Modern Arabic mathematical notation

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 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 نق (Arabic pronunciation: [nɑq]), 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 region to another. In tertiary education, most regions use the Western notation. The notation mainly differs in numeral system used, and in mathematical symbol 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:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang	Template:Rtl-lang
Perso-Arabic variant	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Urdu variant

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. The symbols "٫" and "٬" may be used as the decimal mark and the thousands separator respectively when writing with Eastern Arabic numerals, e.g. Template:Rtl-lang 3.14159265358, Template:Rtl-lang 1,000,000,000. Negative signs are written to the left of magnitudes, e.g. Template:Rtl-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:Rtl-lang 2/7.

Mirrored Latin symbols

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

Latin	Arabic	Persian

$lim x \to\infty x 4$	$Template:Rtl-lang$ ^[a]	$Template:Rtl-lang$ ^[b]

^a Template:Rtl-lang nūn-hāʾ-ʾalif is derived from the first three letters of Arabic Template:Rtl-lang nihāya "limit".
^b حد ḥadd is Persian for "limit".

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

Latin	Arabic	Mirrored Latin

$n \sum x =0 3 \sqrt x$	$Template:Rtl-lang$ ^[c]	$Template:Rtl-lang$

^c Template:Rtl-lang

Template:Rtl-lang maǧmūʿ means "sum" in Arabic language.

However, in Iran, usually Latin symbols are used.

Examples

Mathematical letters

Latin		Notes
$a$	$Template:Rtl-lang$	From the Arabic letter Template:Rtl-lang ʾalif; a and Template:Rtl-lang ʾalif are the first letters of the Latin alphabet and the Arabic alphabet's ʾabjadī sequence respectively
$b$	$Template:Rtl-lang$	A dotless Template:Rtl-lang bāʾ; b and Template:Rtl-lang bāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively
$c$	$Template:Rtl-lang$	From the initial form of Template:Rtl-lang ḥāʾ, or that of a dotless Template:Rtl-lang jīm; c and Template:Rtl-lang jīm are the third letters of the Latin alphabet and the ʾabjadī sequence respectively
$d$	$Template:Rtl-lang$	From the Arabic letter Template:Rtl-lang dāl; d and Template:Rtl-lang dāl are the fourth letters of the Latin alphabet and the ʾabjadī sequence respectively
$x$	$Template:Rtl-lang$	From the Arabic letter Template:Rtl-lang sīn. It is contested that the usage of Latin x in maths is derived from the first letter Template:Rtl-lang šīn (without its dots) of the Arabic word Template:Rtl-lang šayʾ(un) [ʃajʔ(un)], meaning thing.^[1] (X was used in old Spanish for the sound /ʃ/). However, according to others there is no historical evidence for this.^[2]^[3]
$y$	$Template:Rtl-lang$	From the Arabic letter Template:Rtl-lang ṣād
$z$	$Template:Rtl-lang$	From the Arabic letter Template:Rtl-lang ʿayn

Mathematical constants and units

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

Sets and number systems

Description	Latin	Arabic		Notes
Natural numbers	$\mathbb {N}$		$Template:Rtl-lang$	From Template:Rtl-lang ṭāʾ, which is in turn derived from the first letter of the second word of Template:Rtl-langʿadadun ṭabīʿiyyun "natural number"
Integers	$\mathbb {Z}$		$Template:Rtl-lang$	From Template:Rtl-lang ṣād, which is in turn derived from the first letter of the second word of Template:Rtl-lang ʿadadun ṣaḥīḥun "integer"
Rational numbers	$\mathbb {Q}$		$Template:Rtl-lang$	From Template:Rtl-lang nūn, which is in turn derived from the first letter of Template:Rtl-lang nisba "ratio"
Real numbers	$\mathbb {R}$		$Template:Rtl-lang$	From Template:Rtl-lang ḥāʾ, which is in turn derived from the first letter of the second word of Template:Rtl-lang ʿadadun ḥaqīqiyyun "real number"
Imaginary numbers	$\mathbb {I}$		$Template:Rtl-lang$	From Template:Rtl-lang tāʾ, which is in turn derived from the first letter of the second word of Template:Rtl-lang ʿadadun taḫīliyyun "imaginary number"
Complex numbers	$\mathbb {C}$		$Template:Rtl-lang$	From Template:Rtl-lang mīm, which is in turn derived from the first letter of the second word of Template:Rtl-lang ʿadadun markabun "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	$\mathbf {S}$		$Template:Rtl-lang$	From Template:Rtl-lang šīn, which is in turn derived from the first letter of the second word of Template:Rtl-lang majmūʿatun šāmila "universal set"

Arithmetic and algebra

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

Trigonometric and hyperbolic functions

Trigonometric functions

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

Hyperbolic functions

The letter ( $Template:Rtl-lang$ zayn, from the first letter of the second word of Template:Rtl-lang "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to the way $\operatorname {h}$ 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	$\sinh$	$\cosh$	$\tanh$	$\coth$	$\operatorname {sech}$	$\operatorname {csch}$
Arabic	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$

Inverse trigonometric functions

For inverse trigonometric functions, the superscript $Template:Rtl-lang$ 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	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$	$Template:Rtl-lang$

Inverse hyperbolic functions


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

Calculus

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

Complex analysis

Latin	Arabic
$z=x+iy=r(\cos {\varphi }+i\sin {\varphi })=re^{i\varphi }=r\angle {\varphi }$
	$Template:Rtl-lang$

References

^ Moore, Terry. "Why is X the Unknown". Ted Talk.
^ Cajori, Florian (1993). A History of Mathematical Notation. Courier Dover Publications. pp. 382–383. Retrieved 11 October 2012. Nor is there historical evidence to support the statement found in Noah Webster's Dictionary, under the letter x, to the effect that 'x was used as an abbreviation of Ar. shei (a thing), something, which, in the Middle Ages, was used to designate the unknown, and was then prevailingly transcribed as xei.'
^ Oxford Dictionary, 2nd Edition. There is no evidence in support of the hypothesis that x is derived ultimately from the mediaeval transliteration xei of shei "thing", used by the Arabs to denote the unknown quantity, or from the compendium for L. res "thing" or radix "root" (resembling a loosely-written x), used by mediaeval mathematicians.

External links

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

[1] Moore, Terry. "Why is X the Unknown". Ted Talk.

[2] Cajori, Florian (1993). A History of Mathematical Notation. Courier Dover Publications. pp. 382–383. Retrieved 11 October 2012. Nor is there historical evidence to support the statement found in Noah Webster's Dictionary, under the letter x, to the effect that 'x was used as an abbreviation of Ar. shei (a thing), something, which, in the Middle Ages, was used to designate the unknown, and was then prevailingly transcribed as xei.'

[3] Oxford Dictionary, 2nd Edition. There is no evidence in support of the hypothesis that x is derived ultimately from the mediaeval transliteration xei of shei "thing", used by the Arabs to denote the unknown quantity, or from the compendium for L. res "thing" or radix "root" (resembling a loosely-written x), used by mediaeval mathematicians.

[1]

[2]

[3]