We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of the complex numbers. The reals are just the x-axis in the complex plane.
The modulus of the complex number z= a + ib now can be interpreted as the distance from z to the origin in the complex plane.
Since the hypotenuse of a right triangle is longer than the other sides, we have
for every complex number z.
We can also think of the point z= a+ ib as the vector (a,b). From this point of view, the addition of complex numbers is equivalent to vector addition in two dimensions and we can visualize it as laying arrows tail to end. (Picture)
We see in this way that the distance between two points z and w in the complex plane is |z-w|.
The identity is called the parallelogram law because if we think of z and w as vectors (or points) in the plane, then it tells us that the sum of the squares of the lengths of the sides of a parallelogram is equal to the sum of the squares of the lengths of its diagonals.
To prove the identity, just write
The last equality follows from .
Similarly, we have
Adding these gives the identity
We know the inequality when n=1 and when n=2 by the last exercise. We will show that the truth of the inequality for n=k implies it for n=k+1 when k is any integer. That will finish the proof. This is an example of proof by induction.
By the triangle inequality (in the simplest case n=2),
So the inductive hypothesis that
which is the triangle inequality for the case n= k+1.
Describe geometrically, the set of all complex numbers z which satisfy the following condition
Since |z-1| >0, we know the set we want to describe does not contain the point z=1. By the triangle inequality, we have
for all z. So we want to exclude all points from the plane where the equality
holds. That is, we want to exclude any z whose distance from 1 is equal to 1 plus its distance to the origin. This just means we have to exclude the negative real axis and the origin. (Draw a picture.)
We can also see this algebraically. Writing z= x+iy we have
Setting these equal gives
which reduces to
This can only hold if and y=0.
So the set we want is the complex plane with the point z=1 and the segment deleted.
Prove that the medians of a triangle with vertices , and intersect at the point
Using the previous exercise we can write the medians of the triangle as
These segments intersect if and only if there are real numbers in the interval [0,1] such that
It's clear that is the unique solution to this system so the point of intersection is