Kris said:
z1 = −11+2 i and z2 = −1+13 i are given
I need to find the following in the form x + y i.
conjugate of z1 =
conjugate of z1z2=
z1/z2 =
a) how would i go about it
and
b) can someone provide the solutions to the questions if possible
I realize that with the other posts, this will be somewhat redundant, but I hope it will be useful nevertheless to have everything consolidated.
The Question:
Given the numbers $z_1 = -11 + 2i$ and $z_2 = -1 + 13i$, find:
i) the conjugate of $z_1$ (which I'll denote $\overline{z_1}$)
ii) $\overline{z_1 \, z_2}$
iii) $\overline{z_1/z_2}$
The Solution:
We begin by noting that for any complex number in the form $z=a+bi$, we may simply write $\overline{z}=a-bi$. Thus, the answer to i) is simply
$$\overline{z_1}=-11-2i$$
The second bit is a little trickier, because we have to multiply two complex numbers. There are different methods to use here, but I always prefer to "FOIL" it, that is, expand it as the usual product of binomials, remembering that $i^2=1$:
$$z_1\,z_2=(-11 + 2i)(-1 + 13i)=11-22i-143i+26i^2=11-26-22i-143i\\
=-15-165i$$
From there, it's easy to find the conjugate of the result:
$$\overline{z_1\,z_2}=-15+165i$$
See what I did there?
Another way we could have done this is by finding the conjugate of each number, then multiplying. In other words:
$$\overline{z_1\,z_2}=(\overline{z_1})(\overline{z_2})$$
As you will find out for problems more intricate than this, this becomes important later.
Part 3 is tricker still, because now we're dividing fractions. I'll make that its own post