diff --git a/src/1_complex_numbers.md b/src/1_complex_numbers.md
index 3351552810b1eac582afee1a7d417090e7b62c9d..f24ee1e2ce0de1f22f08eefc97964d491884bfff 100644
--- a/src/1_complex_numbers.md
+++ b/src/1_complex_numbers.md
@@ -2,8 +2,9 @@
 title: Complex Numbers
 ---
 
-Complex numbers
-===============
+# Complex numbers
+
+## Definition and basic operations
 
 Complex numbers are numbers of the form $$z = a + b {\rm i}.$$ Here
 $\rm i$ is the square root of -1: $${\rm i} = \sqrt{-1},$$ or,
@@ -21,8 +22,8 @@ Some definitions:
     $$z^* = a - b{{\rm i}},$$ i.e., taking the complex conjugate means
     flipping the sign of the imaginary part.
 
-Addition
---------
+### Addition
+
 
 For two complex numbers, $z_1 = a_1 + b_1 {{\rm i}}$ and
 $z_2 = a_2 + b_2 {{\rm i}}$, the sum $w = z_1 + z_2$ is given as
@@ -35,8 +36,8 @@ in twice the real part of $z$. Similarly, subtracting $z^*$ from $z$
 yields $$z - z^* = a + b {{\rm i}} - a + b {{\rm i}}= 2b{\rm i},$$ i.e.,
 twice the imaginary part of $z$ (times $\rm i$).
 
-Multiplication
---------------
+### Multiplication
+
 
 For the same two complex numbers $z_1$ and $z_2$ as above, their product
 is calculated as
@@ -50,8 +51,7 @@ $$z z^* = (a+b{{\rm i}})(a-b{{\rm i}}) = a^2 + b^2.$$ The square root of
 this number is the *norm* $|z|$ of $z$:
 $$|z| = \sqrt{z z^*} = \sqrt{a^2 + b^2}.$$
 
-Division
---------
+### Division
 
 The quotient $z_1/z_2$ of two complex numbers $z_1$ and $z_2$ as above,
 can be evaluated by multiplying the numerator and denominator by the
@@ -59,8 +59,7 @@ complex conjugate of $z_2$:
 $$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
 Check this!
 
-The complex plane
-=================
+## The complex plane
 
 Complex numbers can be rendered on a two-dimensional (2D) plane, the
 *complex plane*. This plane is spanned by two unit vectors, one
@@ -71,8 +70,7 @@ unit vector represents ${\rm i}$.
 
 Note that the norm of $z$ is the length of this vector.
 
-Addition in the complex plane
------------------------------
+### Addition in the complex plane
 
 Adding two numbers in the complex plane corresponds to adding the
 horizontal and vertical components:
@@ -82,8 +80,8 @@ horizontal and vertical components:
 We see that the sum is found as the diagonal of a parallelogram spanned
 by the two numbers.
 
-Argument and Norm
------------------
+### Argument and Norm
+
 
 A complex number can be represented by two real numbers, $a$ and $b$
 which represent the real and imaginary part of the complex number. An
@@ -115,8 +113,7 @@ $$\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
  -\pi + \arctan(b/a) &{\rm ~ for ~} a<0 {\rm ~ and ~} b<0.
  \end{cases}$$
 
-Complex functions
------------------
+## Complex functions
 
 A complex function $f$ maps any complex number $z$ onto another complex
 number $f(z)$. Just as with real functions, we can define the derivative
@@ -184,8 +181,7 @@ $$f(z) = \sum_{j=0}^\infty d_j (z-a)^j,$$ where
 $$d_j = \frac{1}{j!} f^{(j)} (a).$$ Here, $f^{(j)}(a)$ denotes the
 $j$-th derivative of the function $f$ in the point $a$.
 
-The complex exponential function
---------------------------------
+## The complex exponential function
 
 The exponential function $f(z) = \exp(z) = e^z$ is defined as:
 $$\exp(z) = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
@@ -248,8 +244,8 @@ function helps in re-deriving trigonometric identities.
     and imaginary parts leads to $$\cos'\varphi = - \sin\varphi;$$
     $$\sin'\varphi = \cos\varphi.$$
 
-Hyperbolic functions
---------------------
+## Hyperbolic functions
+
 
 From
 $e^{\rm i \varphi} = \left( \cos\varphi + {\rm i} \sin\varphi\right)$,
@@ -280,8 +276,8 @@ Finally, the hyperbolic tangent is defined as
 $$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}.$$ Its derivative is given as
 $$\tanh'(x) = 1 + \frac{\sinh^2 x}{\cosh^2 (x)} = - \frac{1}{\cosh^2(x)}.$$
 
-Summary
-=======
+## Summary
+
 
 -   A complex number $z$ has the form $$z = a + b \rm i$$ where $a$ and
     $b$ are both real, and $\rm i^2 = 1$. The real number $a$ is called
@@ -331,8 +327,7 @@ Summary
         $$\sinh(z) = \frac{e^{z} - e^{-z}}{2}; \phantom{xxx} \cosh(z) = \frac{e^{z} + e^{-z}}{2}.$$
 
 
-Problems
-========
+## Problems
 
 1.  [:grinning:]  Given $a=1+2\rm i$ and $b=4-2\rm i$, draw in the
     complex plane the numbers $a+b$, $a-b$, $ab$, $a/b$, $e^a$ and