neve domain was triggered too early. This is usually an indicator for some code in the plugin or theme running too early. Translations should be loaded at the init action or later. Please see Debugging in WordPress for more information. (This message was added in version 6.7.0.) in /home/mathze5/public_html/learnermath.com/wp-includes/functions.php on line 6114Before looking at the conjugate and multiplicative inverse of complex numbers, it’s a good idea to look at some standard examples of adding and subtracting complex numbers.\n Adding and subtracting complex numbers in Math is often fairly straight forward.<\/p>\n( a + bi<\/font><\/i> ) + ( c + di<\/font><\/i> ) = ( a + c ) + ( b + d )i<\/font><\/i>\n Not entirely dissimilar to adding and subtracting complex numbers.\n The multiplicative inverse of complex numbers is sometimes referred to as just the inverse of complex numbers, but they mean the same thing.\n
\nWhen faced with complex numbers examples requiring common arithmetic, there are some standard approaches to use for such situations.<\/p>\n
\n<\/a>\n\n\n\nAdding and Subtracting Complex Numbers<\/span><\/h2>\n\n\n\n
\n( a + bi<\/font><\/i> ) − ( c + di<\/font><\/i> ) = ( a − c ) + ( b − d )i<\/font><\/i>\n
\nExamples <\/u><\/font><\/font><\/font><\/b>\n
\n(1.1) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\n( 3 + 4i<\/font><\/i> ) + ( 2 + 3i<\/font><\/i> )<\/b> = ( 3 + 2 ) + ( 4 + 3 )i<\/font><\/i><\/b>\n
\n= 5 + 8i<\/font><\/i><\/b>\n
\n(1.2) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\n( 3 + 4i<\/font><\/i> ) − ( 2 + 3i<\/font><\/i> )<\/b> = ( 3 − 2 ) + ( 4 − 3i<\/font><\/i> )<\/b>\n
\n= 1 + 1i<\/font><\/i><\/b> = 1 + i<\/font><\/i><\/b>\n
\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nMultiplying Complex Numbers<\/span><\/h2>\n\n\n\n
\n
\nMultiplying complex numbers follows standard patterns.<\/p>\n( a + bi<\/font><\/i> ) <\/b>× ( c + di<\/font><\/i> )<\/b> = ( a <\/b>× c ) <\/b>+ ( a <\/b>× di<\/font><\/i> ) <\/b>+ ( bi<\/font><\/i> <\/b>× c ) <\/b>+ ( bi<\/font><\/i> <\/b>× di<\/font><\/i> )<\/b>\n
\n= ac <\/b>+ adi<\/font><\/i> <\/b>+ bci<\/font><\/i> <\/b>+ bdi<\/font><\/i>2<\/font><\/sup><\/b>\n
\nac + adi<\/font><\/i> + bci<\/font><\/i> + bdi<\/font><\/i>2<\/font><\/sup><\/b> = ac + adi<\/font><\/i> + bci<\/font><\/i> − bd<\/b> ( as i<\/font><\/i>2<\/font><\/sup> = -1 )\n
\nNow rewriting slightly:<\/font>\n
\nac <\/b>− bd <\/b>+ adi<\/font><\/i> <\/b>+ bci<\/font><\/i><\/b> = ( ac <\/b>− bd ) <\/b>+ ( ad <\/b>+ bc )i<\/font><\/i><\/b>\n
\nIn summary:<\/font>\n
\n( a + bi<\/font><\/i> )( c + di<\/font><\/i> ) = ( ac − bd ) + ( ad + bc )i<\/font><\/i>\n
\nExamples <\/u><\/font><\/font><\/font><\/b>\n
\n(2.1) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\n( 3 + 5i<\/font><\/i> )( 1 + 2i<\/font><\/i> )<\/b> = ( 3 − 10 ) + ( 6 + 5 )i<\/font><\/i><\/b>\n
\n= -7 + 10i<\/font><\/i><\/b>\n
\n(2.2) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\n( 4 − 2i<\/font><\/i> )( 3 + 3i<\/font><\/i> )<\/b> = ( 12 − (-6) ) + ( 12 + (-6) )i<\/font><\/i><\/b> = 18 + 6i<\/font><\/i><\/b>\n
\n(2.3) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\n3 <\/b>× ( 2 + 2i<\/font><\/i> )<\/b> = 6 + 6i<\/font><\/i><\/b>\n
\n<\/a>\n
\n\n\n\n
\n\n\n\nDivision of Complex Numbers,
Conjugate of a Complex Number<\/span><\/h2>\n\n\n\n
\nDivision with complex numbers is very similar to how we rationalise surds.\n
\nBy which we multiply the top and bottom of the division sum by the conjugate of the denominator.\n
\nThe conjugate of a complex number has the same real part, but the imaginary part will have the opposite sign.\n
\nWe do this multiplication between the two because multiplying a complex number by its conjugate has the following result:\n
\n( a<\/b> + bi<\/i><\/b><\/font> ) × ( a<\/b> − bi<\/i><\/b><\/font> ) = a<\/b>2<\/font><\/sup> + b<\/b>2<\/font><\/sup>\n
\nExample <\/u><\/font><\/font><\/font><\/b>\n
\n(3.1) <\/i><\/b><\/font><\/u><\/font><\/font>\n
\n\\bf{\\frac{3 \\space + \\space 5i}{2 \\space - \\space 3i}}<\/span><\/font> [ CONJUGATE OF<\/font> 2 − 3i<\/i><\/font><\/b> IS<\/font> ( 2 + 3i<\/i><\/font><\/b> ) ]\n
\n=> \\bf{\\frac{3 \\space + \\space 5i}{2 \\space - \\space 3i}}<\/span><\/font> × \\bf{\\frac{2 \\space + \\space 3i}{2 \\space + \\space 3i}}<\/span><\/font> = \\bf{\\frac{(6 - 15) \\space + \\space (9 + 10)i}{4 \\space + \\space 9}}<\/span><\/font>\n
\n= \\bf{\\frac{{\\text{-}}9 \\space + \\space 19i}{13}}<\/span><\/font> = \\bf{\\frac{{\\text{-}}9}{13}}<\/span><\/font> − \\bf{\\frac{19}{13}}<\/span><\/font>i<\/i><\/font><\/b>\n
\n
\n<\/a>\n\n\n\n
\n\n\n\nMultiplicative Inverse of Complex Numbers<\/span><\/h2>\n\n\n\n
\nThe inverse of a complex number is simply the reciprocal of the number, which is the fraction of 1<\/font> over the complex number. \n
\nBut usually it’s good practice for this fraction to be rationalized, which is where we again make use of the complex conjugate.<\/p>\n
\n