document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=564049>Question 564049</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Explain how complex numbers combine under the following operations:<BR>\n" );
document.write( "a.  Addition<BR>\n" );
document.write( "b. Division\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The graphical interpretation should demonstrate how to add and divide complex numbers solely using the graph of each complex number (not based upon the algebraic computation).\r<BR>\n" );
document.write( "\n" );
document.write( "*** I understand the concept of ;\r<BR>\n" );
document.write( "\n" );
document.write( "The standard form of complex number is ;<BR>\n" );
document.write( "(a + bi) + (a + bi)<BR>\n" );
document.write( "(a + a) +(bi +bi)<BR>\n" );
document.write( "2a + 2bi<BR>\n" );
document.write( "Or … (3 + 2i) + (1+i)<BR>\n" );
document.write( "(3+1) + (2i + i) = 4 + 3i\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So WHAT IS THE GRAPHICAL INTERPRETATION; demonstrate how to add and divide complex numbers solely using the graph of each complex number </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #365260 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=365260\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Adding complex numbers is quite simple to show graphically, since complex numbers add just like vectors (add the individual components). The sum of the complex numbers is simply the resultant vector formed by the other numbers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiplying/dividing is a little more difficult to show graphically. Perhaps you could represent two complex numbers z1, z2 by\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z_1 = r_1e^{i\theta_1}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z_2 = r_2e^{i\theta_2}\"> (note that Euler's formula states that e^(ix) = cos x + i sin x, so you can represent any complex number in the above form).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z_1z_2 = r_1r_2e^{i(\theta_1 + \theta_2)}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Graphically, this means that the radius is numerically equal to the product of the radii of r1 and r2, and the angle formed by z1z2 is equal to the sum of the angles formed by z1 and z2. Representing division graphically is similar; think of division as multiplying by 1 over the complex number. Or, find some way to \"undo\" the multiplication. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );