- (i) ± 1
1
2
±
√
3
2
j −
1
2
±
√
3
2
j
(ii) − 21 ±
√
3 j
12.9 Reinforcement
1.Write the following in simplest form in terms of real numbers andj.
(i)
√
− 1 (ii)
√
9 (iii)
√
−9(iv)j^2
(v) −j^2 (vi)
1
j
(vii) (−j)^2
2.Solve the following equations, writing the answer inz=a+jbform:
(i) z^2 + 25 = 0 (ii) z^2 + 4 z+ 5 = 0
(iii) z^4 − 3 z^2 − 4 =0(iv)z^3 +z− 2 = 0
(v) z^3 + 1 =0(vi)z^2 + 2 jz+ 1 = 0
Using equations (i) – (v) verify the result that the equations with real coefficients have
real roots and/or complex roots occurring in conjugate pairs.
3.Express in the forma+jb:
(i) j^3 (ii) j^27 (iii) 3( 1 +j)− 2 ( 1 −j)
(iv) (− 2 j)^6 (v) j(j+ 2 ) (vi) j^3 /j
(vii) 2j(j− 1 )+j^3 ( 2 +j)
4.Find the real and imaginary parts of:
(i) ( 1 −j)( 1 +j) (ii) ( 3 − 4 j)( 1 +j) (iii) ( 4 + 3 j)^2
(iv) ( 4 + 3 j)^3
5.Write down the complex conjugates of:
(i) 5+ 3 j (ii)
1
3 − 4 j
(iii)
j
j+ 2
(iv)
j− 2
3 + 2 j
(v)
(
j− 2
3 + 2 j
) 4
(vi)
1
j( 2 − 5 j)^2
Where appropriate put both the original complex number and its conjugate intoa+jb
form and check your results.
