Checked for typos, rephrased a few sentences,

centered images and moved their description into a <figcaption> (HTML)

src/1_complex_numbers.md

@@ -94,7 +94,7 @@ respective horizontal and vertical components:
 
 <figure markdown>
   ![image](figures/complex_numbers_8_0.svg)
-  <figcaption>The sum is found as the diagonal of a parallelogram spanned by the two numbers.</figcaption>
+  <figcaption>The sum of two complex numbers is found as the diagonal of a parallelogram spanned by the vectors of those two numbers.</figcaption>
 </figure>
 
 ## 1.2. Complex functions
@@ -134,8 +134,8 @@ angle that the vector makes with the horizontal axis:
 <figure markdown>
   ![image](figures/complex_numbers_10_0.svg)
   <figcaption>The angle with the horizontal axis is denoted by $\varphi$ 
-  like the case of conventional polar coordinates, 
-  but in the context of complex numbers, this angle is denoted as the *argument*.</figcaption>
+  like in the case of conventional polar coordinates, 
+  but in the context of complex numbers, this angle is called as the <b>argument</b>.</figcaption>
 </figure>
 
 !!! info "Polar form of complex numbers"
@@ -157,15 +157,12 @@ can be expressed  in polar coordinates as $$a = |z| \cos\varphi$$ $$b = |z| \sin
  
 It turns out that by using the magnitude $|z|$ and phase $\varphi$, we can write any complex number as
 $$z = |z| e^{{\rm i} \varphi}$$
-When increasing $\varphi$ with $2 \pi$, we make a full circle around the origin and reach the same point on the complex plane. In other words, by adding $2 \pi$ to the argument of $z$, we get the same complex number $z$!
-As a result, the argument $\varphi$ is defined up to $2 \pi$, and we are free to make any choice we like, such as
-$$\begin{align}
--\pi < &\varphi < \pi  \textrm{ (left)} \\
--\frac{\pi}{2} < &\varphi < \frac{3 \pi}{2} \textrm{ (right)} \end{align} $$
+By increasing $\varphi$ by $2 \pi$, we make a full circle around the origin and reach the same point on the complex plane. In other words, by adding $2 \pi$ to the argument of $z$, we get the same complex number $z$!
+As a result, the argument $\varphi$ is defined up to $2 \pi$, and we are free to make any choice we like, such as in the examples in the figure below:
 
 <figure markdown>
   ![image](figures/complex_numbers_11_0.svg)
-  <figcaption> $-\pi < \varphi < \pi$ and $-\frac{\pi}{2} < \varphi < \frac{3 \pi}{2}$ </figcaption>
+  <figcaption> $-\pi < \varphi < \pi$ (left) and (right) $-\frac{\pi}{2} < \varphi < \frac{3 \pi}{2}$ </figcaption>
 </figure>
 
 Some useful values of the complex exponential to know by heart are:
@@ -185,18 +182,21 @@ Furthermore, we can define the sine and cosine in terms of complex exponentials:
     $$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}$$
 
 Most operations on complex numbers become easier when complex numbers are converted to their *polar form* using the complex exponential.
-Some operations which are common in real analysis can be easily derived for their complex counterparts:
+Some functions and operations, which are common in real analysis, can be easily derived for their complex counterparts by sustituting the real variable $x$ with the complex variable $z$ in its polar form:
 !!! info "Examples of some complex functions stated using polar form"
     $$z^{n} = \left(r e^{{\rm i} \varphi}\right)^{n} = r^{n} e^{{\rm i} n \varphi}$$
     $$\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \varphi} } = \sqrt[n]{r} e^{{\rm i}\varphi/n} $$
     $$\log(z) = log \left(r e^{{\rm i} \varphi}\right) = log(r) + {\rm i} \varphi$$
     $$z_{1}z_{2} = r_{1} e^{{\rm i} \varphi_{1}} r_{2} e^{{\rm i} \varphi_{2}} = r_{1} r_{2} e^{{\rm i} (\varphi_{1} + \varphi_{2})}$$
 
-As a result of multiplication, the norm of the new number is the *product* of the norms of the multiplied numbers, and its argument is the *sum* of the arguments of the multiplied numbers. 
+Use of polar form lets us notice immediately that for example, as a result of multiplication, the norm of the new number is the *product* of the norms of the multiplied numbers and its argument is the *sum* of the arguments of the multiplied numbers. 
 
 In the complex plane, this looks as follows:
 
-![image](figures/complex_numbers_12_0.svg)
+<figure markdown>
+  ![image](figures/complex_numbers_12_0.svg)
+  <figcaption></figcaption>
+</figure>
 
 !!! check "Example: Find all solutions solving $z^4 = 1$."
     Of course, we know that $z = \pm 1$ are two solutions, but which other solutions are possible? We take a systematic approach:
@@ -219,8 +219,8 @@ We can then regard the complex ${\rm i}$ as another constant, and use our usual
 
 ## 1.4. Bonus: the complex exponential function and trigonometry
 
-Let us show some tricks where the simple properties of the exponential
-function helps in re-deriving trigonometric identities.
+Let us show some tricks in the folloiwing examples where the simple properties of the exponential
+function help in re-deriving trigonometric identities.
 
 !!! example "Properties of the complex exponential function I"
     Take $|z_1| = |z_2| = 1$, and $\arg{(z_1)} = \varphi_1$ and
@@ -234,13 +234,13 @@ function helps in re-deriving trigonometric identities.
     \sin\varphi_1 \cos\varphi_2 \right).
     \end{align}$$
 
-    On the other hand, the right hand side can be written as
+    Also, the right hand side can be written as
     $$\exp[{\rm i} (\varphi_1 + \varphi_2)] = \cos(\varphi_1 + \varphi_2) + {\rm i} \sin(\varphi_1 + \varphi_2).$$
     Comparing the two expressions, equating their real and imaginary parts, we find
     $$\cos(\varphi_1 + \varphi_2) = \cos\varphi_1 \cos\varphi_2 - \sin\varphi_1 \sin\varphi_2;$$
     $$\sin(\varphi_1 + \varphi_2) = \cos\varphi_1 \sin\varphi_2 + 
     \sin\varphi_1 \cos\varphi_2.$$ 
-    Note that we used the resulting formulas immediately in order to derive the properties of the exponential function.
+    Note that we used the Euler formula in order to derive the identities of trigonometric function.
     The point is to show you that you can use the properties of the complex exponential to quickly find the form of trigonometric formulas, which are often easily forgotten.
 
 !!! example "Properties of the complex exponential function II" 
@@ -263,7 +263,7 @@ function helps in re-deriving trigonometric identities.
     $$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
 
 2.  Complex numbers can also be characterised by their *norm*
-    $|z|=\sqrt{a^2+b^2}$ and *argument* $\varphi$. These coordinates
+    $|z|=\sqrt{a^2+b^2}$ and *argument* $\varphi$. These parameters
     correspond to polar coordinates in the complex plane. For a complex
     number $z = a + b {\rm i}$, its real and imaginary parts can be
     expressed as $$a = |z| \cos\varphi$$ $$b = |z| \sin\varphi$$ The
@@ -277,7 +277,7 @@ function helps in re-deriving trigonometric identities.
 
 3.  The most important complex function for us is the complex exponential function, which simplifies many operations on complex numbers
     $$\exp(z) = e^{x + {\rm i}y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
-    where $y$ is defined up to $2 \pi$.
+    where $y$ is defined up to $2 \pi$.\\
     The $\sin$ and $\cos$ can be rewritten in terms of this complex exponential as
     $$\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}$$
     $$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2i}$$