Operations with two complex numbers

Problem

Let $z=2+3i$ and $w=1-i$.

(a) Compute $z+w$ and $z\cdot w$.

(b) Compute $z/w$ in algebraic form.

(c) Compute $|z|$ and $|w|$.

(d) Find $\bar z$ and $1/z$.

(e) Plot the vectors $z$, $w$, $\bar z$ and $z+w$ on the Gauss plane.

Solution

(a) Adding real and imaginary parts gives $z+w=3+2i$. For the product, apply the distributive property, recalling that $i^2=-1$: $z\cdot w=(2+3i)(1-i)=2-2i+3i-3i^2=5+i.$

(b) Multiply numerator and denominator by the conjugate of $w$, namely $1+i$: $\frac zw=\frac{(2+3i)(1+i)}{(1-i)(1+i)}=\frac{-1+5i}{2}=-\frac12+\frac52 i.$

(c) The moduli are obtained as the square root of the sum of the squares of the real and imaginary parts: $|z|=\sqrt{2^2+3^2}=\sqrt{13}\approx3.606,\qquad |w|=\sqrt{1^2+(-1)^2}=\sqrt2\approx1.414.$

(d) The conjugate is obtained by flipping the sign of the imaginary part: $\bar z=2-3i$. The reciprocal is $\frac1z=\frac{\bar z}{|z|^2}=\frac{2-3i}{13}=\frac2{13}-\frac3{13}i.$

(e) In the Gauss plane the image points are $z=(2,3)$, $w=(1,-1)$, $\bar z=(2,-3)$ and $z+w=(3,2)$: four arrows from the origin to these points.

$\boxed{z+w=3+2i,\quad zw=5+i,\quad \frac zw=-\frac12+\frac52 i,\quad |z|=\sqrt{13},\quad |w|=\sqrt2,\quad \bar z=2-3i,\quad \frac1z=\frac{2-3i}{13}}$