1. ## Complex no. problem......

a)Find the complex number z , in the form a + ib that satisfy the equation z^2 = -3 +4i

B)
Give the geometrical representation of the product of the two complex numbers.

c) if zz' - 5iz = 10 - 20i , find the possible values of z.

2. Originally Posted by asnali

a)Find the complex number z , in the form a + ib that satisfy the equation [FONT=&quot] z^2 = -3 +4i
see here

B) Give the geometrical representation of the product of the two complex numbers.
For two complex numbers $\displaystyle z_1$ and $\displaystyle z_2$ we wish to geometrically interpret $\displaystyle z_1z_2$.

This is probably easier to visualize if you write the complex numbers in polar form, but just to describe it to you. This is taken from Bak's Complex Analysis: "If we form a triangle with two sides given by the vectors (originating from 0 to) 1 and $\displaystyle z_1$ and then form a similar triangle with the same orientation and the vector $\displaystyle z_2$ corresponding to the vector 1, the vector which corresponds to $\displaystyle z_1$ will be $\displaystyle z_1z_2$."

c) if zz' - 5iz = 10 - 20i , find the possible values of z.
write $\displaystyle z = x + iy$ so that $\displaystyle \bar z = x - iy$. recall that $\displaystyle z \bar z = |z|^2 = x^2 + y^2$

Thus we have $\displaystyle x^2 + y^2 - 5i(x + iy) = 10 - 20i$

Now expand the brackets on the left, equate coefficients and solve the resulting system of equations to find $\displaystyle x$ and $\displaystyle y$ and hence, $\displaystyle z$