The argument of 1-i 3 1+i 3 is

Step-by-step solution

Let z = 1 - i 3 1 + i 3 . Multiply the numerator and denominator by the conjugate of the denominator, 1 - i 3 : z = (1 - i 3 )(1 - i 3 ) (1 + i 3 )(1 - i 3 ) z = 1 - 2i 3 + (i 3 )^2 1^2 + ( 3 )^2 z = 1 - 2i 3 - 3 1 + 3 z = -2 - 2i 3 4 z = - 1 2 - i 3 2 The complex number z is in the third quadrant since both the real part x = - 1 2 and the imaginary part y = - 3 2 are negative. The argument is given by = - + ⁻¹ ( | y x | ) or simply by observing the angle in the third quadrant. ( ) = | - 3 /2 -1/2 | = 3 = 3 . Since