Three famous mathematical formulas

Prerequisites: Before attempting this assessment you should have already worked through all the articles in this module, and also have an understanding of HTML basics (study Introduction to HTML).
Objective: To have a play with some MathML and test your new-found knowledge.

A small math article

The goal is to rewrite the following math article using HTML and MathML:

Screenshot of the PDF output generated by XeLaTeX

Although you don't need to be familiar with LaTeX, it might be useful to know the LaTeX source from which it was generated:

latex
\documentclass{article}

\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}

To solve the cubic equation $t^3 + pt + q = 0$ (where the real numbers
$p, q$ satisfy ${4p^3 + 27q^2} > 0$) one can use Cardano's formula:

\[
  \sqrt[{3}]{
    -\frac{q}{2}
    +\sqrt{\frac{q^2}{4} + {\frac{p^{3}}{27}}}
  }+
  \sqrt[{3}]{
    -\frac{q}{2}
    -\sqrt{\frac{q^2}{4} + {\frac{p^{3}}{27}}}
  }
\]

For any $u_1, \dots, u_n \in \mathbb{C}$ and
$v_1, \dots, v_n \in \mathbb{C}$, the Cauchy–Bunyakovsky–Schwarz
inequality can be written as follows:

\[
  \left| \sum_{k=1}^n {u_k \bar{v_k}} \right|^2
  \leq
  {
    \left( \sum_{k=1}^n {|u_k|} \right)^2
    \left( \sum_{k=1}^n {|v_k|} \right)^2
  }
\]

Finally, the determinant of a Vandermonde matrix can be calculated
using the following expression:

\[
  \begin{vmatrix}
  1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\
  1 & x_2 & x_2^2 & \dots & x_2^{n-1} \\
  1 & x_3 & x_3^2 & \dots & x_3^{n-1} \\
  \vdots & \vdots & \vdots & \ddots & \vdots \\
  1 & x_n & x_n^2 & \dots & x_n^{n-1} \\
  \end{vmatrix}
  = {\prod_{1 \leq {i,j} \leq n} {(x_i - x_j)}}
\]

\end{document}

Starting point

To get this assessment started, you can rely on our usual HTML template. By default it uses UTF-8 encoding, special Web fonts for the <body> and <math> tags (with similar look & feel as the LaTeX output). The goal is to replace the question marks ??? with actual MathML content.

html
<!doctype html>
<html lang="en-US">
  <head>
    <meta charset="utf-8" />
    <title>Three famous mathematical formulas</title>
    <link
      rel="stylesheet"
      href="https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css" />
  </head>
  <body class="htmlmathparagraph">
    <p>
      To solve the cubic equation ??? (where the real numbers ??? satisfy ???)
      one can use Cardano's formula: ???
    </p>

    <p>
      For any ??? and ???, the Cauchy–Bunyakovsky–Schwarz inequality can be
      written as follows: ???
    </p>

    <p>
      Finally, the determinant of a Vandermonde matrix can be calculated using
      the following expression: ???
    </p>
  </body>
</html>

Hints and tips

  • Start by inserting empty <math> tags, deciding whether they should have a display="block" attribute or not.
  • Check the text used and find their Unicode characters ("−", "ℂ", "∑", ...).
  • Analyze the semantics of each portion of text (variable? operator? number?) and determine the proper token element to use for each of them.
  • Look for advanced constructions (fractions? roots? scripts? matrices?) and determine the proper MathML element to use for each of them.
  • Don't forget to rely on <mrow> for grouping subexpressions.
  • Pay attention to stretchy and large operators!
  • Use the W3C validator to catch unintended mistakes in your HTML/MathML markup.
  • If you are stuck, or realize how painful it is to write MathML by hand, feel free to use tools to help write MathML such as TeXZilla.