$\frac{1}{2}$

\frac{1}{2}

\frac comes from "fraction". The first wave brackets is for the numerator and the second is for the denominator.

$\displaystyle \frac{1}{2}$

\displaystyle \frac{1}{2}

If you add "\displaystyle", it will be displayed larger.

$\dfrac{1}{2}$

\dfrac{1}{2}

\dfrac means \frac in \displaystyle.

$\require{physics} \flatfrac{1}{2}$

\require{physics} \flatfrac{1}{2}

\flatfrac in the physics extension allows you to write fractions on a line. You can also use the sign / to write "1/2".

$\left( -\frac{1}{2} \right)^2$

\left( -\frac{1}{2} \right)^2

To keep the fractions in parentheses, prefix the brackets with \left and \right.

$\require{physics} \qty( -\frac{1}{2} )^2$

\require{physics} \qty( -\frac{1}{2} )^2

\qty in the physics extension allows you to write parenthesized fractions a little simpler. \qty comes from physical quantity.

$\frac{a+b}{c+\frac{d}{e}}$

\frac{a+b}{c+\frac{d}{e}}

Fractions can be nested.

$\cfrac{a+b}{c+\cfrac{d}{e}}$

\cfrac{a+b}{c+\cfrac{d}{e}}

With \cfrac, the size of the fraction will be the same. \cfrac comes from continued fraction. Using \dfrac would produce a similar display.

$0.123$

0.123

A period is used for the decimal point.

$\frac{1}{11} = 0.\dot{0}\dot{9}$

\frac{1}{11} = 0.\dot{0}\dot{9}

To add dot on top of a number, use \dot.

$\pi = 3.14 \ldots$

\pi = 3.14 \ldots

Here's a sample to add three points to the bottom.

$\sqrt{2} = 1.4142 \cdots$

\sqrt{2} = 1.4142 \cdots

Here's a sample to add three points in the middle.

$\infty$

\infty

\infty comes from "infinity".

$|x|$

|x|

Absolute values can be represented by a vertical line symbol.

$\vert x \vert$

\vert x \vert

A vertical line used for absolute values can also be represented by vert, which comes from vertical line.

$\left| \dfrac{x}{2} \right|$

\left| \dfrac{x}{2} \right|

If you add \left and \right before the vertical lines, the vertical lines will be longer to match the size of the fraction.

$\require{physics} \qty|\dfrac{x}{2}|$

\require{physics} \qty|\dfrac{x}{2}|

\qty in the physics extension allows you to write fractions with absolute value symbols a little simpler. \qty comes from physical quantity.

$\require{physics} \abs{ \dfrac{x}{2} }$

\require{physics} \abs{ \dfrac{x}{2} }

\abs in the physics extension allows you to write fractions with absolute value symbols. \abs comes from "absolute value".

$[x]$

[x]

The Gauss brackets can be represented by a square bracket symbol.

$\lbrack x \rbrack$

\lbrack x \rbrack

The Gauss brackets can also be represented with \lbrack and \rblack. They come from the left bracket and the right bracket.

$\lfloor x \rfloor$

\lfloor x \rfloor

They come from "floor function".

$\lceil x \rceil$

\lceil x \rceil

They come from "ceiling function".

$1 + 2$

1 + 2

We use the symbols as it is.

$3 - 1$

3 - 1

We use the symbols as it is.

$2 \times 3$

2 \times 3

It comes from reading "2 x 3" as "2 times 3".

$6 \divisionsymbol 3$

6 \div 3

\div comes from "divide". If you are using the physics extension, the div will be overwritten. So you must duplicate it before it is overwritten.

$\pm 1$

\pm 1

\pm comes from "plus and minus".

$\mp 1$

\mp 1

\mp comes from "minus and plus".

$a \cdot b = ab$

a \cdot b = ab

\cdot comes from "center dot".

$a \divisionsymbol b = \frac{a}{b}$

a \div b = \frac{a}{b}

This is the relationship between division and fractions.

$a \equiv b \mod n$

a \equiv b \mod n

\mod and \equiv come from modular arithmetic and equivalence. Three lines are for describe equivalence.

$a \equiv b \pmod n$

a \equiv b \pmod n

\pmod means "mod with parentheses".

$x \propto y$

x \propto y

\propto comes from "proportional to".

$a \gt b$

a \gt b

In MathJax, we use \gt because > has a special meaning on the web. It comes from "greater than".

$a \geq b$

a \geq b

\geq is a combination of "greater than" and "equal".

$a \geqq b$

a \geqq b

If you use \geqq, you will have one more horizontal line than with \geq.

$a \lt b$

a \lt b

In MathJax, we use \lt because < has a special meaning on the web. It comes from "less than".

$a \leq b$

a \leq b

\leq is a combination of "less than" and "equal".

$a \leqq b$

a \leqq b

If you use \leqq, you will have one more horizontal line than with \leq.

$a = b$

a = b

We use the symbols as it is.

$a \neq b$

a \neq b

\neq comes from "not equal". It can also be written as \ne or \not=.

$a \fallingdotseq b$

a \fallingdotseq b

\fallingdotseq is a combination of "falling dots" and "equal".

$a \sim b$

a \sim b

\sim comes from "similar".

$a \simeq b$

a \simeq b

It is a combination of "similar" and "equal". If you write \eqsim, the symbols above and below will be swapped

$a \approx b$

a \approx b

\approx comes from "approximately".

$a \gg b$

a \gg b

If you write "\gg", two > overlap.. If you write "\ggg", three > overlap.

$a \ll b$

a \ll b

If you write "\ll", two < overlap.. If you write "\lll", three < overlap.

$\max f(x)$

\max f(x)

$\min f(x)$

\min f(x)

$\begin{eqnarray} aaa \\ bbb \end{eqnarray}$

\begin{eqnarray}
aaa \\
bbb
\end{eqnarray}

To display a multi-line expression, we use the eqnarray environment. A line break is not reflected in the expression. To start a new line, we put "\\".

$\begin{eqnarray} aaa \\[5pt] bbb \end{eqnarray}$

\begin{eqnarray}
aaa \\[5pt]
bbb
\end{eqnarray}

You can change the size of a line break by using square brackets after "\\", e.g. \\[5pt].

$\begin{eqnarray} x + 2x &=& 3 \\ x &=& 1 \end{eqnarray}$

\begin{eqnarray}
x + 2x &=& 3 \\
x &=& 1
\end{eqnarray}

You can use "&" to align the position.

$x \in A$

x \in A

\in comes from "x is in A".

$A \ni x$

A \ni x

Reversing \in to \ni changes the direction of the symbol.

$x \notin A$

x \notin A

\notin comes from "x is not in A".

$A \subset B$

A \subset B

It shows A is a subset of B.

$A \subseteq B$

A \subseteq B

\subseteq means "subset or equal".

$A \subseteqq B$

A \subseteqq B

If you repeat q, the two lines are shown on the bottom.

$A \supset B$

A \supset B

This shows A is a superset of B.

$A \supseteq B$

A \supseteq B

\supseteq means "superset or equal".

$A \supseteqq B$

A \supseteqq B

If you repeat q, the two lines are shown on the bottom.

$A \not \subset B$

A \not \subset B

If you combine \not and \subset, the slashed line will be added.

$A \subsetneqq B$

A \subsetneqq B

This shows A is a proper subset of B. If you use \supsetneqq, that shows A is a proper superset of B.

$A \cap B$

A \cap B

$A \cup B$

A \cup B

$\varnothing$

\varnothing

The symbol is for an empty set, which comes from "nothing". It is similar to but different from the Greek letter phi.

$\emptyset$

\emptyset

The symbol is for an empty set, which comes from "empty set". It is similar to but different from the Greek letter phi.

$A^c$

A^c

"c" comes from "complement set".

$\overline{ A }$

\overline{ A }

You can also draw a line over the set to represent the complementary set.

$A \setminus B$

A \setminus B

\setminus means a difference set. It is similar to backslash, but differs that \setminus contains a space before and after it.

$A \triangle B$

A \triangle B

The symmetric difference is represented by a triangle.

$\mathbb{ N }$

\mathbb{ N }

This is the blackboard bold.

$\mathbb{ Z }$

\mathbb{ Z }

$\mathbb{ Q }$

\mathbb{ Q }

$\mathbb{ R }$

\mathbb{ R }

$\mathbb{ C }$

\mathbb{ C }

$\mathbb{ H }$

\mathbb{ H }

$\sup A$

\sup A

$\inf A$

\inf A

$\aleph$

\aleph

It is used to express the cardinality of an infinite set.

$P \implies Q$

P \implies Q

$P \Rightarrow Q$

P \Rightarrow Q

The right arrow is sometimes used to indicate "implication".

$P \to Q$

P \to Q

The single line arrow may be used to indicate "implication".

$P \Leftarrow Q$

P \Leftarrow  Q

$P \gets Q$

P \gets  Q

A single arrow to the left.

$P \iff Q$

P \iff Q

\iff means "if and only if".

$P \Leftrightarrow Q$

P \Leftrightarrow Q

The left right double arrow may be used to indicate "equivalence".

$P \leftrightarrow Q$

P \leftrightarrow Q

The left right arrow may be used to indicate "equivalence".

$P \equiv Q$

P \equiv Q

\equiv comes from "equivalence".

$\therefore$

\therefore

$\because$

\because

$\forall x$

\forall x

This is Turned A.

$\exists x$

\exists x

This is Turned E.

$\nexists$

\nexists

\nexists come from "not" and "exists".

$P \land Q$

P \land Q

\land comes from "And in logic".

$P \lor Q$

P \lor Q

\lor comes from "Or in logic".

$\lnot P$

\lnot P

\lnot comes from "Not in logic".

$\overline{ P }$

\overline{ P }

Another way to show negation is to draw a line over the letter.

$!P$

!P

You can also write an "!" in front of the letter to indicate negation.

$P \oplus Q$

P \oplus Q

A + ("plus") sign in a circle ("O") indicates an exclusive disjunction.

$P \veebar Q$

P \veebar Q

This symbol is a combination of the letters V ("vee") and horizontal line ("bar"), and is sometimes used to represent an exclusive disjunction.

$\top$

\top

This is used to show tautology. There's a horizontal line on top.

$\bot$

\bot

This is used to show contradiction. There's a horizontal line on bottom.

$P \vdash Q$

P \vdash Q

\vdash comes from "vertical line" and "dash".

$P \models Q$

P \models Q

The same notation applies when using \vDash instead of \models.

${}_n \mathrm{ P }_k$

{}_n \mathrm{ P }_k

To write a small letter in the lower left corner, use "{}_". "P" is Roman type.

${}_n \mathrm{ C }_k$

{}_n \mathrm{ C }_k

$n!$

n!

$\binom{ n }{ k }$

\binom{ n }{ k }

\binom comes from "binomial coefficient".係数）に由来しています。

${ n \choose k }$

{ n \choose k }

\choose is used to choose k from n. The braces are necessary to separate it from the preceding and following characters.

$\dbinom{ n }{ k }$

\dbinom{ n }{ k }

This is \binom in displaystyle.

${}_n \mathrm{ H }_k$

{}_n \mathrm{ H }_k

$\sum_{i=1}^{n} a_n$

\sum_{i=1}^{n} a_n

\sum comes from sum. To write expressions below and above the sigma, use "_" and "^".

$\displaystyle \sum_{i=1}^n a_n$

\displaystyle \sum_{i=1}^n a_n

If you use displaystyle, the sigma will be larger. The formula will be placed above and below the sigma.

$\prod_{ i = 0 }^n x_i$

\prod_{ i = 0 }^n x_i

\prod comes from product.

$\displaystyle \prod_{i=0}^n x_i$

\displaystyle \prod_{i=0}^n x_i

$\begin{eqnarray} n! = \prod_{ k = 1 }^n k \end{eqnarray}$

\begin{eqnarray}
n! = \prod_{ k = 1 }^n k
\end{eqnarray}

This is a sample using factorial.

$2^3$

2^3

To write a number in the upper right corner, use "^".

$e^{ i \pi }$

e^{ i \pi }

If you want to write multiple numbers or letters in the upper right corner, put them in braces.

$\exp ( x )$

\exp ( x )

$\sqrt{ 2 }$

\sqrt{ 2 }

\sqrt comes from "square root".

$\sqrt{ \mathstrut a } + \sqrt{ \mathstrut b }$

\sqrt{ \mathstrut a } + \sqrt{ \mathstrut b }

Using \mathstrut, you can align the height of the square root.

$\sqrt[ n ]{ x }$

\sqrt[ n ]{ x }

When writing power roots, use brackets.

$\log x$

\log x

$\log_{ 2 } x$

\log_{ 2 } x

The logarithm base is specified using the "_".

$\ln x$

\ln x

$90^{ \circ }$

90^{ \circ }

The small circle in the upper right corner representing degrees can be represented using \circ, which comes from "circle".

$\frac{ \pi }{ 2 }$

\frac{ \pi }{ 2 }

$\angle A$

\angle A

$AB /\!/ CD$

AB /\!/ CD

In Japan, two "/" signs are used to indicate parallelism. "\!" can be used to close the gap.

$AB \parallel CD$

AB \parallel CD

$AB \perp CD$

AB \perp CD

\perp comes from "perpendicular".

$\triangle ABC$

\triangle ABC

$\Box ABCD$

\Box ABCD

You can use \Box to represent a rectangle. It starts with a capital letter.

$\sin x$

\sin x

$\cos x$

\cos x

$\tan x$

\tan x

$\begin{eqnarray} \sin 45^\circ = \frac{ \sqrt{2} }{ 2 } \end{eqnarray}$

\begin{eqnarray}
\sin 45^\circ
 = \frac{ \sqrt{2} }{ 2 }
\end{eqnarray}

$\begin{eqnarray} \cos \frac{ \pi }{ 3 } = \frac{ 1 }{ 2 } \end{eqnarray}$

\begin{eqnarray}
\cos \frac{ \pi }{ 3 }
 = \frac{ 1 }{ 2 }
\end{eqnarray}

$a+bi$

a+bi

The imaginary unit is commonly denoted by "i."

$\oldRe x$

\Re x

It is derived from the term "real part."

$\require{physics} \Re x$

\require{physics} \Re x

When using the "physics" extension, the text is represented in Roman font.

$\oldIm x$

\Im x

It is derived from the term "imaginary part."

$\require{physics} \Im x$

\require{physics} \Im x

When using the "physics" extension, the text is represented in Roman font.

$\bar{z}$

\bar{z}

A line placed above a complex number represents its complex conjugate.

$\arg (z)$

\arg (z)

It is derived from the concept of the argument.

$\omega$

\omega

The Greek letter "omega" is sometimes used to represent the cube root of 1.

$\mathcal{O}$

\mathcal{O}

In some cases, the symbol for Landau notation is represented using the letter "O" in calligraphy fonts.

$\frac{ dy }{ dx }$

\frac{ dy }{ dx }

This represents the derivative of y with respect to x expressed in fractional form.

$\frac{ \mathrm{ d } y }{ \mathrm{ d } x }$

\frac{ \mathrm{ d } y }{ \mathrm{ d } x }

This is the Roman font representation of "d" used in the previous example.

$\frac{ d^n y }{ dx^n }$

\frac{ d^n y }{ dx^n }

This represents the expression of the nth derivative of y with respect to x in fractional form.

$\require{physics} \dv[n]{f}{x}$

\require{physics} \dv[n]{f}{x}

By using the "physics" extension, we can simplify the notation for the nth derivative.

$\left. \frac{dy}{dx} \right|_{x=a}$

\left. \frac{dy}{dx} \right|_{x=a}

A long vertical line drawn on the right side represents the evaluation of the expression at x=a. It is indicated using a subscript notation.

$\require{physics} \eval{\dv{y}{x}}_{x=a}$

\require{physics} \eval{\dv{y}{x}}_{x=a}

By using the "physics" extension, we can simplify the expression for the value of the derivative at x=a.

$f'$

f'

With the use of the apostrophe symbol (') in notation, we can represent the derivative.

$f^{\prime\prime}$

f^{\prime\prime}

If using two consecutive apostrophe symbols ('') does not display correctly, the notation "\prime" can be used instead to represent the derivative.

$f^{ ( n ) }$

f^{ ( n ) }

$Df$

Df

$D_x f$

D_x f

$D^n f$

D^n f

$\dot{y} = \frac{dy}{dt}$

\dot{y} = \frac{dy}{dt}

There is also a method of representing differentiation by placing a dot above a character.

$\ddddot{ y } = \frac{ d^4 y }{ dt^4 }$

\ddddot{ y } = \frac{ d^4 y }{ dt^4 }

Increasing "d" increases the number of dots.

$f_x$

f_x

$f_{ xy }$

f_{ xy }

$\nabla f$

\nabla f

$\Delta f$

\Delta f

$\int_0^1 f(x) dx$

\int_0^1 f(x) dx

The symbol "\int" is derived from the concept of integration. The limits of integration are represented using a subscript and a superscript.

$\displaystyle \int_{-\infty}^{ \infty } f(x) dx$

\displaystyle \int_{-\infty}^{ \infty } f(x) dx

Adding "\displaystyle" causes the expression to be displayed in a larger format. When including multiple symbols within the integral section, they are enclosed in curly brackets.

$\iint_D f(x,y) dxdy$

\iint_D f(x,y) dxdy

By stacking the symbol "i" (integral) multiple times, such as "\iint" for double integral, "\iiint" for triple integral, and "\iiiint" for quadruple integral, we represent iterated integrals.

$\oint_C f(z) dz$

\oint_C f(z) dz

The symbol "oint" represents a contour integral, where the "o" attached to the "int".

$\vec{ a }$

\vec{ a }

The symbol "vector" is derived from the vectors.

$\overrightarrow{ AB }$

\overrightarrow{ AB }

To place an arrow above multiple characters, the notation "overrightarrow" is used (over + right + arrow).

$\boldsymbol{ A }$

\boldsymbol{ A }

Bold symbols, denoted using "boldsymbol," are sometimes used to represent vectors.

$( a_1, a_2, \ldots, a_n )$

( a_1, a_2, \ldots, a_n )

$\left( \begin{array}{c} a_1 \\ a_2 \\ \vdots \\ a_n \end{array} \right)$

\left(
  \begin{array}{c}
    a_1 \\
    a_2 \\
    \vdots \\
    a_n
  \end{array}
\right)

$\| x \|$

\| x \|

Combining the backslash "\" and the vertical bar "|" creates a double vertical line symbol "\\."

$\require{physics} \norm{ \dfrac{1}{2} }$

\require{physics} \norm{ \dfrac{1}{2} }

By using the "physics" extension, the command "\norm" can be used to create double vertical lines, representing the norm. The length of the vertical lines adjusts automatically based on the size of the contents within them.

$\vec{ a } \cdot \vec{ b }$

\vec{ a } \cdot \vec{ b }

$\vec{ a } \times \vec{ b }$

\vec{ a } \times \vec{ b }

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}

If you want to enclose the matrix in parentheses, you can use the "pmatrix" (p from parentheses +matrix). If you use just "matrix" without adding "p," the matrix will be displayed without parentheses.

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$

\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}

To enclose a matrix in brackets, you can use the "bmatrix" (b from brackets +matrix). However, if you want the brackets to be curly brackets, you can use the "Bmatrix", with a capital "B".

$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$

\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}

To enclose a matrix in vertical lines, you can use the "vmatrix" (v from vertical lines +matrix). However, if you want the vertical lines to be double lines, you can use the "Vmatrix", with a capital "V".

$A^{ \mathrm{ T } }$

A^{ \mathrm{ T } }

To represent the transpose of a matrix, you can use a Roman font uppercase "T" in the top right corner of the matrix.

${}^t \! A$

{}^t \! A

It is also common to represent the transpose of a matrix by writing a lowercase "t" in the top left corner of the matrix.

$\dim$

\dim

The notation "dimension" is derived from the dimension.

$\mathrm{ rank } A$

\mathrm{ rank } A

$\mathrm{ Tr } A$

\mathrm{ Tr } A

$\mathrm{ det }A$

\mathrm{ det }A

$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & I \end{pmatrix}$

\begin{pmatrix}
  a & b & c \\
  d & e & f \\
  g & h & I
\end{pmatrix}

Using the "pmatrix" environment, you can write a 3x3 matrix.

$\begin{eqnarray} \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
\left(
  \begin{array}{ccc}
    a & b & c \\
    d & e & f \\
    g & h & i
  \end{array}
\right)
\end{eqnarray}

Using the "array" environment, you can write a matrix. The "ccc" is for center alignment.

$\begin{eqnarray} \left( \begin{array}{rrr} 111 & 111 & 111 \\ 22 & 0.2 & -2 \\ 3 & 3 & 3 \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
\left(
  \begin{array}{rrr}
    111 & 111 & 111 \\
    22 & 0.2 & -2 \\
    3 & 3 & 3
  \end{array}
\right)
\end{eqnarray}

Using the "array" environment, you can write a matrix with right alignment for each value by specifying "rrr".

$\begin{array}{ccc} xxx & yyy & zzz \\ 1 & 2 & 3 \end{array}$

\begin{array}{ccc}
  xxx & yyy & zzz \\
  1   & 2   & 3
\end{array}

Using the "array" environment, you can create tables. The "ccc" aligns each value in the table cells to the center.

$\begin{array}{|c|c|c|} xxx & yyy & zzz \\ 1 & 2 & 3 \\ \end{array}$

\begin{array}{|c|c|c|}
  xxx & yyy & zzz \\
  1   & 2   & 3 \\
\end{array}

By using "|" like the "c|c|c" specification, you can include vertical lines to separate the columns in the table.

$\begin{array}{ccc} \hline xxx & yyy & zzz \\ \hline 1 & 2 & 3 \\ \hline \end{array}$

\begin{array}{ccc}
  \hline
  xxx & yyy & zzz \\
  \hline
  1   & 2   & 3 \\
  \hline
\end{array}

The command "\hline" is used to insert a horizontal line in a table. It is derived from hozirontal line.

$| x |$

| x |

$\vert x \vert$

\vert x \vert

It is derived from vertical line.

$\{ x \mid x \in A \}$

\{ x \mid x \in A \}

By using the "\mid" command, you can create a vertical line that has spaces before and after it.

$\Vert x \Vert$

\Vert x \Vert

By using the uppercase version of the "\vert" command as "\Vert", you can create a double vertical line in your document.

$AB \parallel CD$

AB \parallel CD

By using the "\parallel" command, you can create a double vertical line that represents parallelism.

$\overline{ A }$

\overline{ A }

$\bar{ A }$

\bar{ A }

$\underline{ A }$

\underline{ A }

$/$

/

The slash symbol ("/") is used as is, without any additional commands or modifications.

$\backslash$

\backslash

In MathJax, the backslash "\" has a special meaning, so if you want to display it as a symbol, you need to write "\backslash".

$\diagdown$

\diagdown

It is a diagonal line that slopes downwards on the right side.

$\diagup$

\diagup

It is a diagonal line that slopes upwards on the right side.

$\cancel{a}$

\cancel{a}

By using the "cancel", you can create a diagonal line to cancel out specific expressions.

$\bcancel{a}$

\bcancel{a}

The command "\bcancel" is similar to "\cancel," but it produces a diagonal line with the opposite direction.

$\xcancel{a}$

\xcancel{a}

The command "\xcancel" creates a cross (x) symbol over the content. It can be seen as a combination of the "\cancel" and "\bcancel".

$\cancelto{A}{a}$

\cancelto{A}{a}

The combination of a diagonal cancel line and an arrow.

$\leftarrow$

\leftarrow

Left awwor.

$\longleftarrow$

\longleftarrow

Long left awwor.

$\rightarrow$

\rightarrow

Right awwor.

$\longrightarrow$

\longrightarrow

$\uparrow$

\uparrow

Up arrow.

$\downarrow$

\downarrow

Down arrow.

$\leftrightarrow$

\leftrightarrow

Left and right arrow (bidirectional arrow).

$\longleftrightarrow$

\longleftrightarrow

$\updownarrow$

\updownarrow

Up and down arrow (vertical bidirectional arrow).

$\Leftarrow$

\Leftarrow

When you capitalize the first "l" to "L" in the command, the line becomes double.

$\Longleftarrow$

\Longleftarrow

$\Rightarrow$

\Rightarrow

$\Longrightarrow$

\Longrightarrow

$\Uparrow$

\Uparrow

$\Downarrow$

\Downarrow

$\Leftrightarrow$

\Leftrightarrow

$\Longleftrightarrow$

\Longleftrightarrow

$\Updownarrow$

\Updownarrow

$\nearrow$

\nearrow

An arrow pointing in the direction of the upper-right or northeast direction.

$\searrow$

\searrow

An arrow pointing in the direction of the down-right or southeast direction.

$\nwarrow$

\nwarrow

An arrow pointing in the direction of the upper-left or northwest direction.

$\swarrow$

\swarrow

An arrow pointing in the direction of the down-left or southwest direction.

$\mapsto$

\mapsto

It is derived from "maps to".

$\longmapsto$

\longmapsto

$\vec{ a }$

\vec{ a }

The arrow symbol used in vectors.

$\overrightarrow{ AB }$

\overrightarrow{ AB }

You can use the notation of placing a large arrow above the character.

$\overleftarrow{ AB }$

\overleftarrow{ AB }

$\circlearrowright$

\circlearrowright

A clockwise circular arrow

$\circlearrowleft$

\circlearrowleft

A counterclockwise circular arrow

$( x )$

( x )

Parentheses, represented by "(" and ")", are used as they are and do not require any special notation or commands.

$[ x ]$

[ x ]

Brackets, represented by "[" and "]", are used as they are and do not require any special notation or commands.

$\lbrack x \rbrack$

\lbrack x \rbrack

The term "brack" is used as a shorthand for brackets "[" and "]".

$\lceil x \rfloor$

\lceil x \rfloor

By combining the ceiling function and the floor function, you can create a notation that represents Japanese brackets.

$\lfloor x \rceil$

\lfloor x \rceil

$\{ x \}$

\{ x \}

Braces have special meaning, so if you want to display them as symbols, you need to use a backslash ("\") before them.

$\lbrace x \rbrace$

\lbrace x \rbrace

You can also use the term "\brace" for braces.

$\langle x \rangle$

\langle x \rangle

By using the "\angle", you can create angle brackets.

$\left[ \frac{ 1 }{ 2 } \right]$

\left[ \frac{ 1 }{ 2 } \right]

When you want to enclose a large expression within parentheses, you can use the "\left" and "\right" before the parentheses.

$\overbrace{ x + y + z }$

\overbrace{ x + y + z }

By using the "\overbrace", you can add a brace above an expression.

$\underbrace{ x + y + z }$

\underbrace{ x + y + z }

By using the "\underbrace", you can add a brace below an expression.。

$\cdot$

\cdot

The center dot.

$\cdots$

\cdots

By adding an "s" to the command "\cdot", you can create multiple dots instead of a single dot.

$\ldots$

\ldots

The lower dot.

$\vdots$

\vdots

The vertical dots.

$\ddots$

\ddots

The diagonal dots.

$\dot{ a }$

\dot{ a }

By using the "\dot", you can place a small dot above a character.

$\ddot{ a }$

\ddot{ a }

By stacking the "d", you can add additional dots above the character. Each stacked "d" adds another dot, allowing you to create a sequence of up to four dots.

$\circ$

\circ

It is derived from the circle.

$\bullet$

\bullet

It is derived from the bullet point used in bullet lists.

$\bigcirc$

\bigcirc

By adding the "big" to certain commands, you can increase their size.

$\oplus$

\oplus

A "+" sign is placed inside the letter "o".

$\ominus$

\ominus

A "-" sign is placed inside the letter "o".

$\otimes$

\otimes

A "times" (multiplication symbol) is placed inside the letter "o".

$\odot$

\odot

A dot is placed inside the letter "o".

$\triangle$

\triangle

$\triangledown$

\triangledown

Adding "down" to the "\triangle" makes it point downward.

$\bigtriangleup$

\bigtriangleup

$\bigtriangledown$

\bigtriangledown

$\triangleleft$

\triangleleft

Adding "left" to the "triangle" indicates that the triangle is facing towards the left side.

$\lhd$

\lhd

The term "lhd" is an abbreviation derived from "left-hand diamond."

$\triangleright$

\triangleright

Adding "right" to the "triangle" indicates that the triangle is facing towards the right side.

$\rhd$

\rhd

The term "rhd" is an abbreviation derived from "right-hand diamond."

$\unlhd$

\unlhd

The "unlhd" is "underline" and "lhd".

$\unrhd$

\unrhd

The "unrhd" is "underline" and "rhd".

$\blacktriangle$

\blacktriangle

Adding "black" to the "triangle," "down triangle," "left triangle," and "right triangle" indicates that these triangle symbols are filled in black.

$\square$

\square

$\Box$

\Box

It is important to note that "Box" is spelled with an uppercase "B".

$\boxplus$

\boxplus

A "+" sign (plus) is placed inside the box.

$\boxminus$

\boxminus

A "-" sign (minus) is placed inside the box.

$\boxtimes$

\boxtimes

A "times" (multiplication symbol) is placed inside the box.

$\boxdot$

\boxdot

A dot is placed inside the box.

$\blacksquare$

\blacksquare

A square shape that is filled in black.

$\diamond$

\diamond

$\Diamond$

\Diamond

When the "d" in "diamond" is capitalized, it represents a larger-sized diamond shape.

$\lozenge$

\lozenge

The "lozenge" is a term used to refer to a diamond.

$\blacklozenge$

\blacklozenge

A lozenge shape that is filled in black.

$\boxed{ abc }$

\boxed{ abc }

The "\boxed" is used to enclose an equation within a box.

$\fbox{ abc }$

\fbox{ abc }

The "\fbox" is used to enclose text within a frame.

$\ast$

\ast

The "\ast" is derived from the "asterisk."

$\star$

\star

$\ltimes$

\ltimes

It is a combination of the left line and the times symbol.

$\rtimes$

\rtimes

It is a combination of the right line and the times symbol.

$\Join$

\Join

It is a symbol used to represent natural join. It can also be represented using the "\bowtie", named after its shape.

$\$$

\$

In MathJax, certain characters like "\$", "#", "%", and "&" have special meaning. To display these characters themselves, you can prefix them with a backslash "\".

$\And$

\And

The ampersand symbol "&" has special meaning in MathJax, so if you want to display the ampersand symbol itself, you can use either "&" or "\And".

$\yen$

\yen

The yen "\" symbol has special meaning in MathJax and is used for commands, so if you want to display the yen symbol, you need to use a specific notation.

$\checkmark$

\checkmark

$\diamondsuit$

\diamondsuit

It is the symbol used in playing card. Compared to a diamond shape, the edges are slightly concave inward.

$\heartsuit$

\heartsuit

It is the symbol used in playing card.

$\clubsuit$

\clubsuit

It is the symbol used in playing card.

$\spadesuit$

\spadesuit

It is the symbol used in playing card.

$\flat$

\flat

$\natural$

\natural

$\sharp$

\sharp

$\dagger$

\dagger

$\ddagger$

\ddagger

$aaa \ bbb$

aaa \ bbb

To represent a space, you can use a combination of "\ " and a space. Using just a space character alone may not display as intended.

$aaa \quad bbb$

aaa \quad bbb

To represent a wider space, you can either repeat the space multiple times or use the "\quad" command. The term "quad" is derived from "quadrat," which refers to a square frame. In applications like Word, a space is sometimes represented as a square shape, and imagining that can provide a clearer understanding of the command.

$aaa \qquad bbb$

aaa \qquad bbb

The more "q"s you add to "quad", the wider the space becomes.

$aaa \hspace{ 10pt } bbb$

aaa \hspace{ 10pt } bbb

The command "\hspace" is used to create horizontal space. It allows you to specify the desired size of the space you want to create.

$aaa \! bbb$

aaa \! bbb

By combining the commands "\" and "!", you can reduce the space between two elements.

$\begin{eqnarray} aaa \\ bbb \end{eqnarray}$

\begin{eqnarray} 
aaa \\ bbb
\end{eqnarray}

By doubling the backslash "\" or using the command "\cr", you can create a line break or new line. The "cr" stands for "carriage return," which represents the control character that moves the cursor to the beginning of the line.

$\begin{eqnarray} aaa \\[5pt] bbb \end{eqnarray}$

\begin{eqnarray} 
aaa \\[5pt] bbb
\end{eqnarray}

After doubling the backslash "\" to indicate a line break, you can specify the width of the line break using additional commands.

$\tiny{ abc ABC }$

\tiny{ abc ABC }

$\scriptsize{ abc ABC }$

\scriptsize{ abc ABC }

$\small{ abc ABC }$

\small{ abc ABC }

$\normalsize{ abc ABC }$

\normalsize{ abc ABC }

$\large{ abc ABC }$

\large{ abc ABC }

$\Large{ abc ABC }$

\Large{ abc ABC }

$\LARGE{ abc ABC }$

\LARGE{ abc ABC }

$\mathrm{ ABC }$

\mathrm{ ABC }

It is derived from the Roman typeface (RoMan).

$\mathtt{ ABC }$

\mathtt{ ABC }

It is derived from typewriter typestyle.

$\mathsf{ ABC }$

\mathsf{ ABC }

It is derived from sans serif.

$\mathcal{ ABC }$

\mathcal{ ABC }

It is derived from calligraphy.

$\mathbf{ ABC }$

\mathbf{ ABC }

It is derived from bold font.

$\mathit{ ABC }$

\mathit{ ABC }

It is derived from italic.

$\mathbb{ ABC }$

\mathbb{ ABC }

It is derived from blackboard bold.

$\mathscr{ ABC }$

\mathscr{ ABC }

It is derived from script.

$\mathfrak{ ABC }$

\mathfrak{ ABC }

It is derived from Fraktur.

$a^{ xy }$

a^{ xy }

${}^{ xy } a$

{}^{ xy } a

$a_{ xy }$

a_{ xy }

${}_{ xy } a$

{}_{ xy } a

$\hat{ a }$

\hat{ a }

$\grave{ a }$

\grave{ a }

$\acute{ a }$

\acute{ a }

$\dot{ a }$

\dot{ a }

$\ddot{ a }$

\ddot{ a }

$\bar{ a }$

\bar{ a }

$\vec{ a }$

\vec{ a }

$\check{ a }$

\check{ a }

$\tilde{ a }$

\tilde{ a }

$\breve{ a }$

\breve{ a }

$\widehat{ AAA }$

\widehat{ AAA }

$\widetilde{ AAA }$

\widetilde{ AAA }

$\forall$

\forall

The symbol consists of an upside-down "A".

$\exists$

\exists

The symbol consists of an upside-down "E".

$\Finv$

\Finv

The symbol consists of an upside-down "F".

$\hbar$

\hbar

The symbol is a combination of the letter "h" and a bar. It is sometimes used to represent the reduced Planck constant, also known as the Dirac constant.

$\imath$

\imath

"i" without a above dot.

$\jmath$

\jmath

"j" without a above dot.

$\Bbbk$

\Bbbk

It is "k" in blackboard bold.

$\ell$

\ell

It is the cursive letter "l".

$\circledR$

\circledR

It is a symbol consisting of the letter "R" inside a circle.

$\circledS$

\circledS

It is a symbol consisting of the letter "S" inside a circle.

$\color{red}{a \times b}$

\color{red}{a \times b}

By using the "\color" command and specifying a color name, you can change the color of mathematical symbols within an equation.

$\color{ #ff0000 }{a \times b}$

\color{ #ff0000 }{a \times b}

With the "\color" command, you can also specify colors using hexadecimal color codes.

$\colorbox{red}{ Important! }$

\colorbox{red}{ Important! }

By using the "\colorbox" command, you can specify the background color for a block of text.

$\colorbox{red}{$a \times b$}$

\colorbox{red}{$a \times b$}

To include mathematical expressions within the "\colorbox" command, you can enclose the mathematical expression in $ symbols to switch to math mode.

$\unicode{x0041}$

\unicode{x0041}

By using the "\unicode" command followed by the Unicode character code, you can display a particular character in your text.

$\S$

\S

$\aleph$

\aleph

The Hebrew letter "Aleph".

$\beth$

\beth

The Hebrew letter "Bet".

$\gimel$

\gimel

The Hebrew letter "Gimel".

$\daleth$

\daleth

The Hebrew letter "Dalet".

$\TeX$

\TeX

It represents the logo of TeX.

$\LaTeX$

\LaTeX

It represents the logo of LaTeX.