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Contents

Formula
Symbol
Text

## Formula

### Number

fraction

$\frac{1}{2}$

\frac{1}{2}

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

fraction large

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

\displaystyle \frac{1}{2}

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

fraction large 2

$\dfrac{1}{2}$

\dfrac{1}{2}

\dfrac means \frac in \displaystyle.

fraction one line

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

\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".

fraction and parentheses

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

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

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

fraction and parentheses 2

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

\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.

continued fraction

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

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

Fractions can be nested.

continued fraction 2

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

\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.

infinite continued fraction

$\begin{eqnarray} 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots}}} = \frac{1}{2} \left( 1+\sqrt{5} \right) \end{eqnarray}$

\begin{eqnarray}
1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots}}}
= \frac{1}{2} \left( 1+\sqrt{5} \right)
\end{eqnarray}

The diagonal dots indicate that it goes on forever.

decimal

$0.123$

0.123

A period is used for the decimal point.

repeating decimal

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

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

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

infinite decimal 1

$\pi = 3.14 \ldots$

\pi = 3.14 \ldots

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

infinite decimal 2

$\sqrt{2} = 1.4142 \cdots$

\sqrt{2} = 1.4142 \cdots

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

infty

$\infty$

\infty

\infty comes from "infinity".

absolute value

$|x|$

|x|

Absolute values can be represented by a vertical line symbol.

absolute value 2

$\vert x \vert$

\vert x \vert

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

fraction and absolute value

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

\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.

fraction and absolute value 2

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

\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.

fraction and absolute value 3

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

\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".

Gaussian brackets

$[x]$

[x]

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

Gaussian brackets 2

$\lbrack x \rbrack$

\lbrack x \rbrack

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

floor function

$\lfloor x \rfloor$

\lfloor x \rfloor

They come from "floor function".

ceiling function

$\lceil x \rceil$

\lceil x \rceil

They come from "ceiling function".

Gaussian brackets sample

$\begin{eqnarray} [x] = \lfloor x \rfloor = \max\{ n\in\mathbb{Z} \mid n \leqq x \} \end{eqnarray}$

\begin{eqnarray}
[x]
= \lfloor x \rfloor
= \max\{ n\in\mathbb{Z} \mid n \leqq x \}
\end{eqnarray}

The definition of the Gaussian brackets.

### Arithmetic Operation

plus

$1 + 2$

1 + 2

We use the symbols as it is.

minus

$3 - 1$

3 - 1

We use the symbols as it is.

times

$2 \times 3$

2 \times 3

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

divide

$6 \divisionsymbol 3$

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.

plus minus

$\pm 1$

\pm 1

\pm comes from "plus and minus".

minus plus

$\mp 1$

\mp 1

\mp comes from "minus and plus".

times dot

$a \cdot b = ab$

a \cdot b = ab

\cdot comes from "center dot".

divide fraction

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

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

This is the relationship between division and fractions.

column multiplication

$\begin{array}{r} 67 \\[-3pt] \underline{\times\phantom{0}63}\\[-3pt] 201 \\[-3pt] \underline{\phantom{0}402\phantom{0}} \\[-3pt] 4221 \end{array}$

\begin{array}{r}
67 \\[-3pt]
\underline{\times\phantom{0}63}\\[-3pt]
201 \\[-3pt]
\underline{\phantom{0}402\phantom{0}} \\[-3pt]
4221
\end{array}

The array environment is used to specify right-justification. If you use underline, a line will be drawn underneath the text. The phantom allocates a space for the characters you specify. -3pt is a way of writing to reduce the width of a line break a little.

column division

$\begin{array}{r} 7.6 \\[-3pt] 25\enclose{longdiv}{190\phantom{0}} \\[-3pt] \underline{175\phantom{.0}} \\[-3pt] 15\phantom{.}0 \\[-3pt] \underline{15\phantom{.}0} \\[-3pt] \phantom{000}0 \end{array}$

\begin{array}{r}
7.6 \\[-3pt]
25\enclose{longdiv}{190\phantom{0}} \\[-3pt]
\underline{175\phantom{.0}} \\[-3pt]
15\phantom{.}0 \\[-3pt]
\underline{15\phantom{.}0} \\[-3pt]
\phantom{000}0
\end{array}

You can use \enclose{longdiv} to represent the long division. The phantom is used to substitute whitespace for the specified characters and align them.

modular equivalence

$a \equiv b \mod n$

a \equiv b \mod n

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

modular equivalence with parentheses

$a \equiv b \pmod n$

a \equiv b \pmod n

\pmod means "mod with parentheses".

modular equivalence like binary operator

$\gcd(a, b) = \gcd(b, a \bmod b)$

\gcd(a, b) = \gcd(b, a \bmod b)

\bmod means "mod used like a binary operator".

proportional

$x \propto y$

x \propto y

\propto comes from "proportional to".

### Greater or Less

greater than

$a \gt b$

a \gt b

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

greater than or equal

$a \geq b$

a \geq b

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

greater than or equal 2

$a \geqq b$

a \geqq b

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

less than

$a \lt b$

a \lt b

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

less than or equal

$a \leq b$

a \leq b

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

less than or equal 2

$a \leqq b$

a \leqq b

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

equal

$a = b$

a = b

We use the symbols as it is.

not equal

$a \neq b$

a \neq b

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

nearly equal

$a \fallingdotseq b$

a \fallingdotseq b

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

nearly equal 2

$a \sim b$

a \sim b

\sim comes from "similar".

nearly equal 3

$a \simeq b$

a \simeq b

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

nearly equal 4

$a \approx b$

a \approx b

\approx comes from "approximately".

much greater than

$a \gg b$

a \gg b

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

much less than

$a \ll b$

a \ll b

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

maximum

$\max f(x)$

\max f(x)
minimum

$\min f(x)$

\min f(x)
max sample

$\begin{eqnarray} \max ( a, b ) = \begin{cases} a & ( a \geqq b ) \\ b & ( a \lt b ) \end{cases} \end{eqnarray}$

\begin{eqnarray}
\max ( a, b )
=
\begin{cases}
a & ( a \geqq b ) \\
b & ( a \lt b )
\end{cases}
\end{eqnarray}

The eqnarray environment is used to display multiple expressions. The cases environment is used to write cases.

### Multiple Line Equations

line break

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

\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 "\\".

big line break

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

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

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

alignment

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

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

You can use "&" to align the position.

simultaneous equations

$\begin{eqnarray} \left\{ \begin{array}{l} x + y = 10 \\ 2x + 4y = 32 \end{array} \right. \end{eqnarray}$

\begin{eqnarray}
\left\{
\begin{array}{l}
x + y = 10 \\
2x + 4y = 32
\end{array}
\right.
\end{eqnarray}

You can use "\left\{" and "\right." to represent the big curly bracket on the left side of the simultaneous equations.

case

$\begin{eqnarray} |x| = \begin{cases} x & ( x \geqq 0 ) \\ -x & ( x \lt 0 ) \end{cases} \end{eqnarray}$

\begin{eqnarray}
|x|
=
\begin{cases}
x & ( x \geqq 0 ) \\
-x & ( x \lt 0 )
\end{cases}
\end{eqnarray}

The cases environment can be used to display for piecewise-defined functions.

### Set

belong to

$x \in A$

x \in A

\in comes from "x is in A".

belong to 2

$A \ni x$

A \ni x

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

not belong to

$x \notin A$

x \notin A

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

subset

$A \subset B$

A \subset B

It shows A is a subset of B.

subset 2

$A \subseteq B$

A \subseteq B

\subseteq means "subset or equal".

subset 3

$A \subseteqq B$

A \subseteqq B

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

superset

$A \supset B$

A \supset B

This shows A is a superset of B.

superset 2

$A \supseteq B$

A \supseteq B

\supseteq means "superset or equal".

superset 3

$A \supseteqq B$

A \supseteqq B

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

not subset

$A \not \subset B$

A \not \subset B

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

proper subset

$A \subsetneqq B$

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.

intersection

$A \cap B$

A \cap B
union

$A \cup B$

A \cup B
empty set

$\varnothing$

\varnothing

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

empty set 2

$\emptyset$

\emptyset

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

complement set

$A^c$

A^c

"c" comes from "complement set".

complement set 2

$\overline{ A }$

\overline{ A }

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

complement set sample

$\overline{ (A\cap B) } = \overline{ A } \cup \overline{ B }$

\overline{ (A\cap B) } = \overline{ A } \cup \overline{ B }

This is De Morgan's Law.

complement set sample 2

$\begin{eqnarray} \left( \bigcup_{\lambda\in\Lambda}A_{\lambda} \right)^c =\bigcap_{\lambda\in\Lambda}A_{\lambda}^c \end{eqnarray}$

\begin{eqnarray}
\left( \bigcup_{\lambda\in\Lambda}A_{\lambda} \right)^c
=\bigcap_{\lambda\in\Lambda}A_{\lambda}^c
\end{eqnarray}

This is De Morgan's Law too.

set difference

$A \setminus B$

A \setminus B

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

set difference sample

$A \setminus B = A \cap B^c = \{ x \in A \mid x \notin B \}$

A \setminus B
= A \cap B^c
= \{ x \in A \mid x \notin B \}

This is the definition of a difference set.

symmetric difference

$A \triangle B$

A \triangle B

The symmetric difference is represented by a triangle.

symmetric difference sample

$A \triangle B = (A \setminus B) \cup (B \setminus A)$

A \triangle B
= (A \setminus B) \cup (B \setminus A)

This is the definition of the symmetric difference.

all natural numbers

$\mathbb{ N }$

\mathbb{ N }

This is the blackboard bold.

all integers

$\mathbb{ Z }$

\mathbb{ Z }
all rational numbers

$\mathbb{ Q }$

\mathbb{ Q }
all real numbers

$\mathbb{ R }$

\mathbb{ R }
all complex numbers

$\mathbb{ C }$

\mathbb{ C }
all quaternions

$\mathbb{ H }$

\mathbb{ H }
supremum

$\sup A$

\sup A
infimum

$\inf A$

\inf A
aleph number

$\aleph$

\aleph

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

### Logic Notation

implication

$P \implies Q$

P \implies Q
implication 2

$P \Rightarrow Q$

P \Rightarrow Q

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

implication 3

$P \to Q$

P \to Q

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

implication reverse

$P \Leftarrow Q$

P \Leftarrow  Q
implication reverse 2

$P \gets Q$

P \gets  Q

A single arrow to the left.

equivalence

$P \iff Q$

P \iff Q

\iff means "if and only if".

equivalence 2

$P \Leftrightarrow Q$

P \Leftrightarrow Q

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

equivalence 3

$P \leftrightarrow Q$

P \leftrightarrow Q

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

equivalence 4

$P \equiv Q$

P \equiv Q

\equiv comes from "equivalence".

therefore

$\therefore$

\therefore
because

$\because$

\because
for all

$\forall x$

\forall x

This is Turned A.

for some

$\exists x$

\exists x

This is Turned E.

not exists

$\nexists$

\nexists

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

quantifier sample

$\begin{eqnarray} & & {}^\forall \varepsilon \gt 0, {}^\exists \delta \gt 0 \mbox{ s.t. } \\ & & {}^\forall x \in \mathbb{ R }, 0 \lt |x - a| \lt \delta \implies |f(x) - b| \lt \varepsilon \end{eqnarray}$

\begin{eqnarray}
& & {}^\forall \varepsilon \gt 0, {}^\exists \delta \gt 0 \mbox{ s.t. } \\
& & {}^\forall x \in \mathbb{ R }, 0 \lt |x - a| \lt \delta
\implies |f(x) - b| \lt \varepsilon
\end{eqnarray}

This is the (ε, δ)-definition of limit.

logical conjunction

$P \land Q$

P \land Q

\land comes from "And in logic".

logical disjunction

$P \lor Q$

P \lor Q

\lor comes from "Or in logic".

negation

$\lnot P$

\lnot P

\lnot comes from "Not in logic".

negation 2

$\overline{ P }$

\overline{ P }

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

negation 3

$!P$

!P

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

exclusive disjunction

$P \oplus Q$

P \oplus Q

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

exclusive disjunction 2

$P \veebar Q$

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.

exclusive disjunction sample

$P \oplus Q = (P \land \lnot Q) \lor (\lnot P \land Q)$

P \oplus Q = (P \land \lnot Q) \lor (\lnot P \land Q)

This is a formula relating exclusive disjunction to logical disjunction, logical product, and negation.

tautology

$\top$

\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.

provable

$P \vdash Q$

P \vdash Q

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

logical consequence

$P \models Q$

P \models Q

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

### Permutation and Combination

permutation

${}_n \mathrm{ P }_k$

{}_n \mathrm{ P }_k

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

combination

${}_n \mathrm{ C }_k$

{}_n \mathrm{ C }_k
factorial

$n!$

n!
binomial coefficient

$\binom{ n }{ k }$

\binom{ n }{ k }

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

binomial coefficient 2

${ n \choose k }$

{ n \choose k }

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

binomial coefficient 3

$\dbinom{ n }{ k }$

\dbinom{ n }{ k }

This is \binom in displaystyle.

repeated combination

${}_n \mathrm{ H }_k$

{}_n \mathrm{ H }_k
combination sample

$\begin{eqnarray} {}_n \mathrm{ C }_k = \binom{ n }{ k } = \frac{ n! }{ k! ( n - k )! } \end{eqnarray}$

\begin{eqnarray}
{}_n \mathrm{ C }_k
= \binom{ n }{ k }
= \frac{ n! }{ k! ( n - k )! }
\end{eqnarray}
permutation sample

$\begin{eqnarray} {}_n \mathrm{ P }_k = n \cdot ( n - 1 ) \cdots ( n - k + 1 ) = \frac{ n! }{ ( n - k )! } \end{eqnarray}$

\begin{eqnarray}
{}_n \mathrm{ P }_k
= n \cdot ( n - 1 ) \cdots ( n - k + 1 )
= \frac{ n! }{ ( n - k )! }
\end{eqnarray}

### Summation and Product

summation

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

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

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

summation large

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

\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.

summation sample

$\begin{eqnarray} \sum_{ k = 1 }^{ n } k^2 = \overbrace{ 1^2 + 2^2 + \cdots + n^2 }^{ n } = \frac{ 1 }{ 6 } n ( n + 1 ) ( 2n + 1 ) \end{eqnarray}$

\begin{eqnarray}
\sum_{ k = 1 }^{ n } k^2
= \overbrace{ 1^2 + 2^2 + \cdots + n^2 }^{ n }
= \frac{ 1 }{ 6 } n ( n + 1 ) ( 2n + 1 )
\end{eqnarray}

Using overbrace, you can display a brace on the top of the formula, and if you set the superscript, you can write text on top of the brace.

product

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

\prod_{ i = 0 }^n x_i

\prod comes from product.

product large

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

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

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

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

This is a sample using factorial.

product sample 2

$\begin{eqnarray} \zeta (s) = \prod_{ p:\mathrm{ prime } } \frac{ 1 }{ 1-p^{-s} } \end{eqnarray}$

\begin{eqnarray}
\zeta (s)
= \prod_{ p:\mathrm{ prime } }
\frac{ 1 }{ 1-p^{-s} }
\end{eqnarray}

This is a sample using the Riemann zeta function.

### Exponent and Logarithm

power

$2^3$

2^3

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

power 2

$e^{ i \pi }$

e^{ i \pi }

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

exponential function

$\exp ( x )$

\exp ( x )
square root

$\sqrt{ 2 }$

\sqrt{ 2 }

\sqrt comes from "square root".

square root same height

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

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

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

nth root

$\sqrt[ n ]{ x }$

\sqrt[ n ]{ x }

When writing power roots, use brackets.

logarithm

$\log x$

\log x
logarithm to base

$\log_{ 2 } x$

\log_{ 2 } x

The logarithm base is specified using the "_".

natural logarithm

$\ln x$

\ln x

### Shape

degree

$90^{ \circ }$

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 symbol

$\angle A$

\angle A
parallel Japanese style

$AB /\!/ CD$

AB /\!/ CD

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

parallel international style

$AB \parallel CD$

AB \parallel CD
perpendicular

$AB \perp CD$

AB \perp CD

\perp comes from "perpendicular".

triangle

$\triangle ABC$

\triangle ABC

$\Box ABCD$

\Box ABCD

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

arc

$\stackrel{\huge\frown}{AB}$

\stackrel{\huge\frown}{AB}

For arcs, since there is no proper command, we combine symbols. \forwn represents the arc symbol (looks like a frown mouth), and \huge makes it bigger. You can use \stackrel to stack the symbols on top of each other.

arc

$\overparen{AB}$

\overparen{AB}

\overparen comes from "over" and "parentheses". It allows you to put round brackets over text. However, it don't display beautifully.

congruence Japanese style

$\triangle ABC \equiv \triangle DEF$

\triangle ABC \equiv \triangle DEF

\equiv comes from "equivalent". In Japan, it is often written like this.

congruence international style

$\triangle ABC \cong \triangle DEF$

\triangle ABC \cong \triangle DEF

\cong comes from "congruent". This is the most common way to write globally.

similar Japanese style

$\triangle ABC \backsim \triangle DEF$

\triangle ABC \backsim \triangle DEF

The commonly used similarity symbol in Japan is obtained by rotating the letter S by 90 degrees. An equivalent symbol for this is "backsim." However, this symbol may feel a bit unfamiliar as it is a reversed tilde.

similar international style

$\triangle ABC \sim \triangle DEF$

\triangle ABC \sim \triangle DEF

It is derived from the concept of similarity. It is more commonly used overseas to represent similarity.

### Trigonometric Function

sin

$\sin x$

\sin x
cos

$\cos x$

\cos x
tan

$\tan x$

\tan x
sin sample

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

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

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

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

$\begin{eqnarray} \tan \theta = \frac{ \sin \theta }{ \cos \theta } \end{eqnarray}$

\begin{eqnarray}
\tan \theta
= \frac{ \sin \theta }{ \cos \theta }
\end{eqnarray}
sec

$\sec x$

\sec x
csc

$\csc x$

\csc x
cot

$\cot x$

\cot x
arcsin

$\arcsin x$

\arcsin x
arccos

$\arccos x$

\arccos x
arctan

$\arctan x$

\arctan x
sinh

$\sinh x$

\sinh x

Hyperbolic functions are not trigonometric functions, but I will introduce them here.

cosh

$\cosh x$

\cosh x
tanh

$\tanh x$

\tanh x
coth

$\coth x$

\coth x

### Complex Number

complex number

$a+bi$

a+bi

The imaginary unit is commonly denoted by "i."

real part

$\oldRe x$

\Re x

It is derived from the term "real part."

real part 2

$\require{physics} \Re x$

\require{physics} \Re x

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

imaginary part

$\oldIm x$

\Im x

It is derived from the term "imaginary part."

imaginary part 2

$\require{physics} \Im x$

\require{physics} \Im x

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

complex conjugate

$\bar{z}$

\bar{z}

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

argument

$\arg (z)$

\arg (z)

It is derived from the concept of the argument.

cube roots of 1

$\omega$

\omega

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

complex number sample

$\begin{eqnarray} z\bar{z} = |z|^2 \end{eqnarray}$

\begin{eqnarray}
z\bar{z} = |z|^2
\end{eqnarray}

### Limit

limit

$\lim_{ x \to +0 } \frac{1}{x} = \infty$

\lim_{ x \to +0 } \frac{1}{x} = \infty

It is derived from the concept of a limit.

limit large

$\displaystyle \lim_{ n \to \infty } f_n(x) = f(x)$

\displaystyle \lim_{ n \to \infty } f_n(x) = f(x)

Adding "\displaystyle" causes the expression to be displayed in a larger format. Subscripts will be positioned below the "lim" symbol.

limit superior

$\limsup_{ n \to \infty } a_n$

\limsup_{ n \to \infty } a_n

It is derived from the limit superior.

limit superior simple

$\varlimsup_{ n \to \infty } a_n$

\varlimsup_{ n \to \infty } a_n
limit inferior

$\liminf_{ n \to \infty } a_n$

\liminf_{ n \to \infty } a_n

It is derived from the limit inferior.

limit inferior simple

$\varliminf_{ n \to \infty } a_n$

\varliminf_{ n \to \infty } a_n
limit superior sample

$\begin{eqnarray} \varlimsup_{ n \to \infty } a_n = \lim_{ n \to \infty } \sup_{ k \geqq n } a_k \end{eqnarray}$

\begin{eqnarray}
\varlimsup_{ n \to \infty } a_n
= \lim_{ n \to \infty } \sup_{ k \geqq n } a_k
\end{eqnarray}

This is an example of the limit superior of a sequence.

limit inferior sample

$\begin{eqnarray} \varliminf_{ n \to \infty } A_n = \bigcup_{ n = 1 }^{ \infty } \bigcap_{ k = n }^{ \infty } A_k = \bigcup_{ n \in \mathbb{ N } } \bigcap_{ k \geqq n } A_k \end{eqnarray}$

\begin{eqnarray}
\varliminf_{ n \to \infty } A_n
= \bigcup_{ n = 1 }^{ \infty } \bigcap_{ k = n }^{ \infty } A_k
= \bigcup_{ n \in \mathbb{ N } } \bigcap_{ k \geqq n } A_k
\end{eqnarray}

Here is an example of the limit inferior of a set.

Big O notation

$\mathcal{O}$

\mathcal{O}

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

### Differentiation

differentiation Leibniz

$\frac{ dy }{ dx }$

\frac{ dy }{ dx }

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

differentiation Leibniz 2

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

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

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

differentiation Leibniz 3

$\require{physics} \dv{y}{x}$

\require{physics} \dv{y}{x}

By using the "physics" extension, we can simplify the notation of "d" in Roman font. The "dv" is derived from "derivative." When we write it with a single curly bracket, only the denominator part remains.

nth differentiation Leibniz

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

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

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

nth differentiation Leibniz 2

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

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

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

differentiation Leibniz at a point

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

\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.

differentiation Leibniz at a point 2

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

\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.

differentiation Lagrange

$f'$

f'

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

second differentiation Lagrange

$f^{\prime\prime}$

f^{\prime\prime}

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

nth differentiation Lagrange

$f^{ ( n ) }$

f^{ ( n ) }
differentiation Euler

$Df$

Df
differentiation Euler 2

$D_x f$

D_x f
nth differentiation Euler

$D^n f$

D^n f
differentiation Newton

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

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

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

4th differentiation Newton

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

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

Increasing "d" increases the number of dots.

differentiation sample

$\begin{eqnarray} f'(x) = \frac{ df }{ dx } = \lim_{ \Delta x \to 0 } \frac{ f(x + \Delta x) - f(x) }{ \Delta x } \end{eqnarray}$

\begin{eqnarray}
f'(x)
= \frac{ df }{ dx }
= \lim_{ \Delta x \to 0 } \frac{ f(x + \Delta x) - f(x) }{ \Delta x }
\end{eqnarray}
partial differentiation

$\frac{ \partial f }{ \partial x }$

\frac{ \partial f }{ \partial x }

It is derived from the partial derivative.

2nd partial differentiation

$\frac{ \partial }{ \partial y } \frac{ \partial }{ \partial x } z$

\frac{ \partial }{ \partial y } \frac{ \partial }{ \partial x } z
nth partial differentiation

$\frac{ \partial^n f}{ \partial x^n }$

\frac{ \partial^n f}{ \partial x^n }
partial differentiation 2

$\require{physics} \pdv{f}{x}$

\require{physics} \pdv{f}{x}

By using the "physics" extension, we can simplify the notation for expressing partial derivatives.

2nd partial differentiation 2

$\require{physics} \pdv{f}{x}{y}$

\require{physics} \pdv{f}{x}{y}

By utilizing the "physics" extension, we can simplify the notation for representing second-order partial derivatives.

nth partial differentiation 2

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

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

By using the "physics" extension, we can simplify the notation for representing nth-order partial derivatives. Simply write "n" inside brackets to denote the order of the partial derivative.

partial differentiation simple

$f_x$

f_x
2nd partial differentiation simple

$f_{ xy }$

f_{ xy }
del

$\nabla f$

\nabla f
lapracian

$\Delta f$

\Delta f
lapracian sample

$\begin{eqnarray} \Delta \varphi = \nabla^2 \varphi = \frac{ \partial^2 \varphi }{ \partial x^2 } + \frac{ \partial^2 \varphi }{ \partial y^2 } + \frac{ \partial^2 \varphi }{ \partial z^2 } \end{eqnarray}$

\begin{eqnarray}
\Delta \varphi
= \nabla^2 \varphi
= \frac{ \partial^2 \varphi }{ \partial x^2 }
+ \frac{ \partial^2 \varphi }{ \partial y^2 }
+ \frac{ \partial^2 \varphi }{ \partial z^2 }
\end{eqnarray}
first derivative test table

$\begin{array}{c|ccccc} x & \cdots & -1 & \cdots & 1 & \cdots \\ \hline f’(x) & + & 0 & – & 0 & + \\ \hline f(x) & \nearrow & e & \searrow & -e & \nearrow \end{array}$

\begin{array}{c|ccccc}
x     & \cdots & -1 & \cdots & 1 & \cdots \\
\hline
f’(x) & + & 0 & – & 0 & + \\
\hline
f(x)  & \nearrow & e & \searrow & -e & \nearrow
\end{array}

### Integral

integral

$\int_0^1 f(x) dx$

\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.

integral large

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

\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.

integral sample

$\begin{eqnarray} \int_0^1 x dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1}{2} \end{eqnarray}$

\begin{eqnarray}
\int_0^1 x dx
= \left[ \frac{x^2}{2} \right]_0^1
= \frac{1}{2}
\end{eqnarray}

Here is an example of a integral calculation:

double integral

$\iint_D f(x,y) dxdy$

\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.

multiple integral

$\idotsint_D f(x_1, x_2, \ldots , x_n) dx_1 \cdots dx_n$

\idotsint_D f(x_1, x_2, \ldots , x_n) dx_1 \cdots dx_n

By combining "int", "dots", and "int" together, we get the symbol "idotsint."

contour integral

$\oint_C f(z) dz$

\oint_C f(z) dz

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

### Vector

vector

$\vec{ a }$

\vec{ a }

The symbol "vector" is derived from the vectors.

vector 2

$\overrightarrow{ AB }$

\overrightarrow{ AB }

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

vector bold

$\boldsymbol{ A }$

\boldsymbol{ A }

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

row vector

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

( a_1, a_2, \ldots, a_n )
column vector

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

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

$\begin{eqnarray} \boldsymbol{ 1 } =( \underbrace{ 1, 1, \ldots, 1 }_{ n } )^{ \mathrm{ T } } =\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
\boldsymbol{ 1 }
=( \underbrace{ 1, 1, \ldots, 1 }_{ n } )^{ \mathrm{ T } }
=\left(
\begin{array}{c}
1 \\
1 \\
\vdots \\
1
\end{array}
\right)
\end{eqnarray}
unit vector sample

$\boldsymbol{ \rm{ e } }_k =( 0, \ldots, 0, \stackrel{k}{ 1 }, 0, \ldots, 0 )^{\mathrm{T}}$

\boldsymbol{ \rm{ e } }_k
=( 0, \ldots, 0, \stackrel{k}{ 1 }, 0, \ldots, 0 )^{\mathrm{T}}
norm

$\| x \|$

\| x \|

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

norm 2

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

\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.

inner product

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

\vec{ a } \cdot \vec{ b }
cross product

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

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

### Matrix

martix parentheses

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

\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.

martix brackets

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

\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".

martix vertical lines

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

\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".

transposed matrix

$A^{ \mathrm{ T } }$

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.

transposed matrix 2

${}^t \! A$

{}^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.

dimension

$\dim$

\dim

The notation "dimension" is derived from the dimension.

matrix rank

$\mathrm{ rank } A$

\mathrm{ rank } A
trace

$\mathrm{ Tr } A$

\mathrm{ Tr } A
determinant

$\mathrm{ det }A$

\mathrm{ det }A
determinant sample

$\begin{eqnarray} \mathrm{ det }A = | A | = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \end{eqnarray}$

\begin{eqnarray}
\mathrm{ det }A
= | A |
= \begin{vmatrix} a & b \\ c & d \end{vmatrix}
\end{eqnarray}
martix large

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

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

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

martix large

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

\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.

right alignment matrix

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

\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".

mxn matrix

$\begin{eqnarray} A = \left( \begin{array}{cccc} a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\ a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ a_{ m1 } & a_{ m2 } & \ldots & a_{ mn } \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
A = \left(
\begin{array}{cccc}
a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\
a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\
\vdots & \vdots & \ddots & \vdots \\
a_{ m1 } & a_{ m2 } & \ldots & a_{ mn }
\end{array}
\right)
\end{eqnarray}

It is an example of representing an m x n matrix using multiple dots.

block matrix

$\begin{eqnarray} \left( \begin{array}{cc|cc} a & b & 0 & 0 \\ c & d & 0 & 0 \\ \hline x & y & 1 & 0 \\ z & w & 0 & 1 \\ \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
\left(
\begin{array}{cc|cc}
a & b & 0 & 0 \\
c & d & 0 & 0 \\
\hline
x & y & 1 & 0 \\
z & w & 0 & 1 \\
\end{array}
\right)
\end{eqnarray}

Using "|" like the "cc|cc", you can draw vertical lines between columns. The "\hline" command can be used to draw horizontal lines.

Jordan block

$\begin{eqnarray} \begin{pmatrix} \lambda & 1 & & & 0 \\ & \lambda & 1 & & \\ & & \ddots & \ddots & \\ & & & \lambda & 1 \\ 0 & & & & \lambda \end{pmatrix} \end{eqnarray}$

\begin{eqnarray}
\begin{pmatrix}
\lambda & 1 &   &  & 0 \\
& \lambda & 1 &   &   \\
&   & \ddots & \ddots &   \\
&   &   & \lambda & 1  \\
0 &   &   &   & \lambda
\end{pmatrix}
\end{eqnarray}
cofactor

$\begin{eqnarray} & & (-1)^{ i+j } \times \\[5pt] & & \quad \begin{vmatrix} a_{1,1} & \ldots & a_{1,j-1} & a_{1,j+1} & \ldots & a_{1,n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{i-1,1} & \ldots & a_{i-1, j-1} & a_{i-1, j+1} & \ldots & a_{i-1, n} \\ a_{i+1,1} & \ldots & a_{i+1, j-1} & a_{i+1, j+1} & \ldots & a_{i+1, n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & \ldots & a_{n, j-1} & a_{n, j+1} & \ldots & a_{n, n} \end{vmatrix} \end{eqnarray}$

\begin{eqnarray}
& & (-1)^{ i+j } \times \\[5pt]
\begin{vmatrix}
a_{1,1} & \ldots & a_{1,j-1} & a_{1,j+1} & \ldots & a_{1,n} \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
a_{i-1,1} & \ldots & a_{i-1, j-1} & a_{i-1, j+1} & \ldots & a_{i-1, n} \\
a_{i+1,1} & \ldots & a_{i+1, j-1} & a_{i+1, j+1} & \ldots & a_{i+1, n} \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
a_{n,1} & \ldots & a_{n, j-1} & a_{n, j+1} & \ldots & a_{n, n}
\end{vmatrix}
\end{eqnarray}

### Table

table simple

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

\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.

table with vertical line

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

\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.

table with horizontal line

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

\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.

table sample

$\begin{array}{c|ccccc} x & \cdots & -1 & \cdots & 1 & \cdots \\ \hline f’(x) & + & 0 & – & 0 & + \\ \hline f(x) & \nearrow & e & \searrow & -e & \nearrow \end{array}$

\begin{array}{c|ccccc}
x     & \cdots & -1 & \cdots & 1 & \cdots \\
\hline
f’(x) & + & 0 & – & 0 & + \\
\hline
f(x)  & \nearrow & e & \searrow & -e & \nearrow
\end{array}

### Commutative Diagram

commutative diagram sample

$\require{AMScd} \begin{CD} A @>{f}>> B\\ @V{gg}VV {\large\circlearrowleft} @VV{hh}V\\ C @>>{k}> D \end{CD}$

\require{AMScd}
\begin{CD}
A @>{f}>> B\\
@V{gg}VV {\large\circlearrowleft} @VV{hh}V\\
C @>>{k}> D
\end{CD}

## Symbol

### Line

vertical line

$| x |$

| x |
vertical line 2

$\vert x \vert$

\vert x \vert

It is derived from vertical line.

vertical line 3

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

\{ x \mid x \in A \}

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

double vertical line

$\Vert x \Vert$

\Vert x \Vert

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

double vertical line 2

$AB \parallel CD$

AB \parallel CD

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

overline

$\overline{ A }$

\overline{ A }
overline 2

$\bar{ A }$

\bar{ A }
underline

$\underline{ A }$

\underline{ A }
slash

$/$

/

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

backslash

$\backslash$

\backslash

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

diagonal line down

$\diagdown$

\diagdown

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

diagonal line up

$\diagup$

\diagup

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

cancel

$\cancel{a}$

\cancel{a}

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

back cancel

$\bcancel{a}$

\bcancel{a}

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

x cancel

$\xcancel{a}$

\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".

cancel arrow

$\cancelto{A}{a}$

\cancelto{A}{a}

The combination of a diagonal cancel line and an arrow.

cancel

$\begin{eqnarray} \frac{\cancel{2}}{\cancel{6}}=\frac{1}{3} \end{eqnarray}$

\begin{eqnarray}
\frac{\cancel{2}}{\cancel{6}}=\frac{1}{3}
\end{eqnarray}

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

cancel arrow

$\begin{eqnarray} \frac{1}{\cancel{3}} \times \frac{\cancelto{2}{6}}{5} \end{eqnarray}$

\begin{eqnarray}
\frac{1}{\cancel{3}} \times \frac{\cancelto{2}{6}}{5}
\end{eqnarray}

The combination of a diagonal cancel line and an arrow.

lower left corner

$\llcorner$

\llcorner

The symbol "llcorner" represents the lower left corner.

lower right corner

$\lrcorner$

\lrcorner

The symbol "lrcorner" represents the lower right corner.

up left corner

$\ulcorner$

\ulcorner

The symbol "ulcorner" represents the up left corner.

up right corner

$\urcorner$

\urcorner

The symbol "urcorner" represents the up right corner.

### Arrow

left arrow

$\leftarrow$

\leftarrow

Left awwor.

long left arrow

$\longleftarrow$

\longleftarrow

Long left awwor.

right arrow

$\rightarrow$

\rightarrow

Right awwor.

long right arrow

$\longrightarrow$

\longrightarrow
up arrow

$\uparrow$

\uparrow

Up arrow.

down arrow

$\downarrow$

\downarrow

Down arrow.

left right arrow

$\leftrightarrow$

\leftrightarrow

Left and right arrow (bidirectional arrow).

long left right arrow

$\longleftrightarrow$

\longleftrightarrow
up down arrow

$\updownarrow$

\updownarrow

Up and down arrow (vertical bidirectional arrow).

double left arrow

$\Leftarrow$

\Leftarrow

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

double long left arrow

$\Longleftarrow$

\Longleftarrow
double right arrow

$\Rightarrow$

\Rightarrow
double long right arrow

$\Longrightarrow$

\Longrightarrow
double up arrow

$\Uparrow$

\Uparrow
double down arrow

$\Downarrow$

\Downarrow
double left right arrow

$\Leftrightarrow$

\Leftrightarrow
double long left right arrow

$\Longleftrightarrow$

\Longleftrightarrow
double up down arrow

$\Updownarrow$

\Updownarrow
north east arrow

$\nearrow$

\nearrow

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

south east arrow

$\searrow$

\searrow

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

north west arrow

$\nwarrow$

\nwarrow

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

south west arrow

$\swarrow$

\swarrow

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

arrow with bar

$\mapsto$

\mapsto

It is derived from "maps to".

long arrow with bar

$\longmapsto$

\longmapsto
over arrow

$\vec{ a }$

\vec{ a }

The arrow symbol used in vectors.

over arrow 2

$\overrightarrow{ AB }$

\overrightarrow{ AB }

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

over left arrow

$\overleftarrow{ AB }$

\overleftarrow{ AB }
clockwise arrow

$\circlearrowright$

\circlearrowright

A clockwise circular arrow

counterclockwise arrow

$\circlearrowleft$

\circlearrowleft

A counterclockwise circular arrow

### Bracket

parenthese

$( x )$

( x )

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

bracket

$[ x ]$

[ x ]

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

bracket 2

$\lbrack x \rbrack$

\lbrack x \rbrack

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

square bracket

$\lceil x \rfloor$

\lceil x \rfloor

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

square bracket 2

$\lfloor x \rceil$

\lfloor x \rceil
brace

$\{ x \}$

\{ x \}

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

brace 2

$\lbrace x \rbrace$

\lbrace x \rbrace

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

angle bracket

$\langle x \rangle$

\langle x \rangle

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

big bracket

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

\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.

over brace

$\overbrace{ x + y + z }$

\overbrace{ x + y + z }

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

over brace and letter

$\overbrace{ a_1 + \cdots + a_n }^{ n }$

\overbrace{ a_1 + \cdots + a_n }^{ n }

By combining the "\overbrace" with the "^" symbol, you can add text above the brace.

under brace

$\underbrace{ x + y + z }$

\underbrace{ x + y + z }

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

under brace and letter

$\underbrace{ a_1 + \cdots + a_n }_{ n }$

\underbrace{ a_1 + \cdots + a_n }_{ n }

By combining the "\underbrace" with the "_" symbol, you can add text below the brace.

### Dot

center dot

$\cdot$

\cdot

The center dot.

center dots

$\cdots$

\cdots

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

low dots

$\ldots$

\ldots

The lower dot.

vertical dots

$\vdots$

\vdots

The vertical dots.

diagonal dots

$\ddots$

\ddots

The diagonal dots.

over dot

$\dot{ a }$

\dot{ a }

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

over dots

$\ddot{ a }$

\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.

### Circle

white circle

$\circ$

\circ

It is derived from the circle.

black circle

$\bullet$

\bullet

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

big circle

$\bigcirc$

\bigcirc

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

circle and plus

$\oplus$

\oplus

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

circle and minus

$\ominus$

\ominus

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

circle and times

$\otimes$

\otimes

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

circle and dot

$\odot$

\odot

A dot is placed inside the letter "o".

### Triangle

triangle

$\triangle$

\triangle
triangle down

$\triangledown$

\triangledown

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

big triangle up

$\bigtriangleup$

\bigtriangleup
big triangle down

$\bigtriangledown$

\bigtriangledown
triangle left

$\triangleleft$

\triangleleft

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

triangle left 2

$\lhd$

\lhd

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

triangle right

$\triangleright$

\triangleright

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

triangle right 2

$\rhd$

\rhd

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

triangle left and underline

$\unlhd$

\unlhd

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

triangle right and underline

$\unrhd$

\unrhd

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

black triangle

$\blacktriangle$

\blacktriangle

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

### Rectangle

square

$\square$

\square
box

$\Box$

\Box

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

box and cross

$\boxplus$

\boxplus

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

box and line

$\boxminus$

\boxminus

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

box and x

$\boxtimes$

\boxtimes

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

box and dot

$\boxdot$

\boxdot

A dot is placed inside the box.

black square

$\blacksquare$

\blacksquare

A square shape that is filled in black.

diamond

$\diamond$

\diamond
diamond 2

$\Diamond$

\Diamond

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

lozenge

$\lozenge$

\lozenge

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

black lozenge

$\blacklozenge$

\blacklozenge

A lozenge shape that is filled in black.

boxed text

$\boxed{ abc }$

\boxed{ abc }

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

boxed text 2

$\fbox{ abc }$

\fbox{ abc }

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

boxed text 3

$\bbox[yellow, 5pt, border: 2px dotted red]{abc}$

\bbox[yellow, 5pt, border: 2px dotted red]{abc}

The "\bbox" is not a native command in $\TeX$, but it can be used in MathJax to finely customize the appearance of a framed box around text or equation. It allows you to specify various settings such as background color, margin, and style by separating them with commas within the brackets, while the equation is written within braces. The "\bbox" is derived from "bounding box".

### Binary Operations

asterisk

$\ast$

\ast

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

star

$\star$

\star
left line and times

$\ltimes$

\ltimes

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

right line and times

$\rtimes$

\rtimes

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

natural join

$\Join$

\Join

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

### General Symbol

dollar sign

$\$$\$

In MathJax, certain characters like "\$", "#", "%", and "&" have special meaning. To display these characters themselves, you can prefix them with a backslash "\". ampersand$\And$\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 sign$\yen$\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. check mark$\checkmark$\checkmark diamond$\diamondsuit$\diamondsuit It is the symbol used in playing card. Compared to a diamond shape, the edges are slightly concave inward. heart$\heartsuit$\heartsuit It is the symbol used in playing card. club$\clubsuit$\clubsuit It is the symbol used in playing card. spade$\spadesuit$\spadesuit It is the symbol used in playing card. flat$\flat$\flat natural$\natural$\natural sharp$\sharp$\sharp dagger$\dagger$\dagger dagger 2$\ddagger$\ddagger ## Text ### Space space$aaa \ bbb$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. wide space$aaa \quad bbb$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. wide space 2$aaa \qquad bbb$aaa \qquad bbb The more "q"s you add to "quad", the wider the space becomes. specified space size$aaa \hspace{ 10pt } bbb$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. no space$aaa \! bbb$aaa \! bbb By combining the commands "\" and "!", you can reduce the space between two elements. new line$\begin{eqnarray} aaa \\ bbb \end{eqnarray}$\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. specified new line size$\begin{eqnarray} aaa \\[5pt] bbb \end{eqnarray}$\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. specified new line size sample$\begin{eqnarray} & & \frac{1}{2} +\frac{1}{3} +\frac{1}{6} \\[ 5pt ] &=& \frac{3}{6} +\frac{2}{6} +\frac{1}{6} \\ &=& 1 \end{eqnarray}$\begin{eqnarray} & & \frac{1}{2} +\frac{1}{3} +\frac{1}{6} \\[ 5pt ] &=& \frac{3}{6} +\frac{2}{6} +\frac{1}{6} \\ &=& 1 \end{eqnarray} ### Letter size tiny$\tiny{ abc ABC }$\tiny{ abc ABC } small size$\scriptsize{ abc ABC }$\scriptsize{ abc ABC } small size 2$\small{ abc ABC }$\small{ abc ABC } normal size$\normalsize{ abc ABC }$\normalsize{ abc ABC } large size$\large{ abc ABC }$\large{ abc ABC } large size 2$\Large{ abc ABC }$\Large{ abc ABC } large size 3$\LARGE{ abc ABC }$\LARGE{ abc ABC } huge size$\huge{ abc ABC }$\huge{ abc ABC } huge size 2$\Huge{ abc ABC }$\Huge{ abc ABC } ### Font roman font$\mathrm{ ABC }$\mathrm{ ABC } It is derived from the Roman typeface (RoMan). type writer font$\mathtt{ ABC }$\mathtt{ ABC } It is derived from typewriter typestyle. sans serif$\mathsf{ ABC }$\mathsf{ ABC } It is derived from sans serif. calligraphy font$\mathcal{ ABC }$\mathcal{ ABC } It is derived from calligraphy. bold font$\mathbf{ ABC }$\mathbf{ ABC } It is derived from bold font. italic$\mathit{ ABC }$\mathit{ ABC } It is derived from italic. blackboard bold$\mathbb{ ABC }$\mathbb{ ABC } It is derived from blackboard bold. script letters$\mathscr{ ABC }$\mathscr{ ABC } It is derived from script. Fraktur$\mathfrak{ ABC }$\mathfrak{ ABC } It is derived from Fraktur. roman font sample$\mathrm{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz }$\mathrm{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz } typewriter font sample$\mathtt{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz }$\mathtt{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz } sans serif sample$\mathsf{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz }$\mathsf{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz } calligraphy font sample$\mathcal{ ABCDEFGHIJKLMNOPQRSTUVWXYZ }$\mathcal{ ABCDEFGHIJKLMNOPQRSTUVWXYZ } bold font sample$\mathbf{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz }$\mathbf{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz } italic sample$\mathit{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz }$\mathit{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz } blackboard bold sample$\mathbb{ ABCDEFGHIJKLMNOPQRSTUVWXYZ }$\mathbb{ ABCDEFGHIJKLMNOPQRSTUVWXYZ } script letters sample$\mathscr{ ABCDEFGHIJKLMNOPQRSTUVWXYZ }$\mathscr{ ABCDEFGHIJKLMNOPQRSTUVWXYZ } Fraktur sample$\mathfrak{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz }$\mathfrak{ ABCDEFGHIJKLMNOPQRSTUVWXYZ \ abcdefghijklmnopqrstuvwxyz } ### Superscript and Subscript superscript$a^{ xy }$a^{ xy } superscript left${}^{ xy } a${}^{ xy } a subscript$a_{ xy }$a_{ xy } subscript left${}_{ xy } a${}_{ xy } a subscript sample$\begin{eqnarray} a_n^2 + a_{ n + 1 }^2 = a_{ 2n + 1 } \end{eqnarray}$\begin{eqnarray} a_n^2 + a_{ n + 1 }^2 = a_{ 2n + 1 } \end{eqnarray} ### Accent hat$\hat{ a }$\hat{ a } grave$\grave{ a }$\grave{ a } acute$\acute{ a }$\acute{ a } dot$\dot{ a }$\dot{ a } double dots$\ddot{ a }$\ddot{ a } bar$\bar{ a }$\bar{ a } arrow$\vec{ a }$\vec{ a } check$\check{ a }$\check{ a } tilde$\tilde{ a }$\tilde{ a } breve$\breve{ a }$\breve{ a } wide hat$\widehat{ AAA }$\widehat{ AAA } wide tilde$\widetilde{ AAA }$\widetilde{ AAA } ### Alphabet upside down A$\forall$\forall The symbol consists of an upside-down "A". upside down E$\exists$\exists The symbol consists of an upside-down "E". upside down F$\Finv$\Finv The symbol consists of an upside-down "F". h bar$\hbar$\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. dotless i$\imath$\imath "i" without a above dot. dotless j$\jmath$\jmath "j" without a above dot. blackboard bold k$\Bbbk$\Bbbk It is "k" in blackboard bold. handwriting-style l$\ell$\ell It is the cursive letter "l". circled R$\circledR$\circledR It is a symbol consisting of the letter "R" inside a circle. circled S$\circledS$\circledS It is a symbol consisting of the letter "S" inside a circle. ### Greek alphabet alpha$\alpha$\alpha beta$\beta$\beta gamma$\gamma$\gamma delta$\delta$\delta epsilon$\epsilon$\epsilon epsilon-2$\varepsilon$\varepsilon zeta$\zeta$\zeta eta$\eta$\eta theta$\theta$\theta theta-2$\vartheta$\vartheta iota$\iota$\iota kappa$\kappa$\kappa lambda$\lambda$\lambda mu$\mu$\mu nu$\nu$\nu xi$\xi$\xi o$o$o It is the same as the alphabet. pi$\pi$\pi pi-2$\varpi$\varpi rho$\rho$\rho rho-2$\varrho$\varrho sigma$\sigma$\sigma sigma-2$\varsigma$\varsigma tau$\tau$\tau upsilon$\upsilon$\upsilon phi$\phi$\phi phi-2$\varphi$\varphi chi$\chi$\chi psi$\psi$\psi omega$\omega$\omega Alpha$A$A It is the same as the alphabet. Beta$B$B It is the same as the alphabet. Gamma$\Gamma$\Gamma Gamma-2$\varGamma$\varGamma Delta$\Delta$\Delta Delta-2$\varDelta$\varDelta Epsilon$E$E It is the same as the alphabet. Zeta$Z$Z It is the same as the alphabet. Eta$H$H It is the same as the alphabet. Theta$\Theta$\Theta Theta-2$\varTheta$\varTheta Iota$I$I It is the same as the alphabet. Kappa$K$K It is the same as the alphabet. Lambda$\Lambda$\Lambda Lambda-2$\varLambda$\varLambda Mu$M$M It is the same as the alphabet. Nu$N$N It is the same as the alphabet. Xi$\Xi$\Xi Xi-2$\varXi$\varXi O$O$O It is the same as the alphabet. Pi$\Pi$\Pi Pi-2$\varPi$\varPi Rho$P$P It is the same as the alphabet. Sigma$\Sigma$\Sigma Sigma-2$\varSigma$\varSigma Tau$T$T It is the same as the alphabet. Upsilon$\Upsilon$\Upsilon Upsilon-2$\varUpsilon$\varUpsilon Phi$\Phi$\Phi Phi-2$\varPhi$\varPhi Chi$X$X It is the same as the alphabet. Psi$\Psi$\Psi Psi-2$\varPsi$\varPsi Omega$\Omega$\Omega Omega-2$\varOmega$\varOmega ### HTML color text$\color{red}{a \times b}$\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 text 2$\color{ #ff0000 }{a \times b}$\color{ #ff0000 }{a \times b} With the "\color" command, you can also specify colors using hexadecimal color codes. color box$\colorbox{red}{ Important! }$\colorbox{red}{ Important! } By using the "\colorbox" command, you can specify the background color for a block of text. color box 2$\colorbox{red}{$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.

color border box

$\fcolorbox{black}{ #00ff00 }{$a \times b$}$

\fcolorbox{black}{ #00ff00 }{$a \times b$}

\fcolorbox is a combination of the frame and colorbox. It allows for specifying the frame color, background color, and text content in that order. To write mathematical expressions, you can use the $ to switch to math mode. color border box 2$\bbox[yellow, 5pt, border: 2px dotted red]{abc}$\bbox[yellow, 5pt, border: 2px dotted red]{abc} The "\bbox" command is for the bounding box. It allows for more detailed customization in blackets, such as background color, margin, and style. The curly brackets is for the equation. The "\bbox" is not a standard command in$\TeX$. unicode$\unicode{x0041}$\unicode{x0041} By using the "\unicode" command followed by the Unicode character code, you can display a particular character in your text. unicode sample$\begin{eqnarray} \unicode{x5F45}\text{は、弓へんに剪。} \end{eqnarray}$\begin{eqnarray} \unicode{x5F45}\text{は、弓へんに剪。} \end{eqnarray} It's a sample using the "\unicode" ### Special character section$\S$\S aleph$\aleph$\aleph The Hebrew letter "Aleph". beth$\beth$\beth The Hebrew letter "Bet". gimel$\gimel$\gimel The Hebrew letter "Gimel". daleth$\daleth$\daleth The Hebrew letter "Dalet". TeX$\TeX$\TeX It represents the logo of TeX. LaTex$\LaTeX\$

\LaTeX

It represents the logo of LaTeX.

Formula
Symbol
Text
Other