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 (), 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)
Perso-Arabic variant
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. 3.14159265358, 1,000,000,000. Negative signs are written to the left of magnitudes, e.g. −3. In-line fractions are written with the numerator and denominator on the left and right of the fraction slash respectively, e.g. 2/7.

Symbols

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

Arabic mathematical limit in different forms
Latin	Arabic	Persian
Template:Underset x4	[a]	[b]

• ^a nūn-hāʾ-ʾalif is derived from the first three letters of Arabic nihāya "limit".
• ^b ḥadd is Persian for "limit".

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

Arabic mathematical sum in different forms
Latin	Arabic	Mirrored Latin and Greek
Template:Underoverset Template:Radic	[c]

• ^c is derived from Arabic maǧmūʿ "sum".

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

Examples

Mathematical letters

Latin	Arabic		Notes
$a$	File:Arabic mathematical alif.PNG		From the Arabic letter ʾalif; a and 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
$b$	File:Arabic mathematical beh.PNG		A dotless bāʾ; b and bāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively
$c$	File:Arabic mathematical geem.PNG		From the initial form of ḥāʾ, or that of a dotless jīm; c and 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
$d$	File:Arabic mathematical dal.PNG		From the Arabic letter dāl; d and 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
$x$	File:Arabic mathematical seen.PNG		From the Arabic letter sīn. It is contested that the usage of Latin x in maths is derived from the first letter šīn (without its dots) of the Arabic word šayʾ(un) , meaning thing. (X was used in old Spanish for the sound /ʃ/). However, according to others there is no historical evidence for this.
$y$	File:Arabic mathematical sad.PNG		From the Arabic letter ṣād
$z$	File:Arabic mathematical ain.PNG		From the Arabic letter ʿayn

Mathematical constants and units

Description	Latin	Arabic		Notes
Euler's number	$e$	File:Arabic mathematical heh.PNG		Initial form of the Arabic letter hāʾ. Both Latin letter e and Arabic letter hāʾ are descendants of Phoenician letter File:Phoenician he.svg hē.
imaginary unit	$i$	File:Arabic mathematical teh.PNG		From tāʾ, which is in turn derived from the first letter of the second word of waḥdaẗun taḫīliyya "imaginary unit"
pi	$\pi$	File:Arabic mathematical tah.PNG		From ṭāʾ; also $\pi$ in some regions
radius	$r$	File:Arabic mathematical radius.PNG		From nūn followed by a dotless qāf, which is in turn derived from nuṣfu l-quṭr "radius"
kilogram	kg	File:Arabic kilogram.PNG		From kāf-jīm-mīm. In some regions alternative symbols like File:Arabic alternative kilogram 2.PNG ( kāf-ġayn) or File:Arabic alternative kilogram 1.PNG ( kāf-lām-ġayn) are used. All three abbreviations are derived from kīlūġrām "kilogram" and its variant spellings.
gram	g	File:Arabic gram.PNG		From jīm-mīm, which is in turn derived from jrām, a variant spelling of ġrām "gram"
metre	m	File:Arabic mathematical meem.PNG		From mīm, which is in turn derived from mitr "metre"
centimetre	cm	File:Arabic cm.PNG		From sīn-mīm, which is in turn derived from "centimetre"
millimetre	mm	File:Arabic mm.PNG		From mīm-mīm, which is in turn derived from millīmitr "millimetre"
kilometre	km	File:Arabic Km.PNG		From kāf-mīm; also File:Arabic alternative km.PNG ( kāf-lām-mīm) in some regions; both are derived from kīlūmitr "kilometre".
second	s	File:Arabic mathematical theh.PNG		From ṯāʾ, which is in turn derived from ṯāniya "second"
minute	min	File:Arabic mathematical Dal large.PNG		From dālʾ, which is in turn derived from daqīqa "minute"; also File:Arabic mathematical qaf.PNG (, i.e. dotless qāf) in some regions
hour	h	File:Arabic mathematical seen.PNG		From sīnʾ, which is in turn derived from sāʿa "hour"
kilometre per hour	km/h	File:Arabic kmph.PNG		From the symbols for kilometre and hour
degree Celsius	°C	File:Arabic celsius degree.PNG		From sīn, which is in turn derived from the second word of darajat sīlsīūs "degree Celsius"; also File:Arabic centegrade degree.PNG () from mīmʾ, which is in turn derived from the first letter of the third word of "degree centigrade"
degree Fahrenheit	°F	File:Arabic fahrenheit degree.PNG		From fāʾ, which is in turn derived from the second word of darajat fahranhāyt "degree Fahrenheit"
millimetres of mercury	mmHg	File:Arabic mmHg.PNG		From mīm-mīm zayn, which is in turn derived from the initial letters of the words "millimetres of mercury"
Ångström	Å	File:Arabic angestrom.PNG		From ʾalif with hamzah and ring above, which is in turn derived from the first letter of "Ångström", variously spelled or

Sets and number systems

Description	Latin	Arabic		Notes
Natural numbers	$\mathbb{\mathbb{N}}$	File:Arabic mathematical tah large.PNG		From ṭāʾ, which is in turn derived from the first letter of the second word of ʿadadun ṭabīʿiyyun "natural number"
Integers	$\mathbb{\mathbb{Z}}$	File:Arabic mathematical Sad large.PNG		From ṣād, which is in turn derived from the first letter of the second word of ʿadadun ṣaḥīḥun "integer"
Rational numbers	$\mathbb{\mathbb{Q}}$	File:Arabic mathematical noon large.PNG		From nūn, which is in turn derived from the first letter of nisba "ratio"
Real numbers	$\mathbb{ℝ}$	File:Arabic mathematical hah large.PNG		From ḥāʾ, which is in turn derived from the first letter of the second word of ʿadadun ḥaqīqiyyun "real number"
Imaginary numbers	$\mathbb{𝕀}$	File:Arabic mathematical teh large.PNG		From tāʾ, which is in turn derived from the first letter of the second word of ʿadadun taḫīliyyun "imaginary number"
Complex numbers	$\mathbb{ℂ}$	File:Arabic mathematical meem large.PNG		From mīm, which is in turn derived from the first letter of the second word of 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	$\mathbf{𝐒}$	File:Arabic mathematical sheen large.PNG		From šīn, which is in turn derived from the first letter of the second word of majmūʿatun šāmila "universal set"

Arithmetic and algebra

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

Trigonometric and hyperbolic functions

Trigonometric functions

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

Hyperbolic functions

The letter File:Arabic mathematical zain.PNG ( zayn, from the first letter of the second word of "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.

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

Inverse trigonometric functions

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

Arabic inverse trigonometric functions
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

Inverse hyperbolic functions

Arabic 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

Calculus

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

Complex analysis

Latin/Greek	Arabic
$z=x+iy=r(\cos \phi +i\sin \phi )=r{e}^{i\phi }=r\mathrm{\angle }\phi$	File:Arabic mathematical complex analysis.PNG

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.