document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171585>Question 1171585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The complex numbers z and w satisfy |z| = |w| = 1 and zw is not equal to -1.<BR>\n" );
document.write( "(a) Prove that \overline{z} = {1}/{z} and \overline{w} = {1}/{w}.<BR>\n" );
document.write( "(b) Prove that {z + w}/{zw + 1} is a real number.<BR>\n" );
document.write( "Can you please explain in detail? I'm super confused, can someone please help? Thank you so much. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #796489 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=796489\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
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document.write( "Let <BR>\n" );
document.write( "z = a+bi<BR>\n" );
document.write( "w = c+di<BR>\n" );
document.write( "where a,b,c,d are real numbers and i = sqrt(-1) or i^2 = -1\r<BR>\n" );
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document.write( "Since |z| = 1, this means <BR>\n" );
document.write( "|z| = sqrt(a^2+b^2)<BR>\n" );
document.write( "1 = sqrt(a^2+b^2)<BR>\n" );
document.write( "1^2 = (sqrt(a^2+b^2))^2<BR>\n" );
document.write( "1 = a^2 + b^2<BR>\n" );
document.write( "a^2 + b^2 = 1\r<BR>\n" );
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document.write( "Through similar algebraic steps, we can say,<BR>\n" );
document.write( "|w| = 1 <BR>\n" );
document.write( "leads to<BR>\n" );
document.write( "c^2 + d^2 = 1\r<BR>\n" );
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document.write( "---------------------------------\r<BR>\n" );
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document.write( "Using our definition of z, let's find 1/z<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+a%2Bbi ALIGN=MIDDLE  ALT=\"z+=+a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+1%2F%28a%2Bbi%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+1%2F%28a%2Bbi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+%281%2F%28a%2Bbi%29%29%2A%28%28a-bi%29%2F%28a-bi%29%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+%281%2F%28a%2Bbi%29%29%2A%28%28a-bi%29%2F%28a-bi%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > This step is done to turn the denominator into a real number.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+%281%2A%28a-bi%29%29%2F%28%28a%2Bbi%29%2A%28a-bi%29%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+%281%2A%28a-bi%29%29%2F%28%28a%2Bbi%29%2A%28a-bi%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+%28a-bi%29%2F%28a%5E2-%28bi%29%5E2%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+%28a-bi%29%2F%28a%5E2-%28bi%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Difference of squares rule\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+%28a-bi%29%2F%28a%5E2-b%5E2i%5E2%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+%28a-bi%29%2F%28a%5E2-b%5E2i%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+%28a-bi%29%2F%28a%5E2-b%5E2%28-1%29%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+%28a-bi%29%2F%28a%5E2-b%5E2%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in i^2 = -1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+%28a-bi%29%2F%28a%5E2%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+%28a-bi%29%2F%28a%5E2%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+%28a-bi%29%2F%281%29 ALIGN=MIDDLE  ALT=\"1%2Fz+=+%28a-bi%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a^2+b^2 = 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz+=+a-bi ALIGN=MIDDLE  ALT=\"1%2Fz+=+a-bi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{z} = \overline{z}\">\r<BR>\n" );
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document.write( "This proves that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fz ALIGN=MIDDLE  ALT=\"1%2Fz\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to the complex conjugate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The steps to proving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fw ALIGN=MIDDLE  ALT=\"1%2Fw\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to the conjugate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w ALIGN=MIDDLE  ALT=\"w\"   DISABLED_style= \"max-width:90%; height:auto;\" > are effectively the same as shown above. I'll let you do this part.\r<BR>\n" );
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document.write( "We'll use the fact that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large z*\frac{1}{z}=z*\overline{z}=1\"> along with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large w*\frac{1}{w}=w*\overline{w}=1\"> in part (b).\r<BR>\n" );
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document.write( "==========================================================================\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Let<BR>\n" );
document.write( "x = complex conjugate of z = a-bi<BR>\n" );
document.write( "y = complex conjugate of w = c-di<BR>\n" );
document.write( "I'm using x and y instead of the overbar notation because I think the overbar notation is a bit clunky, especially when mixed with fraction bars.\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "z+x = (a+bi)+(a-bi) = 2a<BR>\n" );
document.write( "w+y = (c+di)+(c-di) = 2c<BR>\n" );
document.write( "both are real values. \r<BR>\n" );
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document.write( "Furthermore,<BR>\n" );
document.write( "z*x = 1<BR>\n" );
document.write( "w*y = 1<BR>\n" );
document.write( "was proven earlier in part (a), just with different notation.<BR>\n" );
document.write( "This indicates that zx*wy = 1*1 = 1.\r<BR>\n" );
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document.write( "Now if<BR>\n" );
document.write( "z = a+bi<BR>\n" );
document.write( "w = c+di<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "z*w = (a+bi)*(c+di)<BR>\n" );
document.write( "z*w = a*(c+di)+bi*(c+di)<BR>\n" );
document.write( "z*w = ac+adi+bci+bdi^2<BR>\n" );
document.write( "z*w = ac+adi+bci+bd(-1)<BR>\n" );
document.write( "z*w = ac+adi+bci-bd<BR>\n" );
document.write( "z*w = (ac-bd)+(adi+bci)<BR>\n" );
document.write( "z*w = (ac-bd)+(ad+bc)i\r<BR>\n" );
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document.write( "Through very similar steps we can say,<BR>\n" );
document.write( "x = a-bi<BR>\n" );
document.write( "y = c-di<BR>\n" );
document.write( "x*y = (ac-bd)-(ad+bc)i\r<BR>\n" );
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document.write( "As you can see:<BR>\n" );
document.write( "zw+xy = [(ac-bd)+(ad+bc)i] + [(ac-bd)-(ad+bc)i]<BR>\n" );
document.write( "zw+xy = [(ac-bd)+(ac-bd)]+[(ad+bc)i-(ad+bc)i]<BR>\n" );
document.write( "zw+xy = [2(ac-bd)]+[0(ad+bc)i]<BR>\n" );
document.write( "zw+xy = 2(ac-bd)+0i<BR>\n" );
document.write( "zw+xy = 2(ac-bd)<BR>\n" );
document.write( "which is a real result\r<BR>\n" );
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document.write( "To summarize everything so far for part (b), we can add any complex number to its conjugate to get a real result. Similarly, we can multiply any complex number with its conjugate to get a real result. Lastly, zw+xy sorta involves both concepts going on at once which means zw+xy is also a real number.\r<BR>\n" );
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document.write( "-------------------------\r<BR>\n" );
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document.write( "We build up those statements to be able to say the following\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%2Bw%29%2F%28zw%2B1%29 ALIGN=MIDDLE  ALT=\"%28z%2Bw%29%2F%28zw%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bw%29%28xy%2B1%29%29%2F%28%28zw%2B1%29%2A%28xy%2B1%29%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bw%29%28xy%2B1%29%29%2F%28%28zw%2B1%29%2A%28xy%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply numerator and denominator by xy+1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bw%29%28xy%2B1%29%29%2F%28%28zw%2Axy%29%2Bzw%2Bxy%2B1%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bw%29%28xy%2B1%29%29%2F%28%28zw%2Axy%29%2Bzw%2Bxy%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand out the denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bw%29%28xy%2B1%29%29%2F%28%28zx%2Awy%29%2B%28zw%2Bxy%29%2B1%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bw%29%28xy%2B1%29%29%2F%28%28zx%2Awy%29%2B%28zw%2Bxy%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group up the terms like such\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bw%29%28xy%2B1%29%29%2F%281%2B%28zw%2Bxy%29%2B1%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bw%29%28xy%2B1%29%29%2F%281%2B%28zw%2Bxy%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in zx*wy = 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bw%29%28xy%2B1%29%29%2F%281%2B2%28ac-bd%29%2B1%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bw%29%28xy%2B1%29%29%2F%281%2B2%28ac-bd%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in zw+xy = 2(ac-bd)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bw%29%28xy%2B1%29%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bw%29%28xy%2B1%29%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "The denominator 2(ac-bd)+2 is some real number. Let's expand out the numerator and see what we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bw%29%28xy%2B1%29%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bw%29%28xy%2B1%29%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28zxy%2Bz%2Bwxy%2Bw%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%28zxy%2Bz%2Bwxy%2Bw%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28zx%2Ay%2Bz%2Bwy%2Ax%2Bw%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%28zx%2Ay%2Bz%2Bwy%2Ax%2Bw%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281y%2Bz%2B1x%2Bw%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%281y%2Bz%2B1x%2Bw%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in zx = 1 and wy = 1. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2Bz%2Bx%2Bw%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%28y%2Bz%2Bx%2Bw%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28z%2Bx%29%2B%28w%2By%29%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%28%28z%2Bx%29%2B%28w%2By%29%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group up the conjugate pairs\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a%2B2c%29%2F%282%28ac-bd%29%2B2%29 ALIGN=MIDDLE  ALT=\"%282a%2B2c%29%2F%282%28ac-bd%29%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in z+x = 2a and w+y = 2c\r<BR>\n" );
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document.write( "The numerator 2a+2c is a real number.\r<BR>\n" );
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document.write( "We could divide every term by 2 to simplify further, but at this point we're effectively done with the proof. \r<BR>\n" );
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document.write( "Both the numerator and denominator are real values, so overall <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%2Bw%29%2F%28zw%2B1%29 ALIGN=MIDDLE  ALT=\"%28z%2Bw%29%2F%28zw%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a real number.\r<BR>\n" );
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document.write( "Note: The condition <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zw+%3C%3E+-1 ALIGN=MIDDLE  ALT=\"zw+%3C%3E+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is to prevent the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zw%2B1 ALIGN=MIDDLE  ALT=\"zw%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > from being zero. \r<BR>\n" );
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document.write( "Further Reading:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://math.stackexchange.com/questions/427663/prove-if-z-w-1-and-1zw-neq-0-then-zw-over-1zw-is-a-real\">https://math.stackexchange.com/questions/427663/prove-if-z-w-1-and-1zw-neq-0-then-zw-over-1zw-is-a-real</a><BR>\n" );
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