A complex number z is a number of the form z = x + yi . Its conjugate is a number of the form = x - yi . The complex number and its conjugate have the same real part. Re( z ) = Re( ). The sign of the imaginary part of the conjugate complex number is reversed. Im( z ) = - Im( ).
The conjugate numbers have the same modulus and opposite arguments. | z | = | |, arg( z ) = - arg( ). Any complex number multiplied by its complex conjugate is a real number, equal to the square of the modulus of the complex numbers z . z = ( x + yi )( x - yi ) = x 2+ y 2 = | z |2
Division Of Complex Numbers Let a + bi and c + di be complex numbers. Multiply the numerator and denominator of the fraction by the Complex Conjugate of the Denominator . Then to perform the operation 2+6i x 4-i = (2+6i) (4-i) = 14+22i = 14 + 22 i 4+i 4-i (4+i) (4-i) 17 17 17
Real Axis Imaginary Axis y x The angle formed from the real axis and a line from the origin to ( x , y ) is called the argument of z , with requirement that 0 < 2 . modified for quadrant and so that it is between 0 and 2 Let a complex number be Z such that : z Modulus and Argument of Complex Numbers The magnitude or modulus of z denoted by z is the distance from the origin to the point ( x , y ).
The Principal Argument is between - and Real Axis Imaginary Axis y x z = r 1 The unique value of θ such that – π < θ < π is called principle value of the argument. but in Quad II
The magnitude or modulus of z is the same as r. We can take complex numbers given as and convert them to polar form : Real Axis Imaginary Axis y x z = r Plot the complex number: Find the polar form of this number. 1 factor r out but in Quad II