If a problem is solved. It is not 'the' answer.
No attempt is made to search for the most elegant answer.
I highly recommend that you at least try to solve the
problem before you read the solution.
(4i7)+(1i) = 3i6 (5i) = 5+i i.(i1) = 1i 1  = i i 1 = 1 i = 1 4 = 4
1+i  = ? 1i 
i^{2012} = ? 
solve z^{2} = 4i 
Show that for each complex number z _ z . z = a real number 
Calculate the conjugate complex number of z = a + bi 2 a  bi 2 () + () a  bi a + bi 
Solve : ix^{2} +(15i)x 1+8i=0 
Find the polar representation of (isqrt(3)) 
Simple calculations 
2.(cos(1) +i sin(1)).5.(cos(2) +i sin(2))= 10.(cos(3) +i sin(3)) 6.(cos(5) +i sin(5)) = 2.(cos(3) +i sin(3)) 3.(cos(2) +i sin(2)) (2.(cos(3) +i sin(3)))^{5} = 32.(cos(15) +i sin(15))
Find all z so that z^{4} = 8(isqrt(3)) 
Given : z=cos(3)+ i sin(3) _ Prove that 1 + z = (1 + z )z 
Given : u = 1+i.sqrt(3) and v = sqrt(3) + i
Calculate u^{3} / v^{4}

Show that the equation has a real root.
4z^{3}  6i sqrt(3) z^{2}  3(3 + i sqrt(3)) z  4 = 0

Find
(1+i)^{17}  (1i)^{16} 
Given: z not real and z= 1 z1 Show that w =  is a pure imaginary number. z+1 
Prove that in C, there are no divisors of zero. That is, z.z'=0 => (z=0 or z'=0) 
Calculate ( cos(2)+ i sin(2) + 1)^{n} 
The image point of z = a + bi in the Gaussplane is p. We rotate p about o and the angle of the rotation is pi/3. The new position of p is p'. Calculate the coordinates of p'. 
a, b, c are real numbers in the polynomial p(z) = 2 z^{4} + a z^{3} + b z^{2} + c z + 3 . Find a such that the numbers 2 and i are roots of p(z) = 0. 
Given: n is a positive integer. z is a complex number with modulus 1, such that z^{2n} is not 1. z^{n} Show that  is a real number 1 + z^{2n} 
Calculate all integers n such that z_{n} = (1 + i sqrt(3))^{n} is a real number. 
Calculate the real values of x and y such that (x + iy)^{3} is real and x + i y is higher than 8. 
Given: complex number z = cos(2t) + i sin(2t)
Show that 2/(1+z) = 1  i tan(t) 
Find the real value of m such that the equation 2 z^{2}  ( 3+ 8i )z  ( m + 4i) = 0 has a real root. Then find the roots. 
Find real values of the number a for which a.i is a solution
of the polynomial equation z^{4}  2z^{3} + 7z^{2}  4z + 10 = 0. Then find all roots of this equation. 
u,v and w are the three roots of the equation z^{3}  1 = 0 .
Calculate u.v + v.w + w.u without calculating the 3 roots. 
Calculate all solutions of z1.z1=1 
The equation
z^{3}  (n + i) z + m + 2 i = 0has three roots. n and m are real constants. a) Calculate m such that the modulus of the product of the roots is 5. b) Calculate the modulus of the sum of the roots. 
Let z' the conjugate complex number of z. Now find z such that
z^{2} + z'^{2} = 0 
In the following equation, m is a real number.
z^{2}  (3 + i) z + m + 2 i = 0Calculate the values of m such that the equation has a real root. Calculate the second root. 
The number t is real and not an integer multiple of (pi/2). The complex numbers x_{1} and x_{2} are the roots of the equation tan^{2} (t).x^{2} + tan(t).x + 1 = 0 Show that (x_{1})^{n} + (x_{2})^{n} = 2 cos(2 n pi/3) cot^{n}(t) 
Calculate the values of m such that the roots x_{1} and x_{2} of
x^{2}  2m x + m = 0 satisfy the condition x_{1}^{3} + x_{2}^{3} = x_{1}^{2} + x_{2}^{2}. Calculate the roots for those mvalues and check the condition. 
Find all avalues such that the following statement is true. In C, the set of all roots of z^{8}  1 = 0 is { a^{k}  k in {1,2,3,4,5,6,7,8} }. 
The equation z^{3}  i. 4 sqrt(3) = 4 has a solution z_{1} = 2(cos(pi/9) + i sin(pi/9)) Find the other roots z_{2} and z_{3}. 
The number u, different from 1, is a solution of z^{3}=1. Find the value of the determinant D =
1 u u^{2} u u^{2} 1 u^{2} u 1 