I am new to Complex Analysis. The author says that multiplication of complex numbers is a rotation composed with a dilation but I can't understand why that was true.
Take $z=re^{i\theta},w=se^{i\phi}$
How does it show that multiplication of complex numbers is a rotation composed with a dilation?
Think of $z$ as a constant and $w$ as a variable. Then multiplication by $z$ is a function that takes the variable $se^{i\phi}$ to $(rs)e^{i(\theta+\phi)}$.
That is, if you start with $w$, you can find $zw$ by first rotating $w$ through the angle $\theta$ (replacing $\phi$ with $\phi+\theta$) and then stretching by the factor $r$ (replacing $s$ with $rs$).
Of course, just a formula is not a proof of geometric behavior if it does not mention geometry at all -- it is rather a proof by circle.
It is not easy to prove the geometric property from the algebraic definition of complex numbers multiplication, i.e. if $x=a+bi$ and $y=c+di$ then $xy = (ac-bd)+i(ad+bc)$. It can be derived from laws of sine and cosines but I consider such mind process absurd since the best way of proving laws of sine and cosine uses properties of complex numbers.
I provide you another approach which may look as a cheat at the first glance but it is not: The geometric properties are true by definition. I define multiplication of complex numbers as multiplicating their lengths and summing their oriented angles. If we use such definition of complex multiplication, it is obvious that multiplication by a fixed complex number stretches the whole plane by its length and rotates it by its angle. The "only" unclear thing is what does this definition have in common with the algebraic multiplication.
It suffices to prove that both definitions are equivalent, in other words it suffices to show that my geometric definition satisfies the formula $$(a+bi)(c+di) = (ac-bd)+i(ad+bc).$$ Of course, this formula was not created by magic but by simple algebraic expansion: $$(a+bi)(c+di) = (a+bi)c+(a+bi)di = ac+bci+adi+bdi^2 = (ac-bd)+i(ad+bc).$$ We just have to verify that this process is correct if we use the geometric definition of multiplication. Commutativity and associativity are obvious as well as equivalence with real multiplication and $i^2=-1$. The only not so clear fact is distributivity -- that we can expand a complex product $u(x+y)$ to $ux+uy$. To see that, we use the observation, that multiplying by $x$ is just rotating and stretching the plane. So we are just checking the fact that if $x+y=z$ then the equation remains valid after rotating and stretching of the complex plane.
Alternative, less elementary but still very nice way to define complex numbers result comes from linear algebra. You can define complex numbers as certain linear transformations of the the 2-dimensional space -- as all spiral similarities around 0. They can be added point-wise, multiplication is defined as composition and the algebraic properties are for free from the theory of linear algebra. Some people need to mention that this mean matrices of form $$\pmatrix{\hphantom{-}a&b\cr -b&a}$$ and it is probably useful for proving that it is closed under addition, but in general, you know, proofs involving matrices can be shortened by 50% if one throws the matrices out.
