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:
Although you don't need to be familiar with LaTeX , it might be useful to know the LaTeX source from which it was generated:
\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.
<!
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 adisplay="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 .