While it makes no sense to think about ordering the complex numbers, it is meaningful to talk about distances between them.
The distance between the two complex numbers $z=a+bi$z=a+bi and $w=c+di$w=c+di is the non-negative real number $\left|z-w\right|=\sqrt{\left(a-c\right)^2+\left(b-d\right)^2}$|z−w|=√(a−c)2+(b−d)2 in accordance with Pythagoras' Theorem.
So, for example the distance between $z=2-5i$z=2−5i and $w=-1-i$w=−1−i is given by $\left|z-w\right|=\sqrt{\left(2+1\right)^2+\left(-5+1\right)^2}=\sqrt{9+16}=5$|z−w|=√(2+1)2+(−5+1)2=√9+16=5.
Similarly, we can determine the midpoint of any two complex numbers $z=a+bi$z=a+bi and $w=c+di$w=c+di by using the midpoint rule from coordinate geometry.
Thus midpoint $m$m is the complex number given by $m=\frac{a+c}{2}+\frac{b+d}{2}i$m=a+c2+b+d2i.
Using the previous complex numbers, we have that the midpoint between $z=2-5i$z=2−5i and $w=-1-i$w=−1−i is the complex number $m=\frac{2-1}{2}+\frac{-5-1}{2}i=\frac{1}{2}-3i$m=2−12+−5−12i=12−3i.
Find the distance between the midpoint of the two complex numbers $z_1=5+12i$z1=5+12i and $z_2=7-24i$z2=7−24i and the complex number $z_3=14+9i$z3=14+9i.
Then the midpoint, say $z_m$zm, is the complex number $z_m=\frac{5+7}{2}+\frac{12-24}{2}i=6-6i$zm=5+72+12−242i=6−6i. The distance between $z_m$zm and $z_3$z3 is then $\left|z_3-z_m\right|=\sqrt{\left(14-6\right)^2+\left(9+6\right)^2}=\sqrt{64+225}=17$|z3−zm|=√(14−6)2+(9+6)2=√64+225=17.
Find the distance between $9+4i$9+4i and $1-2i$1−2i.
Find the midpoint between $-5+8i$−5+8i and $7-2i$7−2i.