12.04 Distance between and midpoints of complex numbers

 

Distance between two complex numbers

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.

 

The mid-point between two complex numbers

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+d2​i.   

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​+−5−12​i=12​−3i. 

 

An example 

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​+12−242​i=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. 

 

Worked Examples

Question 1

Question 2