Formatting

mechanical/mmme1026_maths_for_engineering.md

@@ -19,8 +19,8 @@ tags: [ uni, nottingham, mmme1026, maths, complex_numbers ]
 
   Where:
 
-  - $x$ is the real part of $z$ (Re($z$))
+  - $x$ is the real part of $z$ (which you may seen written as $\Re(z) = x$ or Re$(z) = x$)
-  - $y$ is the imaginary part of $z$(Im($z$))
+  - $y$ is the imaginary part of $z$ (which you may seen written as $\Im(z) = y$ or Im$(z) = y$)
 
 - Two complex numbers are equal if both their real and imaginary parts are equal
 
@@ -222,15 +222,24 @@ z_3 = z_1 z_2 &= (r_1e^{i\theta_1})(r_2e^{i\theta_2}) \\
 
 Let $z = re^{i\theta}$. Consider $z^n$.
 
+Since $z = r(\cos\theta + i\sin\theta)$,
 \begin{align*}
-\text{Since } z = r(\cos\theta + i\sin\theta) \\
 z^n &= r^n(\cos\theta + i\sin\theta)^n &\text{(1)} \\
-\text{But also} \\
+\end{align*}
+
+But also
+
+\begin{align*}
 z^n &= (re^{i\theta})^n \\
     &= r^n(e^{i\theta})^n \\
     &= r^ne^{in\theta} \\
     &= r^n(\cos{n\theta} + i\sin{n\theta}) &\text{(2)} \\
-\text{By equating (1) and (2), we find:}\\
+\end{align*}
+
+By equating (1) and (2), we find de Moivre's theorem:
+
+\begin{align*}
+r^n(\cos\theta +i\sin\theta)^n &= r^n(\cos{n\theta} + i\sin{n\theta}) \\
 (\cos\theta +i\sin\theta)^n &= (\cos{n\theta} + i\sin{n\theta})
 \end{align*}