Senior Syntax

Tables

Use $$\begin{array}{column_styles}...\end{array}$$ to create tables.

The column styles can be c, l, or r, which stand for center, left, and right alignment, respectively. You can also use | to add a vertical line.

Rows in the table are separated by \\, and columns by &.

Use \hline to insert a horizontal line before the current row.

Example:

$$
\begin{array}{c|lcr}
n & \text{Left} & \text{Center} & \text{Right} \\
\hline
1 & 0.24 & 1 & 125 \\
2 & -1 & 189 & -8 \\
3 & -20 & 2000 1+10i \\
\end{array}
$$

Output:
$\begin{array}{clcr} n & Left & Center & Right \\ 1 & 0.24 & 1 & 125 \\ 2 & - 1 & 189 & - 8 \\ 3 & - 20 & 2000 & 1 + 10 i \end{array}$

Matrices

Basic usage
Use $$\begin{matrix}...\end{matrix}$$ to create matrices. Separate rows with \\ and columns with &.
- Example:
```
$$
\begin{matrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{matrix}
$$
```
- Output:
  $\begin{matrix} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{matrix}$

Adding brackets

To add brackets around matrices, use \left and \right.

You can also use special matrix types like pmatrix, bmatrix, Bmatrix, vmatrix, or Vmatrix.

Example:

$$
\begin{pmatrix}
1 & 2 \\
3 & 4 \\
\end{pmatrix}
$$

$$
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
\end{bmatrix}
$$

$$
\begin{Bmatrix}
1 & 2 \\
3 & 4 \\
\end{Bmatrix}
$$

$$
\begin{vmatrix}
1 & 2 \\
3 & 4 \\
\end{vmatrix}
$$

$$
\begin{Vmatrix}
1 & 2 \\
3 & 4 \\
\end{Vmatrix}
$$

Output:
$(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix})$ $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$ ${\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}}$ $| \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} |$ $‖ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} ‖$

Omitted Elements

You can use \cdots, \ddots, and \vdots to omit elements in a matrix.

Example:

$$
\begin{matrix}
1 & a_1 & a_1^2 & \cdots & a_1^n \\
1 & a_2 & a_2^2 & \cdots & a_2^n \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & a_m & a_m^2 & \cdots & a_m^n \\
\end{matrix}
$$

Output:
$\begin{matrix} 1 & a_{1} & a_{1}^{2} & \dots & a_{1}^{n} \\ 1 & a_{2} & a_{2}^{2} & \dots & a_{2}^{n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & a_{m} & a_{m}^{2} & \dots & a_{m}^{n} \end{matrix}$

Augmented Matrices

Augmented matrices are created using the array environment.

Example:

$$
\left[
  \begin{array}{cc|c}
  1 & 2 & 3 \\
  4 & 5 & 6 \\
  \end{array}
\right]
$$

Output:
$[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}]$

Aligned Equations

Sometimes, you may want to align equations, particularly at the equals sign. This can be done using the \begin{align}...\end{align} format, where & is used to specify the alignment.

Example:

$$
\begin{align}
\sqrt{37}&=\sqrt{\frac{73^2-1}{12^2}} \\
&=\sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\
&=\sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\
&=\frac{73}{12}\sqrt{1-\frac{1}{73^2}} \\
&\approx\frac{73}{12}\left(1-\frac{1}{2\cdot73^2}\right)
\end{align}
$$

Output:
$\begin{aligned} \sqrt{37} & = \sqrt{\frac{73^{2} - 1}{12^{2}}} \\ = \sqrt{\frac{73^{2}}{12^{2}} \cdot \frac{73^{2} - 1}{73^{2}}} \\ = \sqrt{\frac{73^{2}}{12^{2}}} \sqrt{\frac{73^{2} - 1}{73^{2}}} \\ = \frac{73}{12} \sqrt{1 - \frac{1}{73^{2}}} \\ \approx \frac{73}{12} (1 - \frac{1}{2 \cdot 73^{2}}) \end{aligned}$

Piecewise Expressions

When defining a function with multiple cases, use \begin{cases}...\end{cases}. Rows are separated by \\ and & indicates alignment.

Example:

$$
f(n)=
\begin{cases}
n/2,&\text{if $n$ is even} \\
3n+1,&\text{if $n$ is odd}
\end{cases}
$$

Output:
$f (n) = {\begin{cases} n / 2, & if n is even \\ 3 n + 1, & if n is odd \end{cases}$

To increase vertical spacing between cases, add \\[2ex] at the end of a row (3ex, 4ex, etc., can also be used, 1ex being the default).

Output:
$f (n) = {\begin{cases} \frac{n}{2}, & if n is even \\ 3 n + 1, & if n is odd \end{cases}$

Spacing Issues

There are certain issues in $L A T E X$ formulas that don't affect correctness but make the output look unpolished.

Avoid using \frac in exponents or integrals
Using \frac in such expressions reduces clarity. It’s rarely used in professional mathematical typesetting. Instead, / should be used.
- Example:
```
$$
\begin{array}{c|c}
\mathrm{Bad} & \mathrm{Better} \\
\hline \\
e^{i\frac{\pi}2} & e^{i\pi/2} \\
\int^\frac\pi2_{-\frac\pi2}\sin x \,dx & \int^{\pi/2}_{\pi/2}\sin x \,dx
\end{array}
$$
```
- Output:
  $\begin{array}{cc} Bad & Better \\ e^{i \frac{π}{2}} & e^{i π / 2} \\ \int_{- \frac{π}{2}}^{\frac{π}{2}} \sin x d x & \int_{π / 2}^{π / 2} \sin x d x \end{array}$
Use \mid instead of | as a separator
When | is used as a separator, there’s an issue with spacing. Use \mid instead.
- Example:
```
$$
\begin{array}{c|c}
\mathrm{Bad}&\mathrm{Better}\\
\hline\\
\{x|x^2\in\Bbb Z\}&\{x\mid x^2\in\Bbb Z\}
\end{array}
$$
```
- Output:
  $\begin{array}{cc} Bad & Better \\ {x | x^{2} \in Z} & {x ∣ x^{2} \in Z} \end{array}$
Multiple integrals
For multiple integrals, avoid using \int\int. Instead, use specific commands like \iint, \iiint, etc.
- Example:
```
$$
\begin{array}{c|c}
\mathrm{Bad}&\mathrm{Better}\\
\hline\\
\int\int_S f(x)\,dy\,dx&\iint_S f(x)\,dy\,dx\\
\int\int\int_V f(x)\,dz\,dy\,dx&\iiint_V f(x)\,dz\,dy\,dx
\end{array}
$$
```
- Output:
  $\begin{array}{cc} Bad & Better \\ \int \int_{S} f (x) d y d x & \iint_{S} f (x) d y d x \\ \int \int \int_{V} f (x) d z d y d x & ∭_{V} f (x) d z d y d x \end{array}$

In differential expressions, use \, to add a small amount of space; otherwise, $L A T E X$ will tightly arrange the terms.

Example:

$$
\begin{array}{c|c}
\mathrm{Bad}&\mathrm{Better}\\
\hline\\
\iiint_V f(x)dz dy dx&\iiint_V f(x)\,dz\,dy\,dx
\end{array}
$$

Output:
$\begin{array}{cc} Bad & Better \\ ∭_{V} f (x) d z d y d x & ∭_{V} f (x) d z d y d x \end{array}$

Continued Fractions

When writing continued fractions, use \cfrac instead of \frac or \over.

Example:

$$
x=a_0+\cfrac{1^2}{a_1
+\cfrac{2^2}{a_2
+\cfrac{3^2}{a^3
+\cfrac{4^2}{a_4+\cdots}}}}
$$

Output:
$x = a_{0} + \frac{1^{2}}{a_{1} + \frac{2^{2}}{a_{2} + \frac{3^{2}}{a^{3} + \frac{4^{2}}{a_{4} + \dots}}}}$

Systems of Equations

Use \begin{array}...\end{array} and \left\{...\right. to represent systems of equations.

Example:

$$
\left\{
\begin{array}{c}
a_1x+b_1y+c_1z=d_1 \\
a_2x+b_2y+c_2z=d_2 \\
a_3x+b_3y+c_3z=d_3
\end{array}
\right.
$$

Output:
${\begin{matrix} a_{1} x + b_{1} y + c_{1} z = d_{1} \\ a_{2} x + b_{2} y + c_{2} z = d_{2} \\ a_{3} x + b_{3} y + c_{3} z = d_{3} \end{matrix}$

Alternatively, you can use \begin{cases}...\end{cases} for the same system of equations:

$$
\begin{cases}
a_1x+b_1y+c_1z=d_1 \\
a_2x+b_2y+c_2z=d_2 \\
a_3x+b_3y+c_3z=d_3
\end{cases}
$$

To align equals signs in equations, use \begin{aligned}...\end{aligned}.

Example:

$$
\left\{
\begin{aligned}{c}
a_1x+b_1y+c_1z=d_1 \\
a_2x+b_2y=d_2 \\
a_3x+b_3y+c_3z=d_3
\end{aligned}
\right.
$$

Output:
${\begin{array}{r} c a_{1} x + b_{1} y + c_{1} z = d_{1} \\ a_{2} x + b_{2} y = d_{2} \\ a_{3} x + b_{3} y + c_{3} z = d_{3} \end{array}$

If you need to align equations and terms, use \begin{array}{column_styles}...\end{array}.

Example:

$$
\left\{
\begin{array}{ll}
a_1x+b_1y+c_1z&=d_1 \\
a_2x+b_2y&=d_2 \\
a_3x+b_3y+c_3z&=d_3
\end{array}
\right.
$$

Output:
${\begin{cases} a_{1} x + b_{1} y + c_{1} z & = d_{1} \\ a_{2} x + b_{2} y & = d_{2} \\ a_{3} x + b_{3} y + c_{3} z & = d_{3} \end{cases}$

Additional Decorations

\overline{A}\;\overline{AA}\;\overline{AAA}：
$\overset{―}{A} \overset{―}{A A} \overset{―}{A A A}$
\underline{B}\;\underline{BB}\;\underline{BBB}：
$\underset{―}{B} \underset{―}{B B} \underset{―}{B B B}$
\widetilde{C}\;\widetilde{CC}\;\widetilde{CCC}：
$\tilde{C} \tilde{C C} \tilde{C C C}$
\widehat{D}\;\widehat{DD}\;\widehat{DDD}：
$\hat{D} \hat{D D} \hat{D D D}$
\fbox{E}\;\fbox{EE}\;\fbox{EEE}：
$E EE EEE$
\underleftarrow{F}\;\underleftarrow{FF}\;\underleftarrow{FFF}：
$\underset{\leftarrow}{F} \underset{\leftarrow}{F F} \underset{\leftarrow}{F F F}$
\underrightarrow{G}\;\underrightarrow{GG}\;\underrightarrow{GGG}：
$\underset{\to}{G} \underset{\to}{G G} \underset{\to}{G G G}$
\underleftrightarrow{H}\;\underleftrightarrow{HH}\;\underleftrightarrow{HHH}：
$\underset{\leftrightarrow}{H} \underset{\leftrightarrow}{H H} \underset{\leftrightarrow}{H H H}$
\overbrace{(n-2)+\overbrace{(n_1)+n+(n+1)}+(n+2)}：
$\overset{⏞}{(n - 2) + \overset{⏞}{(n_{1}) + n + (n + 1)} + (n + 2)}$
\underbrace{(n-2)+\underbrace{(n_1)+n+(n+1)}+(n+2)}：
$\underset{⏟}{(n - 2) + \underset{⏟}{(n_{1}) + n + (n + 1)} + (n + 2)}$
\overbrace and \underbrace can be annotated with superscripts and subscripts, such as \underbrace{a\cdot a\cdots a}_{b_\text{times}}:
$\underset{b_{times}}{\underset{⏟}{a \cdot a \dots a}}$
Phonetic symbols, like \check{I}: $\overset{ˇ}{I}$ , \acute{J}: $\overset{´}{J}$ , \grave{K}: $\overset{`}{K}$ .

Commutative Diagrams

Use \begin{CD}...\end{CD} to represent commutative diagrams.

Example:

$$
\begin{CD}
A@>a>>B \\
@VbVV=@VVcV \\
c@>>d>D
\end{CD}
$$

Output:
$\begin{matrix} A & \overset{a}{\to} & B \\ b ↓ & = & ↓ c \\ c & \to_{d}^{} & D \end{matrix}$

In the diagram, @>>> represents a right arrow, @<<< represents a left arrow, @AAA represents an upward arrow, @VVV represents a downward arrow, @= represents a horizontal double line, @| represents a vertical double line, and @. represents no arrow.

Example:

$$
\begin{CD}
A@>>>B@>{\text{very long label}}>>C \\
@.@AAA@|\\
D@=E@<<<F
\end{CD}
$$

Output:
$\begin{matrix} A & \overset{}{\to} & B & \overset{very long label}{\to} & C \\ ↑ & ∥ \\ D & = & E & \overset{}{\leftarrow} & F \end{matrix}$

Colors

There are two ways to color text.

\textcolor{color-name}{text}:
Here, color-name is the system-defined color, and text is the content to be colored.
For example, \textcolor{hotpink}{E=mc^2}: $E = m c^{2}$ .
\textcolor[rgb]{r,g,b}{text} 或 \textcolor[RGB]{R,G,B}{text}:
In rgb, values range from 0 to 1, and in RGB, values range from 0 to 255.
For example, \textcolor[rgb]{0.5,0.8,0.7}{E=mc^2}: $E = m c^{2}$ , or \textcolor[RGB]{202,12,22}{E=mc^2}: $E = m c^{2}$ .

You can also define custom colors using \definecolor{ColorName}{rgb}{r,g,b}.

Example:

$$
\definecolor{mycolor}{rgb}{0.1,0.5,0.8}
\textcolor{mycolor}{E=mc^2}
$$

Output:
$E = m c^{2}$

Highlighting Equations

Use \bbox to highlight an equation.

Example:

\bbox[blue]{\textcolor{white}{e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n\qquad(1)}}

Output:
$e^{x} = lim_{n \to \infty} {(1 + \frac{x}{n})}^{n} (1)$

You can also add padding (similar to padding in CSS) by specifying numerical values.

Example:

\bbox[blue,10px]{\textcolor{white}{e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n\qquad(1)}}

Output:
$e^{x} = lim_{n \to \infty} {(1 + \frac{x}{n})}^{n} (1)$

Borders can also be added.

Example:

\bbox[10px,border:2px solid red]{{e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n\qquad(1)}}

Output:
$e^{x} = lim_{n \to \infty} {(1 + \frac{x}{n})}^{n} (1)$

Additionally, you can set both background color and border at the same time.

Example:

\bbox[10px,hotpink,border:2px dashed blue]{{e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n\qquad(1)}}

Output:
$e^{x} = lim_{n \to \infty} {(1 + \frac{x}{n})}^{n} (1)$

Senior Syntax ​

Tables ​

Matrices ​

Aligned Equations ​

Piecewise Expressions ​

Spacing Issues ​

Continued Fractions ​

Systems of Equations ​

Additional Decorations ​

Commutative Diagrams ​

Colors ​

Highlighting Equations ​

Senior Syntax

Tables

Matrices

Aligned Equations

Piecewise Expressions

Spacing Issues

Continued Fractions

Systems of Equations

Additional Decorations

Commutative Diagrams

Colors

Highlighting Equations