About the tutorial
This tutorial explains how to write tutorials or courses that have math expressions, or scientific notations as often seen in scientific publications.
This tutorial is for learners at level of University Year 2
Most learning contents in science contain mathematical expressions and scientific notations. Hence, it is important to know how to write tutorials with math-rich content.
KBIES that empowers TrustOpen Education accept most LaTeX, MathML, and AsciiMath commands/notations for typesetting math expressions, thanks to MathJax JavaScript library. In this tutorial, our focus will be on how to use LaTeX environments and commands to present mathematical expressions.
Math expressions may appear inline within other learning content, or displayed on a line or paragraph alone, so-called display expressions.
To present an inline math expression, encolse the expression in a \\( ... \\) environment, as shown in the following example:
The sum of all integers from 1 to n can be written as \(\sum_{i=1}^n i \), which is equal to
\({{n+1}\over{2}} n\), or \( {{n^2+n}\over 2} \) by simplification.
which will be rendered as
To present a display math expression, enclose the expression within \$\$ ... \$\$ , which is called a \(LaTeX\) environment. The entire expression above can be presented as a display expression as follows:
$$\sum_{i=1}^n i = {{n+1}\over 2}\times{n} = {{n^2+n}\over 2} $$
which will be rendered as
\[\sum_{i=1}^n i = {{n+1}\over 2}\times{n} = {{n^2+n}\over 2} \]
which will be rendered the same as
Please note the differences between inline math expressions and display math expressions:
$$
x_1+{1\over x_3+
{1\over x_5+
{1 \over x_7 +
{1 \over x_9}}}}\label{name11}
$$
which will be rendered as
\[
x_1+{1\over x_3+
{1\over x_5+
{1 \over x_7 +
{1 \over x_9}}}}
\]
which will be rendered the same as expression (\ref{name11}) above.
For better readability and control, when presenting more complicated math expressions, especially those taking multiple lines, instead of using \$\$ ... \$\$ or \\[ ... \\] to enclose the expression, you can use either \\begin{equation} ... \\end{equation} or \\begin{equation*} ... \\end{equation*} environments. In \(\LaTeX\) term, the former is called equation environment, and the latter is called starred equation environment.
The only difference between equation environment and equation* environment is that the math expression within equation environment will be numbered, whereas diplayed expression within equation* environment will not.
Here is an example using equation environment within which the expression will be numbered:
\begin{equation}
x+1\over\sqrt{1-x^2}
\end{equation}
which will be rendered as
\begin{equation*}
x+1\over\sqrt{1-x^2}
\end{equation*}
which will be rendered as
From the examples above we have seen expressions with a number attached. By default, all display expressions, except those within a stared environment such as \\begin{equation*} ... \\end{equation*} environment, will be automatically numbered. If you really don't want a display expression to be numbered, you need explicitly tell that by adding a \nonumber command to the expression, as shown below:
$$\nonumber\sum_{i=0}^n i = {{n^2}\over 2} $$
which will be rendered without numbering, as shown below:
Why do we want a display expression numbered? The main reason is that we need to refer to the math expression within the context so that it is easy to tell which math expression, formula or equation we are talking about. Because numbers for numbered math expressions are automatically assigned by the system in order, and the number previously assigned to an expression will change if a new numbered expression is added before it within the same document, it is necesssary to give a unique name to each numbered display math expression so that the number attached to a math expression can be reliably and correctly retrieved with the unique name. In \(\LaTeX\) this is done by adding a \\label{ref_name} command to the expression, as shown in the following code:
$$x+1\over\sqrt{1-x^2}\label{num1}$$
will produce a numbered equation as
$$x+1\over\sqrt{1-x^2}\label{num1}$$
which can be referred as expression No.\ref{num1}
There are two ways to refer a display math expression. One is to use \\ref{...} defined in the LaTeX kernel, whereas the other is to use \\eqref{...} which is defined in the amsmath extension. The difference is that the latter will automatically add parentheses around the reference, whereas the former will only display the referred number. If you want the number to be in a parentheses or squared brackets you will have to add that explicitly, as shown in the example below:
Both [\ref{ref2}] and \eqref{ref2} can be used to refer to the same equation
which will be rendered as
You may have noted that a referred math expression may appear after where it is referred. This is called a forward reference. You may safely ignore the jargon. However,you must make sure that the referred label must defined somewhere within the context, and the label must be unique. Otherwise, no right reference number will show up, as shown in the following example.
You may use equation \eqref(ref9) to solve the problem.
which will be rendered as
When an expression runs across multiple-line, most time it is desirable to make the lines aligned as desired somehow. This done by using the \\begin{align}...\\end{align} and & and \cr command, as shown below:
\begin{align}
a&=2\times b\label{ref3}\cr
&=c+d
\end{align}
which will be rendered as
\begin{align*}
a&=2\times b\label{ref3}\cr
&=c+d
\end{align*}
which will be rendered as:
Here is another example:
The area A of a circle with radius R is as follows:
\begin{equation} \label{eq1}
\begin{split}
A & = \frac{\pi r^2}{2} \\
& = \frac{1}{2} \pi r^2
\end{split}
\end{equation}
which will be rendered as:
\begin{align*}
d&=2r & P &=2\pi r & A&=\frac{1}{2}\pi r^2\\
C^2&=A^2+B^2 & c^n&=a^n+b^n & ax^2+bx+c&=0\\
y&=ax+b & y&=ax^2+bx+c & y&=\frac{1}{x}
\end{align*}
which is the result of the code below:
$$ a+3 \over b\times{5} \over c\div{7} $$
because it can be interpreted as {a+3} over b\times{5} \over c\div{7}, or a+3 \over b\times{5} over c\div{7}
To make it clear and avoid ambiguity, we need to use curly braces {...} to group certain items of an expression. In the case above, we can put all the items to be denominator or numerator inside a curly braces as a group, as shown below:
$$ a+3 \over {b\times{5} \over c\div{7}} $$
which will be rendered as
$$ a+3 \over {b\times{5} \over c\div{7}} $$
If you actually want a+3 \over b\times{5} over c\div{7}, the expression should be grouped as follows:
$$ {a+3 \over b\times{5}} \over c\div{7} $$
which will be rendered as
$$ {a+3 \over b\times{5}} \over c\div{7} $$ All examples above are about grouping items within a math expression. If you want to group some math expressions together to display, using the \\begin{gather} ... \\end{gather} or \\begin{gather*} ... \\end{gather*}. The following example shows two version of well-known Euler formula. Formula #\ref{euler0} is the original one while Formula #\ref{euler1} is derived from the first when \(x=\pi\):
\begin{gather}
e^{i\,x} = \cos\,x + i\,\sin\,x\label{euler0} \\
e^{i\pi} + 1= 0\label{euler1}
\end{gather}
which will be rendered as:
Sometimes we want to change spaces between items within a math expression, either adding more spaces or making the items stay tighter. The following are the commands for this purpose.
| Name | \(\LaTeX\) Code |
|---|---|
| Negative space | \! |
| Thinnest space | \, |
| Thin space | \: |
| Medium space | \; |
| 1em space | \quad |
| 2em space | \qquad |
In the examples given in the previous section, spacing command \\, has been used two to add spaces.
In later sections, we will show more examples where grouping is needed.
Mathematical expressions are made of numbers, constants such as \(\pi, i, e\), variables and operators. We will begin by explaining how to present/typeset numbers, constants, and variables in mathematical expressions.
Most people writing learning materials in math and science should befamiliar with the numbers used in math and science expressions. For integers and real numbers, one may use the numbers directly in a form they are already used to, such as \(198 -\!36\,\), \(1.25 +\!3.78\,\). Please note that in the two inline math expressions spacing commands we discussed in previous section are used to adjust the spaces between items of the expression.
For numbers in scientific notation such as \(5.12 \cdot 10^3\), specific \(\LaTeX\) command \\cdot for multiplication, and notation ^ for superscript or exponentiation of math need to be used to make proper rendering. The code for the scientific number in the pagraph above is \\(5.12 \cdot 10^3\\) , for example.
The coding for constant \(\pi\), for \(3.1415926\) is \\(\pi\\), whereas for contants \(e\), for \(2.71828182\) and \(i\), for \(\sqrt{-1}\), you can simply use letter e and i as needed.
Different from variables in computer programs, variables in math and science are single letters such as \(x, y, z\), or single letters with subscripts such as \(x_1, y_i, z_{jk}\).
Other than letters in English or one's native language, Greek letters are commonly used in math expressions. Hence, it is important to know how to present them.LaTeX has a command for each and every Greek letter. The following table shows commands for lowercase Greek letters.
| Command | Letter | Command | letter | Command | letter |
|---|---|---|---|---|---|
| \\alpha | \(\alpha\) | \\beta | \(\beta \) | \\gamma | \(\gamma\) |
| \\delta | \(\delta \) | \\epsilon | \(\epsilon \) | \\varepsilon | \(\varepsilon\) |
| \\zeta | \( \zeta \) | \\eta | \(\eta \) | \\theta | \(\theta\) |
| \\vartheta | \(\vartheta \) | \\iota | \(\iota \) | \\kappa | \(\kappa\) |
| \\lambda | \(\lambda \) | \\mu | \(\mu \) | \\nu | \(\nu\) |
| \\xi | \(\xi \) | \\pi | \(\pi \) | \\varpi | \(\varpi\) |
| \\rho | \(\rho \) | \\varrho | \(\varrho \) | \\sigma | \(\sigma\) |
| \\tau | \(\tau \) | \\upsilon | \(\upsilon \) | \\phi | \(\phi\) |
| \\varphi | \(\varphi \) | \\chi | \(\chi \) | \\psi | \(\psi\) |
| \\omega | \( \omega \) |
The following table is a list of uppercase Greek letters. Please note that some uppercase Greek letters are the same as in English, and some Greek letters are different in lowercase but the same in uppercase.
| Command | Letter | Command | letter | Command | letter |
|---|---|---|---|---|---|
| A | \(A\) | B | \(B \) | \\Gamma | \(\Gamma\) |
| \\Delta | \(\Delta \) | E | \(E \) | E | \(E\) |
| Z | \( Z \) | H | \(H \) | \\Theta | \(\Theta\) |
| \\Theta | \(\Theta \) | I | \(I \) | K | \(K\) |
| \\Lambda | \(\Lambda \) | M | \(M \) | N | \(N\) |
| \\Xi | \(\Xi \) | \\Pi | \(\Pi \) | \\Phi | \(\Phi\) |
| P | \(P \) | P | \(P \) | \\Sigma | \(\Sigma\) |
| \\Tau | \(\tau \) | \\Upsilon | \(\Upsilon \) | \\Phi | \(\Phi\) |
| \\Phi | \(\Phi \) | X | \(X \) | \\Psi | \(\Psi\) |
| \\Omega | \( \Omega \) |
The fundamental math operations include addition, subtraction, mutiplication, and division. We use + for addition and use - for sutraction. For multiplication, there are three ways to present.
\[
A = {\pi\cdot r^2 \over 2} = \frac{1}{2}\cdot\pi\cdot r^2
\]
which will be rendered as
\[
A = {\pi\cdot r^2 \over 2} = \frac{1}{2}\cdot\pi\cdot r^2
\]
$$ a+b/\!c-d\times 5\div 6 $$
which will be rendered as
$$ a+b/\!c-d\times 5\div 6 $$
Here are some examples using \frac{...}{...} command:
$$ \frac{a+b}{c+d} = \frac{b+a}{c+d} $$
$$ \frac{a+b}{c+d} = \frac{b+a}{d+c} $$
which will be rendered as
As can be seen, everything inside first curly braces will be the numerator of the fraction, whereas everything inside the second curly braces will be the denominator of the fraction.
The following are the same fractions presented using {... \over ...} command, where inside the pair of curly braces everything on the left side of \over will be taken as nominator whereas everything on the right will be taken as denominator of the fraction:
$$ {a+b\over c+d} = {b+a\over c+d} $$
$$ {a+b\over c+d} = {b+a\over d+c} $$
which will have the same result:
$$E = mc^2$$
which will be rendered beautifully as
$$E = mc^2$$
and, the famous Fermat's Last Theorem can be presented with
$$ a^n + b^n = c^n $$
which will be rendered as
$$ a^n + b^n = c^n $$
As a special case when \( n=2 \), the equation $$ a^2 + b^2 = c^2 $$ defines all Pythagorean triples \( (a, b, c) \), where \( a, b, c\) are all positive integers such that \( a^2 + b^2 = c^2 \).
| Name | Rendered | \(\LaTeX\) Code | Used for |
|---|---|---|---|
| Multiplication dot | \(\cdot\) | \cdot | Multiplication |
| Three baseline dots | \(\ldots\) | \ldots | General inline ellipsis |
| Three centered dots | \(\cdots\) | \cdots | Ellipsis in horizontal direction |
| Three vertical dots | \(\vdots\) | \vdots | Ellipsis in vertical direction |
| Three diagonal dots | \(\ddots\) | \ddots | Ellipsis in diagonal direction |
Rather than being used as functions, in form of f(x), many commonly used math functions are used as operator in form of \(fx\) in mathematical expressions. The following table shows most of these functions used as operators in mathematical expressions.
| \(\LaTeX\) code | Rendered |
|---|---|
| \sin x | \(\sin x\) |
| \cos x | \(\cos x\) |
| \arcsin x | \(\arcsin x\) |
| \arccos x | \(\arccos x\) |
| \sinh x | \(\sinh x\) |
| \cosh x | \(\cosh x\) |
| \tan x | \(\tan x\) |
| \arctan x | \(\arctan x\) |
| \log x | \(\log x\) |
| \ln x | \(\ln x\) |
| \max x | \(\max x\) |
| \min x | \(\min x\) |
| \sup x | \(\sup x\) |
| \inf x | \(\inf x\) |
| \tanh x | \(\tanh x\) |
| \det x | \(\det x\) |
| \csc x | \(\csc x\) |
| \sec x | \(\sec x\) |
| \cot x | \(\cot x\) |
| Rendered | \(\LaTeX\) Code | Operation |
|---|---|---|
| \(\displaystyle \int_{a}^{b}\) | \int_{a}^{b} | Integration |
| \(\displaystyle \sum_{k=0}^{n}\) | \sum_{k=0}^{n} | Summation of sequence |
| \(\displaystyle \prod_{k=0}^{n}\) | \prod_{k=0}^{n} | Multiplication of sequence |
| \(\displaystyle \lim_{x \to 0}\) | \lim_{x \to 0} | Limit of sequence |
As mentioned in previous subsection, a simple integral operation can be presented using the \\int_{lower}^{upper} command. For single variable integral, the command will be \\int_{lower}^{upper} f(x)dx . The following is a single variable integral expression:
\[ \int_{a}^{b} x^2 \,dx \]For two-variable integrals, use \\int_{lower}^{upper}\\int_{lower}^{upper} f(x,y) dxdy command, as shown in the following example:
\[ \int_{a}^{b}\int_{c}^{d} \mu{x,y} \,dx\,dy \]
which will be rendered as
\[ \int_{a}^{b}\int_{c}^{d} \mu(x,y) \,dx\,dy \]
Similarly, multiple-variable integrals can be presented, if lower-bounds and upper-bounds are known for all varialbles. Otherwise, The following shortcuts for mutiple-variable integrals \\iinit_V, \\iiinit_V, \\iiiinit_V ... \\idotsint_V need to be used, as shown in the examples below:
\[\iint_V \mu(u,v) \,du\,dv \]
\[\iiint_V \mu(u,v,w) \,du\,dv\,dw \]
\[\iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw \]
\[\idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k \]
which will rendered respectively as:
\[\iint_V \mu(u,v) \,du\,dv \]
\[\iiint_V \mu(u,v,w) \,du\,dv\,dw \]
\[\iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw \]
\[\idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k \]
| Rendered | \(\LaTeX\) Code | Meaning |
|---|---|---|
| \(a \bmod b\) | a \bmod b | The remainder of integer a divided by integer b |
| \(a \equiv b \pmod{m}\) | a \equiv b \pmod{m} | integer \(a\) and \(b\) have the same remainder when divided by the same \(m\) |
| Rendered | \(\LaTeX\) Code |
|---|---|
| \(\lvert x \rvert\) | \lvert x \rvert |
| \(\lVert x \rVert\) | \lVert x \rVert |
| Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code |
|---|---|---|---|---|---|
| \(\uparrow\) | \uparrow | \(\downarrow\) | \downarrow | \(\updownarrow\) | \updownarrow |
| \(\Uparrow\) | \Uparrow | \(\Downarrow\) | \Downarrow | \(\Updownarrow\) | \Updownarrow |
| \(\leftarrow\) | \leftarrow or \gets | \(\rightarrow\) | \rightarrow or \to | \(\leftrightarrow\) | \leftrightarrow |
| \(\Leftarrow\) | \Leftarrow | \(\Rightarrow\) | \Rightarrow | \(\Leftrightarrow\) | \Leftrightarrow |
| \(\mapsto\) | \mapsto | \(\longleftarrow\) | \longleftarrow | \(\longrightarrow\) | \longrightarrow |
| \(\longleftrightarrow\) | \longleftrightarrow | \(\Longleftarrow\) | \Longleftarrow | \(\Longrightarrow\) | \Longrightarrow |
| \(\Longleftrightarrow\) | \Longleftrightarrow | \(\longmapsto\) | \longmapsto |
| Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code |
|---|---|---|---|---|---|
| \(\ne\) | \ne | \(\le\) | \le | \(\ge\) | \ge |
| \(\equiv\) | \equiv | \(\ll\) | \ll | \(\gg\) | \gg |
| \(\doteq\) | \doteq | \(\sim\) | \sim | \(\simeq\) | \simeq |
| \(\subset\) | \subset | \(\supset\) | \supset | \(\approx\) | \approx |
| \(\subseteq\) | \subseteq | \(\supseteq\) | \supseteq | \(\cong\) | \cong |
| \(\in\) | \in | \(\ni\) | \ni | \(\propto\) | \propto |
| \(\mid\) | \mid | \(\parallel\) | \parallel | \(\perp\) | \perp |
| Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code |
|---|---|---|---|---|---|
| \(\pm\) | \pm | \(\mp\) | \mp | \(\cdot\) | \cdot |
| \(\div\) | \div | \(\times\) | \times | \(\setminus\) | \setminus |
| \(\star\) | \star | \(\cup\) | \cup | \(\cap\) | \cap |
| \(\ast\) | \ast | \(\circ\) | \circ | \(\bullet\) | \bullet |
| \(\oplus\) | \oplus | \(\ominus\) | \ominus | \(\odot\) | \odot |
| \(\oslash\) | \oslash | \(\otimes\) | \otimes | \(\smallsetminus\) | \smallsetminus |
| Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code | Rendered | \(\LaTeX\) Code |
|---|---|---|---|---|---|
| \(\lor\) | \lor | \(\land\) | \land | \(\neg\) | \neg |
| \(\exists\) | \exists | \(\nexists\) | \nexists | \(\forall\) | \forall |
| \(\implies\) | \implies | \(\iff\) | \iff | \(\models\) | \models |
| Symbol | Rendered | \(\LaTeX\) Code |
|---|---|---|
| Infinity | \(\infty\) | \infty |
| Partial derivative | \(\partial\) | \partial |
| Empty set | \(\emptyset\) | \emptyset |
| Nabla | \(\nabla\) | \nabla |
| Angle brackets | \(\langle x \rangle\) | \langle x \rangle |
Array is a type of structure for organizing data. An array can be 1 dimension or n-dimension. To present array with \(\LaTeX\), using the array environment., as shown in the following example, where \\\\ is used to separate rows, and & is used to separate elements of each row. To produce large delimiters around the array, use \left and \right followed by the desired delimiter.:
\[\left(
\begin{array}{lcr}
a & b & c \\
1 & 2 & 3 \\
Apple & Pear & Orange
\end{array}
\right)\]
\[\left(
\begin{array}{lcr}
a & b & c \\
1 & 2 & 3 \\
Apple & Pear & Orange
\end{array}
\right)\]
Of the lcr in the code above, each letter tells how elements in the corresponding column should be aligned in accordance with the following
| l | left aligned text |
| c | centered text |
| r | right aligned text |
In math and science it is often needed to enumerate cases when presenting expressions and formulas, such as a mathematical function. In \(\LaTeX\) this is done by using the cases environment, as hown in the following example, where \\\\ is used to separate different cases, and & is used for correct alignment.
\[f(x) =
\begin{cases}
\sqrt{x} & \text{if } x > 0 \\
0 & \text{if } x = 0 \\
x^2+5 & \text{if } x < 0
\end{cases}\]
which will be rendered as follows:
In math and science, matrices are commonly used, and a matrix may be presented with different delimiter. In \LaTeX, this is done by using one of the following environments.
| matrix | No delimiter |
| pmatrix | \((...)\) as delimiter |
| bmatrix | \([...]\) as delimiter |
| Bmatrix | \(\{...\}\) as delimiter |
| vmatrix | \(\lvert ... \lvert\) as delimiter |
| Vmatrix | \(\lVert ... \lVert\) delimiter |
Use \\\\ to separate different rows, and use & to separate elements of each row.
\[\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
\end{bmatrix}\]
In mathematics, vectors are special type of matrices where it has only one row, or one column, as shown in the example below:
\[\begin{bmatrix}
1 & 2 & 3 & 4 & 5 & 6
\end{bmatrix}\]
which is rendered as a vector in a row.