Complex numbers are important in applied mathematics. Problems and questions on complex numbers with detailed solutions are presented.

 Evaluate the following expressions a) (3 + 2i) - (8 - 5i) b) (4 - 2i)*(1 - 5i) c) (- 2 - 4i) / i d) (- 3 + 2i) / (3 - 6i) If (x + yi) / i = ( 7 + 9i ) , where x and y are real, what is the value of (x + yi)(x - yi)? Determine all complex number z that satisfy the equation z + 3 z' = 5 - 6i where z' is the complex conjugate of z. Find all complex numbers of the form z = a + bi , where a and b are real numbers such that z z' = 25 and a + b = 7 where z' is the complex conjugate of z. The complex number 2 + 4i is one of the root to the quadratic equation x2 + bx + c = 0, where b and c are real numbers. a) Find b and c b) Write down the second root and check it. Find all complex numbers z such that z2 = -1 + 2 sqrt(6) i. Find all complex numbers z such that (4 + 2i)z + (8 - 2i)z' = -2 + 10i, where z' is the complex conjugate of z. Given that the complex number z = -2 + 7i is a root to the equation: z3 + 6 z2 + 61 z + 106 = 0 find the real root to the equation. a) Show that the complex number 2i is a root of the equation z4 + z3 + 2 z2 + 4 z - 8 = 0 b) Find all the roots root of this equation. P(z) = z4 + a z3 + b z2 + c z + d is a polynomial where a, b, c and d are real numbers. Find a, b, c and d if two zeros of polynomial P are the following complex numbers: 2 - i and 1 - i. Solutions to the Above Questions a) -5 + 7i b) -6 - 22i c) -4 + 2i d) -7/15 - 4i/15 (x + yi) / i = ( 7 + 9i ) (x + yi) = i(7 + 9i) = -9 + 7i (x + yi)(x - yi) = (-9 + 7i)(-9 - 7i) = 81 + 49 = 130 Let z = a + bi , z' = a - bi ; a and b real numbers. Substituting z and z' in the given equation obtain a + bi + 3*(a - bi) = 5 - 6i a + 3a + (b - 3b) i = 5 - 6i 4a = 5 and -2b = -6 a = 5/4 and b = 3 z = 5/4 + 3i z z' = (a + bi)(a - bi) = a2 + b2 = 25 a + b = 7 gives b = 7 - a Substitute above in the equation a2 + b2 = 25 a2 + (7 - a)2 = 25 Solve the above quadratic function for a and use b = 7 - a to find b. a = 4 and b = 3 or a = 3 and b = 4 z = 4 + 3i and z = 3 + 4i have the property z z' = 25. a) Substitute solution in equation: (2 + 4i)2 + b(2 + 4i) + c = 0 Expand terms in equation and rewrite as: (-12 + 2b + c) + (16 + 4b)i = 0 Real part and imaginary part equal zero. -12 + 2b + c = 0 and 16 + 4b = 0 Solve for b: b = -4 , substitute and solve for c: c = 20 b) Since the given equation has real numbers, the second root is the complex conjugate of the given root: 2 - 4i is the second solution. Check: (2 - 4i)2 - 4 (2 - 4i) + 20 (Expand) = 4 - 16 - 16i - 8 + 16i + 20 = (4 - 16 - 8 + 20) + (-16 + 16)i = 0 Let z = a + bi Substitute into given equation: (a + bi)2 = -1 + 2 sqrt(6) i Expand: a2 - b2 + 2 ab i = - 1 + 2 sqrt(6) i Real part and imaginary parts must be equal. a2 - b2 = - 1 and 2 ab = 2 sqrt(6) Equation 2 ab = 2 sqrt(6) gives: b = sqrt(6) / a Substitute: a2 - ( sqrt(6) / a )2) = - 1 a4 - 6 = - a2 Solve above equation and select only real roots: a = sqrt(2) and a = - sqrt(2) Substitute to find b and write the two complex numbers that satisfies the given equation. z1 = sqrt(2) + sqrt(3) i , z2 = - sqrt(2) - sqrt(3) i Let z = a + bi where a and b are real numbers. The complex conjugate z' is written in terms of a and b as follows: z'= a - bi. Substitute z and z' in the given equation (4 + 2i)(a + bi) + (8 - 2i)(a - bi) = -2 + 10i Expand and separate real and imaginary parts. (4a - 2b + 8a - 2b) + (4b + 2a - 8b - 2a )i = -2 + 10i Two complex numbers are equal if their real parts and imaginary parts are equal. Group like terms. 12a - 4b = -2 and - 4b = 10 Solve the system of the unknown a and b to find: b = -5/2 and a = -1 z = -1 - (5/2)i Since z = -2 + 7i is a root to the equation and all the coefficients in the terms of the equation are real numbers, then z' the complex conjugate of z is also a solution. Hence z3 + 6 z2 + 61 z + 106 = (z - (-2 + 7i))(z - (-2 - 7i)) q(z) = (z2 + 4z + 53) q(z) q(z) = [ z3 + 6 z2 + 61 z + 106 ] / [ z2 + 4z + 53 ] = z + 2 Z + 2 is a factor of z3 + 6 z2 + 61 z + 106 and therefore z = -2 is the real root of the given equation. a) (2i)4 + (2i)3 + 2 (2i)2 + 4 (2i) - 8 = 16 - 8i - 8 + 8i - 8 = 0 b) 2i is a root -2i is also a root (complex conjugate because all coefficients are real). z4 + z3 + 2 z2 + 4 z - 8 = (z - 2i)(z + 2i) q(z) = (z2 + 4)q(z) q(z) = z2 + z - 2 The other two roots of the equation are the roots of q(z): z = 1 and z = -2. Since all coefficients of polynomial P are real, the complex conjugate to the given zeros are also zeros of P. Hence P(z) = (z - (2 - i))(z - (2 + i))(z - (1 - i))(z - (1 + i)) = = z4 - 6 z3 + 15 z2 - 18 z + 10 Hence: a = -6, b = 15, c = -18 and d = 10.