https://www.explainxkcd.com/wiki/index.php?action=history&feed=atom&title=179%3A_e_to_the_pi_times_i 179: e to the pi times i - Revision history 2026-05-23T02:09:57Z Revision history for this page on the wiki MediaWiki 1.30.0 https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=378243&oldid=prev DKMell: italics 2025-05-20T21:44:59Z <p>italics</p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 21:44, 20 May 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8" >Line 8:</td> <td colspan="2" class="diff-lineno">Line 8:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that <ins class="diffchange diffchange-inline">''</ins>e&lt;sup&gt;iπ&lt;/sup&gt;<ins class="diffchange diffchange-inline">'' </ins>+ 1 = 0. Therefore, <ins class="diffchange diffchange-inline">''</ins>e&lt;sup&gt;iπ&lt;/sup&gt;<ins class="diffchange diffchange-inline">'' </ins>= &amp;minus;1.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and <del class="diffchange diffchange-inline">the identity of </del>i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video]. If you know some basic calculus, then [https://www.youtube.com/watch?v=v0YEaeIClKY this video] gives a geometric intuition.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between <ins class="diffchange diffchange-inline">''</ins>e<ins class="diffchange diffchange-inline">''</ins>, <ins class="diffchange diffchange-inline">''</ins>π<ins class="diffchange diffchange-inline">''</ins>, and <ins class="diffchange diffchange-inline">''</ins>i<ins class="diffchange diffchange-inline">'' </ins>(the square root of &amp;minus;1). <ins class="diffchange diffchange-inline">''</ins>e<ins class="diffchange diffchange-inline">'' </ins>is the mathematical identity of which the derivative of <ins class="diffchange diffchange-inline">''</ins>e&lt;sup&gt;x&lt;/sup&gt;<ins class="diffchange diffchange-inline">'' </ins>with respect to <ins class="diffchange diffchange-inline">''</ins>x<ins class="diffchange diffchange-inline">'' </ins>is still <ins class="diffchange diffchange-inline">''</ins>e&lt;sup&gt;x&lt;/sup&gt;<ins class="diffchange diffchange-inline">''</ins>, while <ins class="diffchange diffchange-inline">''</ins>π<ins class="diffchange diffchange-inline">'' </ins>is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of <ins class="diffchange diffchange-inline">''</ins>i<ins class="diffchange diffchange-inline">'' </ins>in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video]. If you know some basic calculus, then [https://www.youtube.com/watch?v=v0YEaeIClKY this video] gives a geometric intuition.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): <ins class="diffchange diffchange-inline">''</ins>e&lt;sup&gt;ix&lt;/sup&gt;<ins class="diffchange diffchange-inline">'' </ins>= cos(<ins class="diffchange diffchange-inline">''</ins>x<ins class="diffchange diffchange-inline">''</ins>) + <ins class="diffchange diffchange-inline">''</ins>i<ins class="diffchange diffchange-inline">'' </ins>sin(<ins class="diffchange diffchange-inline">''</ins>x<ins class="diffchange diffchange-inline">''</ins>).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>From ''[[xkcd: volume 0]]'':</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>From ''[[xkcd: volume 0]]'':</div></td></tr> </table> DKMell https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=367796&oldid=prev ISaveXKCDpapers: added 3Blue1Brown video 2025-03-04T03:24:47Z <p>added 3Blue1Brown video</p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 03:24, 4 March 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video]<ins class="diffchange diffchange-inline">. If you know some basic calculus, then [https://www.youtube.com/watch?v=v0YEaeIClKY this video] gives a geometric intuition</ins>.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</div></td></tr> </table> ISaveXKCDpapers https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=361838&oldid=prev Bautistaarian at 18:23, 12 January 2025 2025-01-12T18:23:52Z <p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 18:23, 12 January 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l26" >Line 26:</td> <td colspan="2" class="diff-lineno">Line 26:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Comics featuring Cueball]]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Comics featuring Cueball]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Math]]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Math]]</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Multiple Cueballs]]</ins></div></td></tr> </table> Bautistaarian https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=347527&oldid=prev 172.70.135.31: Removing incorrect equation 2024-07-29T21:56:12Z <p>Removing incorrect equation</p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 21:56, 29 July 2024</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Therefore, e&lt;sup&gt;i&lt;/sup&gt; = &amp;minus;1/pi.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> </table> 172.70.135.31 https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=347428&oldid=prev 172.68.51.20: /* Explanation */ 2024-07-28T17:49:51Z <p>‎<span dir="auto"><span class="autocomment">Explanation</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 17:49, 28 July 2024</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9" >Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Explanation==</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Therefore, e&lt;sup&gt;i&lt;/sup&gt; = &amp;minus;1/pi.</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> </table> 172.68.51.20 https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=339443&oldid=prev Lettherebedarklight: /* Explanation */ from volume 0 2024-04-12T07:09:19Z <p>‎<span dir="auto"><span class="autocomment">Explanation: </span> from volume 0</span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 07:09, 12 April 2024</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13" >Line 13:</td> <td colspan="2" class="diff-lineno">Line 13:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">From ''[[xkcd: volume 0]]'':</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{quote|I initially got the sign wrong here, and got 40 emails in 3 hours.}}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Transcript==</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Transcript==</div></td></tr> </table> Lettherebedarklight https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=324117&oldid=prev IdiosyncraticLawyer: Spacing 2023-09-21T01:29:57Z <p>Spacing</p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 01:29, 21 September 2023</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10" >Line 10:</td> <td colspan="2" class="diff-lineno">Line 10:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive <del class="diffchange diffchange-inline"> </del>that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</div></td></tr> </table> IdiosyncraticLawyer https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=283792&oldid=prev Theusaf: Reverted edits by Donald Trump (talk) to last revision by CRLF 2022-05-26T20:01:00Z <p>Reverted edits by <a href="/wiki/index.php/Special:Contributions/Donald_Trump" title="Special:Contributions/Donald Trump">Donald Trump</a> (<a href="/wiki/index.php?title=User_talk:Donald_Trump&amp;action=edit&amp;redlink=1" class="new" title="User talk:Donald Trump (page does not exist)">talk</a>) to last revision by <a href="/wiki/index.php/User:CRLF" title="User:CRLF">CRLF</a></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 20:01, 26 May 2022</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap 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crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{comic</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| number&#160; &#160; = 179</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| date&#160; &#160; &#160; = November 3, 2006</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| title&#160; &#160; = e to the pi times i</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| image&#160; &#160; = e_to_the_pi_times_i.png</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| titletext = I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">}}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">==Explanation==</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive&#160; that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">==Transcript==</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:[Two Cueballs are standing at a board with writing on. One Cueball is pointing at the board.]</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Cueball: Numbers of the form n√-1 are &quot;imaginary,&quot; but can still be used in equations.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Friend: Okay.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Cueball: And e^(π√-1)=-1.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Friend: Now you're just fucking with me.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{comic discussion}}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[[Category:Comics featuring Cueball]]</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[[Category:Math]]</ins></div></td></tr> </table> Theusaf https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=281543&oldid=prev Donald Trump: Reverted edit by anti-crap user 2022-05-26T18:56:48Z <p>Reverted edit by anti-crap user</p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 18:56, 26 May 2022</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">{{comic</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap 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crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap crap</ins></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">| number&#160; &#160; = 179</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">| date&#160; &#160; &#160; = November 3, 2006</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">| title&#160; &#160; = e to the pi times i</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">| image&#160; &#160; = e_to_the_pi_times_i.png</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">| titletext = I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">}}</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">==Explanation==</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive&#160; that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">==Transcript==</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:[Two Cueballs are standing at a board with writing on. One Cueball is pointing at the board.]</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:Cueball: Numbers of the form n√-1 are &quot;imaginary,&quot; but can still be used in equations.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:Friend: Okay.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:Cueball: And e^(π√-1)=-1.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:Friend: Now you're just fucking with me.</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">{{comic discussion}}</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">[[Category:Comics featuring Cueball]]</del></div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">[[Category:Math]]</del></div></td><td colspan="2">&#160;</td></tr> </table> Donald Trump https://www.explainxkcd.com/wiki/index.php?title=179:_e_to_the_pi_times_i&diff=280761&oldid=prev CRLF: Reverted vandalism with User:CRLF/OneClickUndo.js 2022-05-26T18:04:08Z <p>Reverted vandalism with <a href="/wiki/index.php/User:CRLF/OneClickUndo.js" title="User:CRLF/OneClickUndo.js">User:CRLF/OneClickUndo.js</a></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 18:04, 26 May 2022</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange 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crap crap crap crap</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{comic</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| number&#160; &#160; = 179</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| date&#160; &#160; &#160; = November 3, 2006</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| title&#160; &#160; = e to the pi times i</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| image&#160; &#160; = e_to_the_pi_times_i.png</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| titletext = I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">}}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">==Explanation==</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The comic largely references {{w|Euler's identity}}. This identity states that e&lt;sup&gt;iπ&lt;/sup&gt; + 1 = 0. Therefore, e&lt;sup&gt;iπ&lt;/sup&gt; = &amp;minus;1.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The humor from this comic is because of the seemingly arbitrary relationship between e, π, and the identity of i (the square root of &amp;minus;1). e is the mathematical identity of which the derivative of e&lt;sup&gt;x&lt;/sup&gt; with respect to x is still e&lt;sup&gt;x&lt;/sup&gt;, while π is the relationship between the circumference of a circle divided by its diameter. Taking these two values and applying them to the value of i in such a manner makes it seem counter-intuitive&#160; that it would yield &amp;minus;1 from basic analysis. The above linked Wikipedia page goes into good detail of how to derive this identity, as does [https://www.youtube.com/watch?v=-dhHrg-KbJ0 this YouTube video].</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The title text refers to how Euler's identity is called upon in complex form (separating real and imaginary numbers): e&lt;sup&gt;ix&lt;/sup&gt; = cos(x) + i sin(x).</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">==Transcript==</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:[Two Cueballs are standing at a board with writing on. One Cueball is pointing at the board.]</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Cueball: Numbers of the form n√-1 are &quot;imaginary,&quot; but can still be used in equations.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Friend: Okay.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Cueball: And e^(π√-1)=-1.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:Friend: Now you're just fucking with me.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{comic discussion}}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[[Category:Comics featuring Cueball]]</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">[[Category:Math]]</ins></div></td></tr> </table> CRLF