Modern Arabic mathematical notation

Template:Short description Modern Arabic mathematical notation is a mathematical notation based on the Arabic script, used especially at pre-university levels of education. Its form is mostly derived from Western notation, but has some notable features that set it apart from its Western counterpart. The most remarkable of those features is the fact that it is written from right to left following the normal direction of the Arabic script. Other differences include the replacement of the Greek and Latin alphabet letters for symbols with Arabic letters and the use of Arabic names for functions and relations.

Features

• It is written from right to left following the normal direction of the Arabic script. Other differences include the replacement of the Latin alphabet letters for symbols with Arabic letters and the use of Arabic names for functions and relations.
• The notation exhibits one of the very few remaining vestiges of non-dotted Arabic scripts, as dots over and under letters (i'jam) are usually omitted.
• Letter cursivity (connectedness) of Arabic is also taken advantage of, in a few cases, to define variables using more than one letter. The most widespread example of this kind of usage is the canonical symbol for the radius of a circle Template:Lang (Template:IPA), which is written using the two letters nūn and qāf. When variable names are juxtaposed (as when expressing multiplication) they are written non-cursively.

Variations

Notation differs slightly from one region to another. In tertiary education, most regions use the Western notation. The notation mainly differs in numeral system used, and in mathematical symbols used.

Numeral systems

There are three numeral systems used in right to left mathematical notation.

• "Western Arabic numerals" (sometimes called European) are used in western Arabic regions (e.g. Morocco)
• "Eastern Arabic numerals" are used in middle and eastern Arabic regions (e.g. Egypt and Syria)
• "Eastern Arabic-Indic numerals" are used in Persian and Urdu speaking regions (e.g. Iran, Pakistan, India)

European (descended from Western Arabic)	0	1	2	3	4	5	6	7	8	9
Arabic-Indic (Eastern Arabic)	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang
Perso-Arabic variant	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang	Template:Lang
Urdu variant	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral	Template:Urdu numeral

Written numerals are arranged with their lowest-value digit to the right, with higher value positions added to the left. That is identical to the arrangement used by Western texts using Hindu-Arabic numerals even though Arabic script is read from right to left: Indeed, Western texts are written with the ones digit on the right because when the arithmetical manuals were translated from the Arabic, the numerals were treated as figures (like in a Euclidean diagram), and so were not flipped to match the Left-Right order of Latin text. The symbols "٫" and "٬" may be used as the decimal mark and the thousands separator respectively when writing with Eastern Arabic numerals, e.g. Template:Lang 3.14159265358, Template:Lang 1,000,000,000. Negative signs are written to the left of magnitudes, e.g. Template:Lang −3. In-line fractions are written with the numerator and denominator on the left and right of the fraction slash respectively, e.g. Template:Lang 2/7.Template:Cn

Symbols

Sometimes, symbols used in Arabic mathematical notation differ according to the region:

Latin	Arabic	Persian
Template:Math	Template:MathTemplate:Ref	Template:MathTemplate:Ref

• Template:Note Template:Lang nūn-hāʾ-ʾalif is derived from the first three letters of Arabic Template:Lang nihāya "limit".
• Template:Note Template:Lang ḥadd is Persian for "limit".

Sometimes, mirrored Latin and Greek symbols are used in Arabic mathematical notation (especially in western Arabic regions):

Latin	Arabic	Mirrored Latin and Greek
Template:Math	Template:MathTemplate:Ref	Template:Math

• Template:Note Template:Lang is derived from Arabic Template:Lang maǧmūʿ "sum".

However, in Iran, usually Latin and Greek symbols are used.

Examples

Mathematical letters

Latin	Arabic		Notes
$a$		Template:Math	From the Arabic letter Template:Lang ʾalif; a and Template:Lang ʾalif are the first letters of the Latin alphabet and the Arabic alphabet's ʾabjadī sequence respectively, and the letters also share a common ancestor and the same sound
$b$		Template:Math	A dotless Template:Lang bāʾ; b and Template:Lang bāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively
$c$		Template:Math	From the initial form of Template:Lang ḥāʾ, or that of a dotless Template:Lang jīm; c and Template:Lang jīm are the third letters of the Latin alphabet and the ʾabjadī sequence respectively, and the letters also share a common ancestor and the same sound
$d$		Template:Math	From the Arabic letter Template:Lang dāl; d and Template:Lang dāl are the fourth letters of the Latin alphabet and the ʾabjadī sequence respectively, and the letters also share a common ancestor and the same sound
$x$		Template:Math	From the Arabic letter Template:Lang sīn. It is contested that the usage of Latin x in maths is derived from the first letter Template:Lang šīn (without its dots) of the Arabic word Template:Lang šayʾ(un) Template:IPA, meaning thing. (X was used in old Spanish for the sound /ʃ/). However, according to others there is no historical evidence for this.
$y$		Template:Math	From the Arabic letter Template:Lang ṣād
$z$		Template:Math	From the Arabic letter Template:Lang ʿayn

Mathematical constants and units

Description	Latin	Arabic		Notes
Euler's number	$e$		Template:Math	Initial form of the Arabic letter Template:Lang hāʾ. Both Latin letter e and Arabic letter Template:Lang hāʾ are descendants of Phoenician letter hē.
imaginary unit	$i$		Template:Math	From Template:Lang tāʾ, which is in turn derived from the first letter of the second word of Template:Lang waḥdaẗun taḫīliyya "imaginary unit"
pi	$\pi$		Template:Math	From Template:Lang ṭāʾ; also $\pi$ in some regions
radius	$r$		Template:Math	From Template:Lang nūn followed by a dotless Template:Lang qāf, which is in turn derived from Template:Lang nuṣfu l-quṭr "radius"
kilogram	kg		Template:Math	From Template:Lang kāf-jīm-mīm. In some regions alternative symbols like (Template:Math kāf-ġayn) or (Template:Math kāf-lām-ġayn) are used. All three abbreviations are derived from Template:Lang kīlūġrām "kilogram" and its variant spellings.
gram	g		Template:Math	From Template:Lang jīm-mīm, which is in turn derived from Template:Lang jrām, a variant spelling of Template:Lang ġrām "gram"
metre	m		Template:Math	From Template:Lang mīm, which is in turn derived from Template:Lang mitr "metre"
centimetre	cm		Template:Math	From Template:Lang sīn-mīm, which is in turn derived from Template:Lang "centimetre"
millimetre	mm		Template:Math	From Template:Lang mīm-mīm, which is in turn derived from Template:Lang millīmitr "millimetre"
kilometre	km		Template:Math	From Template:Lang kāf-mīm; also (Template:Math kāf-lām-mīm) in some regions; both are derived from Template:Lang kīlūmitr "kilometre".
second	s		Template:Math	From Template:Lang ṯāʾ, which is in turn derived from Template:Lang ṯāniya "second"
minute	min		Template:Bigmath	From Template:Lang dālʾ, which is in turn derived from Template:Lang daqīqa "minute"; also (Template:Math, i.e. dotless Template:Lang qāf) in some regions
hour	h		Template:Math	From Template:Lang sīnʾ, which is in turn derived from Template:Lang sāʿa "hour"
kilometre per hour	km/h		Template:Math	From the symbols for kilometre and hour
degree Celsius	°C		Template:Math	From Template:Lang sīn, which is in turn derived from the second word of Template:Lang darajat sīlsīūs "degree Celsius"; also (Template:Math) from Template:Lang mīmʾ, which is in turn derived from the first letter of the third word of Template:Lang "degree centigrade"
degree Fahrenheit	°F		Template:Math	From Template:Lang fāʾ, which is in turn derived from the second word of Template:Lang darajat fahranhāyt "degree Fahrenheit"
millimetres of mercury	mmHg		Template:Math	From Template:Lang mīm-mīm zayn, which is in turn derived from the initial letters of the words Template:Lang "millimetres of mercury"
Ångström	Å		Template:Math	From Template:Lang ʾalif with hamzah and ring above, which is in turn derived from the first letter of "Ångström", variously spelled Template:Lang or Template:Lang

Sets and number systems

Description	Latin	Arabic		Notes
Natural numbers	$\mathbb{N}$		Template:Bigmath	From Template:Lang ṭāʾ, which is in turn derived from the first letter of the second word of Template:Langʿadadun ṭabīʿiyyun "natural number"
Integers	$\mathbb{Z}$		Template:Bigmath	From Template:Lang ṣād, which is in turn derived from the first letter of the second word of Template:Lang ʿadadun ṣaḥīḥun "integer"
Rational numbers	$\mathbb{Q}$		Template:Bigmath	From Template:Lang nūn, which is in turn derived from the first letter of Template:Lang nisba "ratio"
Real numbers	$ℝ$		Template:Bigmath	From Template:Lang ḥāʾ, which is in turn derived from the first letter of the second word of Template:Lang ʿadadun ḥaqīqiyyun "real number"
Imaginary numbers	$𝕀$		Template:Bigmath	From Template:Lang tāʾ, which is in turn derived from the first letter of the second word of Template:Lang ʿadadun taḫīliyyun "imaginary number"
Complex numbers	$ℂ$		Template:Bigmath	From Template:Lang mīm, which is in turn derived from the first letter of the second word of Template:Lang ʿadadun murakkabun "complex number"
Empty set	$\varnothing$	$\varnothing$	∅
Is an element of	$\in$	$\ni$	∈	A mirrored ∈
Subset	$\subset$	$\supset$	⊂	A mirrored ⊂
Superset	$\supset$	$\subset$	⊃	A mirrored ⊃
Universal set	$𝐒$		Template:Bigmath	From Template:Lang šīn, which is in turn derived from the first letter of the second word of Template:Lang majmūʿatun šāmila "universal set"

Arithmetic and algebra

Description	Latin/Greek	Arabic		Notes
Percent	%		Template:Math	e.g. 100% "Template:Lang"
Permille	‰		Template:Math	Template:Math is an Arabic equivalent of the per ten thousand sign ‱.
Is proportional to	$\propto$		∝	A mirrored ∝
n th root	$\sqrt[n]{}$		Template:Math	Template:Lang is a dotless Template:Lang nūn while √ is a mirrored radical sign √
Logarithm	$\log$		Template:Math	From Template:Lang lām-wāw, which is in turn derived from لوغاريتم Template:Transl "logarithm"
Logarithm to base b	${\log }_b$		Template:Math
Natural logarithm	$\ln$		Template:Math	From the symbols of logarithm and Euler's number
Summation	$\sum$		Template:Math	Template:Lang mīm-medial form of jīm is derived from the first two letters of Template:Lang majmūʿ "sum"; also (∑, a mirrored summation sign ∑) in some regions
Product	$\prod$		Template:Math	From Template:Lang jīm-ḏāl. The Arabic word for "product" is جداء jadāʾun. Also $\prod$ in some regions.
Factorial	$n!$		Template:Math	Also (Template:Math) in some regions
Permutations	${}^n{𝐏}_r$		Template:Math	Also (Template:Math) is used in some regions as $𝐏(n,r)$
Combinations	${}^n{𝐂}_k$		Template:Math	Also (Template:Math) is used in some regions as $𝐂(n,k)$ and (Template:Math ) as the binomial coefficient $(\genfrac{}{}{0ex}{}{n}{k})$

Trigonometric and hyperbolic functions

Trigonometric functions

Description	Latin	Arabic		Notes
Sine	$\sin$		Template:Math	from Template:Lang ḥāʾ (i.e. dotless Template:Lang jīm)-ʾalif; also (Template:Math jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "sine" is Template:Lang jayb
Cosine	$\cos$		Template:Math	from Template:Lang ḥāʾ (i.e. dotless Template:Lang jīm)-tāʾ-ʾalif; also (Template:Math tāʾ-jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "cosine" is Template:Lang
Tangent	$\tan$		Template:Math	from Template:Lang ṭāʾ (i.e. dotless Template:Lang ẓāʾ)-ʾalif; also (Template:Math ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "tangent" is Template:Lang ẓill
Cotangent	$\cot$		Template:Math	from Template:Lang ṭāʾ (i.e. dotless Template:Lang ẓāʾ)-tāʾ-ʾalif; also (Template:Math tāʾ-ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "cotangent" is Template:Lang
Secant	$\sec$		Template:Math	from Template:Lang dotless Template:Lang qāf-ʾalif; Arabic for "secant" is Template:Lang
Cosecant	$\csc$		Template:Math	from Template:Lang dotless Template:Lang qāf-tāʾ-ʾalif; Arabic for "cosecant" is Template:Lang

Hyperbolic functions

The letter (Template:Math zayn, from the first letter of the second word of Template:Lang "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to the way $\mathrm{h}$ is added to the end of trigonometric functions in Latin-based notation.

Description	Hyperbolic sine	Hyperbolic cosine	Hyperbolic tangent	Hyperbolic cotangent	Hyperbolic secant	Hyperbolic cosecant
Latin	$\sinh$	$\cosh$	$\tanh$	$\coth$	$\mathrm{sech}$	$\mathrm{csch}$
Arabic	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math

Inverse trigonometric functions

For inverse trigonometric functions, the superscript Template:Math in Arabic notation is similar in usage to the superscript $-1$ in Latin-based notation.

Description	Inverse sine	Inverse cosine	Inverse tangent	Inverse cotangent	Inverse secant	Inverse cosecant
Latin	${\sin }^{-1}$	${\cos }^{-1}$	${\tan }^{-1}$	${\cot }^{-1}$	${\sec }^{-1}$	${\csc }^{-1}$
Arabic	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math

Inverse hyperbolic functions

Description	Inverse hyperbolic sine	Inverse hyperbolic cosine	Inverse hyperbolic tangent	Inverse hyperbolic cotangent	Inverse hyperbolic secant	Inverse hyperbolic cosecant
Latin	${\sinh }^{-1}$	${\cosh }^{-1}$	${\tanh }^{-1}$	${\coth }^{-1}$	${\mathrm{sech}}^{-1}$	${\mathrm{csch}}^{-1}$
Arabic	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math

Calculus

Description	Latin	Arabic		Notes
Limit	$\lim$		Template:Math	Template:Lang nūn-hāʾ-ʾalif is derived from the first three letters of Arabic Template:Lang nihāya "limit"
Function	$𝐟(x)$		Template:Math	Template:Lang dāl is derived from the first letter of Template:Lang "function". Also called Template:Lang, Template:Lang for short, in some regions.
Derivatives	${𝐟}^{\prime }(x),\frac{dy}{dx},\frac{d^{2}y}{d{x}^2},\frac{\partial y}{\partial x}$		Template:Math	‵ is a mirrored prime ′ while ، is an Arabic comma. The Template:Math signs should be mirrored: ∂.
Integrals	$\int ,\iint ,\iiint ,\oint$		‪∮ ،∭ ،∬ ،∫‬	Mirrored ∫, ∬, ∭ and ∮

Complex analysis

Latin/Greek	Arabic
$z=x+iy=r(\cos \phi +i\sin \phi )=r{e}^{i\phi }=r\mathrm{\angle }\phi$
	Template:Math

See also

• Mathematical notation
• Arabic Mathematical Alphabetic Symbols

External links

• Multilingual mathematical e-document processing
• Arabic mathematical notation - W3C Interest Group Note.
• Arabic math editor - by WIRIS.