document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=230237>Question 230237</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a complex number a+bi such that radical i is equal to a+bi. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #170501 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=170501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(\frac{\pi}{2}\right)\ =\ 0\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(\frac{\pi}{2}\right)\ =\ 1\">, therefore we can say that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ i\ =\ \cos\left(\frac{\pi}{2}\right)\ +\ i\sin\left(\frac{\pi}{2}\right)\\">\r<BR>\n" );
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document.write( "But we know from Euler's Formula that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^{i\Theta}\ =\ \cos\Theta\ +\ i\sin\Theta\">\r<BR>\n" );
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document.write( "Hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^{i\left(\frac{\pi}{2}\right)}\ =\ i\">\r<BR>\n" );
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document.write( "Taking the square root of both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^{i\left(\frac{\pi}{4}\right)}\ =\ i^{\frac{1}{2}}\">\r<BR>\n" );
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document.write( "But\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^{i\left(\frac{\pi}{4}\right)}\ =\ \pm\sqrt{i}\ =\ \cos\left(\frac{\pi}{4}\right)\ +\ i\sin\left(\frac{\pi}{4}\right)\\">\r<BR>\n" );
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document.write( "Finally:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \pm\sqrt{i}\ =\ \frac{1}{\pm\sqrt{2}}\ +\ \frac{i}{\pm\sqrt{2}}\ =\ \pm\frac{\sqrt{2}}{2}\left(1\ +\ i\right)\">\r<BR>\n" );
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document.write( "after rationalizing the denominators.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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