date incompatibility (help) <a href="978-0-486-49729-7" target="_blank">978-0-486-49729-7</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>"Why is $0^0$ also known as indeterminate?". Mathematics Stack Exchange. Retrieved 2024-12-02. <a href="https://math.stackexchange.com/questions/1028244/why-is-00-also-known-as-indeterminate" target="_blank">https://math.stackexchange.com/questions/1028244/why-is-00-also-known-as-indeterminate</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Jena, Sisir Kumar (2022). C programming: learn to code (1st ed.). Boca Raton, FL: Chapman & Hall/CRC Press. p. 19. ISBN 978-1-032-03625-0. <a href="978-1-032-03625-0" target="_blank">978-1-032-03625-0</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>"What is 0^0". cs.uwaterloo.ca. Retrieved 2024-12-02. <a href="https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html" target="_blank">https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>"Zero to the zero power – is $0^0=1$?". Mathematics Stack Exchange. Retrieved 2024-12-02. <a href="https://math.stackexchange.com/questions/11150/zero-to-the-zero-power-is-00-1" target="_blank">https://math.stackexchange.com/questions/11150/zero-to-the-zero-power-is-00-1</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Vaughan, Lena (April 1903). "A History of i = \sqrt 1". Mathematical Supplement of School Science. 1 (1): 173–175 – via Google Books. <a href="https://books.google.com/books?id=N-buAAAAMAAJ&pg=PA173" target="_blank">https://books.google.com/books?id=N-buAAAAMAAJ&pg=PA173</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>McCallum, William G.; Hughes-Hallet, Deborah; Gleason, Andrew M. (October 2012). Calculus: Single and Multivariable (6th ed.). Wiley. p. 40. ISBN 978-1-118-54785-4. <a href="978-1-118-54785-4" target="_blank">978-1-118-54785-4</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Davis, Brent; Renert, Moshe (2013). The math teachers know: profound understanding of emergent mathematics. New York: Routledge. pp. 77–79. ISBN 978-1-135-09779-0. <a href="978-1-135-09779-0" target="_blank">978-1-135-09779-0</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> </ol>