Thousand Words
Here the diagram says it all. Can you find the diagram?
Age
16 to 18
Challenge level
Problem
Draw a simple diagram which makes it clear that the following inequality holds for any complex numbers $z$ and $w$ $$|z-w|\geq |z| -|w|$$
Now draw another simple diagram which makes it clear that the following inequality holds for any real numbers $\alpha> \beta$
$$|e^{i\alpha}-e^{i\beta}|\leq \alpha - \beta$$
Getting Started
The whole point of the question is for you to find the right
diagrams. You need to know that the distance between two points $a$
and $b$ in the complex plane (Argand Diagram) is given by
$|a-b|$.
By definition $e^{i\alpha}= \cos \alpha + i\sin \alpha$ so
where does this lie in the complex plane? Now what about
$e^{i\beta}$?
Student Solutions
Thank you to Roy M. from Allerton High School and Andrei L. from Tudor Vianu National College, Bucharest, Romania for their solutions.
Draw two concentric circles with centres at the origin and radii $|z|$ and $|w|$ and suppose, without loss of generality, that $|z|> |w|$ then $z$ lies somewhere on the outer circle and $w$ lies somewhere on the inner circle.
Image
In this representation, working with complex numbers is similar to working with vectors. The two complex numbers can be added and subtracted as vectors. So, to subtract them, it is the same as adding one to the opposite of the other. The result is represented in the figure. The distance between the two points representing the complex numbers $z$ and $w$ is the modulus of $z-w$ written as
$|z-w|$. The lengths of the sides of the triangle in the diagram are $|z|$, $|w|$ and $|z-w|$.
The shortest distance between any two points where one is on each circle is given by the difference of the radii of the two circles $|z|-|w|$. This would be the distance between the points $z$ and $w$ if they had the same argument. If $z$ and $w$ have different arguments then the distance between the points will be $|z- w|$ and hence $|z-w|\geq |z| - |w|$.
For the second part, mark points $A$ and $B$ on the unit circle to represent $e^{i\alpha*}$ and $e^{i\beta*}$ respectively where $\alpha*$ and $\beta*$ lie between 0 and $2\pi$. To include all possible values of $\alpha$ and $\beta$ we have $\alpha =\alpha* +2m\pi$ and $\beta = \beta* + 2n\pi$ for some integers $m$ and $n$. Clearly $A$ and $B$ also represent $e^{i\alpha}$ and
$e^{i\beta}.$
Image
The chord length $AB$ is less than or equal to the arc length $AB$. Hence $$|e^{i\alpha}-e^{i\beta}|\leq \alpha - \beta.$$
Teachers' Resources
Care needs to be taken over values of $\alpha$ and $\beta$ greater than $2\pi$.