Transcription of Problem Set 8 Solutions - Open Yale Courses
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Problem Set 8 Solutions - Open Yale Courses
Problem Set 8 the real part, imaginary part, modulus, complex conjugate, and inverse of the following numbers: (i)23+4i,(ii)(3 + 4i)2, (iii)3+4i3 4i, (iv)1+ 2i1 3i, and (v)cos +isin .To find the quantities we are looking for, we need to put the complex number into the formz=a+bi. Then,the modulus is|z|= a2+b2, the complex conjugate isz =a bi, and the inverse can be found using theprevious quantities, as shown belowz 1=1z=1z(z z )=z |z|2(i)z=23 + 4i=23 + 4i(3 4i3 4i)=225(3 4i) = Re(z) =625, Im(z) = 825|z|=225 32+ 42=25z =225(3 + 4i)z 1=12(3 + 4i)(ii)z= (3 + 4i)2= 7 + 24i= Re(z) = 7, Im(z) = 24|z|= 72+ 242= 25z = 7 24iz 1= 1252(7 + 24i) = 1625(7 + 24i)(iii)z=3 + 4i3 4i=3 + 4i3 4i(3 + 4i3 + 4i)=125( 7 + 24i) = Re(z) = 725, Im(z) =2425|z|=125 72+ 242= 1z = 125(7 + 24i)z 1= 125(7 + 24i)(iv)z=1 + 2i1 3i=1 + 2i1 3i(1 + 3i1 + 3i)=14((1 6) +i( 2 + 3))= Re(z) =1 64, Im(z) = 2 + 34|z|=14 (1 6)2+ ( 2 + 3)2= 124= 32z =14((1 6) i( 2 + 3))z 1=13((1 6) i( 2 + 3))
Problem Set 8 Solutions 1. Find the real part, imaginary part, modulus, complex conjugate, and inverse of the following numbers: (i) 2 3+4i, (ii) (3+4i) 2, (iii) 3+4i 3−4i, (iv) 1+ √ i 1− √ 3i, and (v) cosθ +isinθ. To find the quantities we are looking for, we need to put the complex number into the form z = a + bi. Then, the modulus ...
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