Merge branch 'maciejedits' into 'master'

Formatting problems sections (complex numbers)

See merge request !18

src/1_complex_numbers.md

@@ -287,27 +287,33 @@ function help in re-deriving trigonometric identities.
 
 ## 1.6. Problems
 
-1.  [:grinning:]  Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the
-    complex plane the numbers $a+b$, $ab$, and $b/a$.
-
-2.  [:grinning:] Evaluate (a) $\rm i^{1/4}$, (b)
-    $\left(1+\rm i \sqrt{3}\right)^{1/2}$, (c) $\exp(2\rm i^3)$.
-
-3.  [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$ (i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively).
-
-4.  [:grinning:] (a) Find the real and imaginary part of 
-    $$ \frac{1+ {\rm i}}{2+3{\rm i}}$$
-    (b) Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
-    for real $a$ and $b$.
-
-5.  [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geometric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
-
-6.  [:grinning:] Compute (a) 
-    $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
-    and (b) calculate the real part of
-    $$\int_0^\infty e^{-\gamma t  +\rm i \omega t} dt$$($k$, $x$, $\omega$, $t$ and
-    $\gamma$ are real; $\gamma$ is positive).
-
-7.  [:smirk:] Compute
+1.  [:grinning:] Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the complex plane the numbers:
+    1.  $a+b$,
+    2.  $ab$, 
+    3.  $b/a$.
+
+2.  [:grinning:] Evaluate:  
+    1. $\rm i^{1/4}$, 
+    2. $\left(1+\rm i \sqrt{3}\right)^{1/2}$,
+    3. $\exp(2\rm i^3)$.
+
+3.  [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$. </br>
+    (i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively).
+
+4.  [:grinning:] *Quotients*</br>
+    1. Find the real and imaginary part of $$ \frac{1+ {\rm i}}{2+3{\rm i}} \, .$$
+    2. Evaluate for real $a$ and $b$:$$\left| \frac{a+b\rm i}{a-b\rm i} \right| \, .$$ 
+
+5.  [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. 
+    1.  Visualize taking the inverse in the complex plane. 
+    2.  What geometric operation does taking the inverse correspond to? </br>
+    (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
+
+6.  [:grinning:]  *Differentation and integration* </br>
+    1. Compute $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
+    2. Calculate the real part of $$\int_0^\infty e^{-\gamma t  +\rm i \omega t} dt$$ 
+    ($k$, $x$, $\omega$, $t$ and $\gamma$ are real; $\gamma$ is positive).
+
+7.  [:smirk:] Compute by making use of the Euler identity.
     $$\int_{0}^{\pi}\cos(x)\sin(2x)dx$$
-    by making use of the Euler identity.
+