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 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 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 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)	٠	١	٢	٣	٤	٥	٦	٧	٨	٩
Perso-Arabic variant	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Urdu variant	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹

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.^[1] The symbols "٫" and "٬" may be used as the decimal mark and the thousands separator respectively when writing with Eastern Arabic numerals, e.g. ٣٫١٤١٥٩٢٦٥٣٥٨ 3.14159265358, ١٬٠٠٠٬٠٠٠٬٠٠٠ 1,000,000,000. Negative signs are written to the left of magnitudes, e.g. ٣− −3. In-line fractions are written with the numerator and denominator on the left and right of the fraction slash respectively, e.g. ٢/٧ 2/7.^{[citation needed]}

Symbols

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

Latin	Arabic	Persian
Arabic mathematical limit in different forms
$lim x \to\infty x 4$	$س ٤ نهــــــــــــا س\leftarrow\infty‏$ ^[a]	$س ۴ حــــــــــــد س\leftarrow\infty‏$ ^[b]

^a نهــــا nūn-hāʾ-ʾalif is derived from the first three letters of Arabic نهاية nihāya "limit".
^b حد 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
Arabic mathematical sum in different forms
$n \sum x =0 3 \sqrt x$	$٣ ‭\sqrt‬ س ں مجــــــــــــ س=٠$ ^[c]	$‪ \sqrt 3 ‬ س ں ‭\sum‬ س=0$

^c مجــــ is derived from Arabic مجموع maǧmūʿ "sum".

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

Examples

Mathematical letters

Latin	Arabic		Notes
$a$	File:Arabic mathematical alif.PNG	$ا$	From the Arabic letter ا ʾalif; a and ا 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	$ٮ$	A dotless ب bāʾ; b and ب bāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively
$c$	File:Arabic mathematical geem.PNG	$حــــ$	From the initial form of ح ḥāʾ, or that of a dotless ج jīm; c and ج 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	$د$	From the Arabic letter د dāl; d and د 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	$س$	From the Arabic letter س sīn. It is contested that the usage of Latin x in maths is derived from the first letter ش šīn (without its dots) of the Arabic word شيء šayʾ(un) [ʃajʔ(un)], meaning thing.^[2] (X was used in old Spanish for the sound /ʃ/). However, according to others there is no historical evidence for this.^[3]^[4]
$y$	File:Arabic mathematical sad.PNG	$ص$	From the Arabic letter ص ṣād
$z$	File:Arabic mathematical ain.PNG	$ع$	From the Arabic letter ع ʿayn

Mathematical constants and units

Description	Latin	Arabic		Notes
Euler's number	$e$	File:Arabic mathematical heh.PNG	$ھ$	Initial form of the Arabic letter ه hāʾ. Both Latin letter e and Arabic letter ه hāʾ are descendants of Phoenician letter File:Phoenician he.svg hē.
imaginary unit	$i$	File:Arabic mathematical teh.PNG	$ت$	From ت tāʾ, which is in turn derived from the first letter of the second word of وحدة تخيلية waḥdaẗun taḫīliyya "imaginary unit"
pi	$π$	File:Arabic mathematical tah.PNG	$ط$	From ط ṭāʾ; also $π$ in some regions
radius	$r$	File:Arabic mathematical radius.PNG	$نٯ$	From ن nūn followed by a dotless ق qāf, which is in turn derived from نصف القطر nuṣfu l-quṭr "radius"
kilogram	kg	File:Arabic kilogram.PNG	$كجم$	From كجم kāf-jīm-mīm. In some regions alternative symbols like File:Arabic alternative kilogram 2.PNG ( $كغ$ kāf-ġayn) or File:Arabic alternative kilogram 1.PNG ( $كلغ$ kāf-lām-ġayn) are used. All three abbreviations are derived from كيلوغرام kīlūġrām "kilogram" and its variant spellings.
gram	g	File:Arabic gram.PNG	$جم$	From جم jīm-mīm, which is in turn derived from جرام jrām, a variant spelling of غرام ġrām "gram"
metre	m	File:Arabic mathematical meem.PNG	$م$	From م mīm, which is in turn derived from متر mitr "metre"
centimetre	cm	File:Arabic cm.PNG	$سم$	From سم sīn-mīm, which is in turn derived from سنتيمتر "centimetre"
millimetre	mm	File:Arabic mm.PNG	$مم$	From مم mīm-mīm, which is in turn derived from مليمتر millīmitr "millimetre"
kilometre	km	File:Arabic Km.PNG	$كم$	From كم kāf-mīm; also File:Arabic alternative km.PNG ( $كلم$ kāf-lām-mīm) in some regions; both are derived from كيلومتر kīlūmitr "kilometre".
second	s	File:Arabic mathematical theh.PNG	$ث$	From ث ṯāʾ, which is in turn derived from ثانية ṯāniya "second"
minute	min	File:Arabic mathematical Dal large.PNG	$د$	From د dālʾ, which is in turn derived from دقيقة daqīqa "minute"; also File:Arabic mathematical qaf.PNG ( $ٯ$ , i.e. dotless ق qāf) in some regions
hour	h	File:Arabic mathematical seen.PNG	$س$	From س sīnʾ, which is in turn derived from ساعة sāʿa "hour"
kilometre per hour	km/h	File:Arabic kmph.PNG	$كم/س$	From the symbols for kilometre and hour
degree Celsius	°C	File:Arabic celsius degree.PNG	$°س$	From س sīn, which is in turn derived from the second word of درجة سيلسيوس darajat sīlsīūs "degree Celsius"; also File:Arabic centegrade degree.PNG ( $°م$ ) from م mīmʾ, which is in turn derived from the first letter of the third word of درجة حرارة مئوية "degree centigrade"
degree Fahrenheit	°F	File:Arabic fahrenheit degree.PNG	$°ف$	From ف fāʾ, which is in turn derived from the second word of درجة فهرنهايت darajat fahranhāyt "degree Fahrenheit"
millimetres of mercury	mmHg	File:Arabic mmHg.PNG	$مم‌ز$	From مم‌ز mīm-mīm zayn, which is in turn derived from the initial letters of the words مليمتر زئبق "millimetres of mercury"
Ångström	Å	File:Arabic angestrom.PNG	$أْ$	From أْ ʾalif with hamzah and ring above, which is in turn derived from the first letter of "Ångström", variously spelled أنغستروم or أنجستروم

Sets and number systems

Description	Latin	Arabic		Notes
Natural numbers	$ℕ$	File:Arabic mathematical tah large.PNG	$ط$	From ط ṭāʾ, which is in turn derived from the first letter of the second word of عدد طبيعيʿadadun ṭabīʿiyyun "natural number"
Integers	$ℤ$	File:Arabic mathematical Sad large.PNG	$ص$	From ص ṣād, which is in turn derived from the first letter of the second word of عدد صحيح ʿadadun ṣaḥīḥun "integer"
Rational numbers	$ℚ$	File:Arabic mathematical noon large.PNG	$ن$	From ن nūn, which is in turn derived from the first letter of نسبة nisba "ratio"
Real numbers	$ℝ$	File:Arabic mathematical hah large.PNG	$ح$	From ح ḥāʾ, which is in turn derived from the first letter of the second word of عدد حقيقي ʿadadun ḥaqīqiyyun "real number"
Imaginary numbers	$𝕀$	File:Arabic mathematical teh large.PNG	$ت$	From ت tāʾ, which is in turn derived from the first letter of the second word of عدد تخيلي ʿadadun taḫīliyyun "imaginary number"
Complex numbers	$ℂ$	File:Arabic mathematical meem large.PNG	$م$	From م mīm, which is in turn derived from the first letter of the second word of عدد مركب ʿadadun murakkabun "complex number"
Empty set	$\emptyset$	$\emptyset$	∅
Is an element of	$\in$	$∋$	∈	A mirrored ∈
Subset	$\subset$	$\supset$	⊂	A mirrored ⊂
Superset	$\supset$	$\subset$	⊃	A mirrored ⊃
Universal set	$𝐒$	File:Arabic mathematical sheen large.PNG	$ش$	From ش šīn, which is in turn derived from the first letter of the second word of مجموعة شاملة majmūʿatun šāmila "universal set"

Arithmetic and algebra

Description	Latin/Greek	Arabic		Notes
Percent	%	File:Arabic percent.PNG	$٪$	e.g. 100% "٪١٠٠"
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	$ں ‭\sqrt‬ ‏$	ں is a dotless ن nūn while √ is a mirrored radical sign √
Logarithm	$\log$	File:Arabic mathematical log.PNG	$لو$	From لو lām-wāw, which is in turn derived from لوغاريتم lūġārītm "logarithm"
Logarithm to base b	$\log_{b}$	File:Arabic mathematical log b.PNG	$لو ٮ$
Natural logarithm	$\ln$	File:Arabic mathematical ln.PNG	$لو ھ$	From the symbols of logarithm and Euler's number
Summation	$\sum$	File:Arabic mathematical sum.PNG	$مجــــ$	مجـــ mīm-medial form of jīm is derived from the first two letters of مجموع majmūʿ "sum"; also File:Arabic mathematical mirrored sum.PNG (∑, a mirrored summation sign ∑) in some regions
Product	$\prod$	File:Arabic mathematical product.PNG	$جــــذ$	From جذ jīm-ḏāl. The Arabic word for "product" is جداء jadāʾun. Also $\prod$ in some regions.
Factorial	$n!$	File:Arabic mathematical factorial.PNG	$ں$	Also File:Arabic mathematical fact.PNG ( $ں!$ ) in some regions
Permutations	$^{n} 𝐏_{r}$	File:Arabic mathematical nPr.PNG	$ں ل ر$	Also File:Arabic mathematical P(n,r).PNG ( $ل(ں، ر)$ ) is used in some regions as $𝐏 (n, r)$
Combinations	$^{n} 𝐂_{k}$	File:Arabic mathematical nCk.PNG	$ں ٯ ك$	Also File:Arabic mathematical C(n,k).PNG ( $ٯ(ں، ك)$ ) is used in some regions as $𝐂 (n, k)$ and File:Arabic mathematical b(n,k).PNG ( $⎛ ⎝ ں ك ⎞ ⎠$ ) as the binomial coefficient $(\binom{n}{k})$

Trigonometric and hyperbolic functions

Trigonometric functions

Description	Latin	Arabic		Notes
Sine	$\sin$	File:Arabic mathematical sin.PNG	$حا$	from حاء ḥāʾ (i.e. dotless ج jīm)-ʾalif; also File:Arabic mathematical sins.PNG ( $جب$ jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "sine" is جيب jayb
Cosine	$\cos$	File:Arabic mathematical cos.PNG	$حتا$	from حتا ḥāʾ (i.e. dotless ج jīm)-tāʾ-ʾalif; also File:Arabic mathematical coss.PNG ( $تجب$ tāʾ-jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "cosine" is جيب تمام
Tangent	$\tan$	File:Arabic mathematical tan.PNG	$طا$	from طا ṭāʾ (i.e. dotless ظ ẓāʾ)-ʾalif; also File:Arabic mathematical tans.PNG ( $ظل$ ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "tangent" is ظل ẓill
Cotangent	$\cot$	File:Arabic mathematical cot.PNG	$طتا$	from طتا ṭāʾ (i.e. dotless ظ ẓāʾ)-tāʾ-ʾalif; also File:Arabic mathematical cots.PNG ( $تظل$ tāʾ-ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "cotangent" is ظل تمام
Secant	$\sec$	File:Arabic mathematical sec.PNG	$ٯا$	from ٯا dotless ق qāf-ʾalif; Arabic for "secant" is قاطع
Cosecant	$\csc$	File:Arabic mathematical csc.PNG	$ٯتا$	from ٯتا dotless ق qāf-tāʾ-ʾalif; Arabic for "cosecant" is قاطع تمام

Hyperbolic functions

The letter File:Arabic mathematical zain.PNG ( $ز$ zayn, from the first letter of the second word of دالة زائدية "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to the way $h$ 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	$\sinh$	$\cosh$	$\tanh$	$\coth$	$sech$	$csch$
Arabic	$حاز$	$حتاز$	$طاز$	$طتاز$	$ٯاز$	$ٯتاز$

Inverse trigonometric functions

For inverse trigonometric functions, the superscript $-١$ 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	$حا -١$	$حتا -١$	$طا -١$	$طتا -١$	$ٯا -١$	$ٯتا -١$

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	$\sinh^{- 1}$	$\cosh^{- 1}$	$\tanh^{- 1}$	$\coth^{- 1}$	${sech}^{- 1}$	${csch}^{- 1}$
Arabic	$حاز -١$	$حتاز -١$	$طاز -١$	$طتاز -١$	$ٯاز -١$	$ٯتاز -١$

Calculus

Description	Latin	Arabic		Notes
Limit	$\lim$	File:Arabic mathematical limit.PNG	$نهــــا$	نهــــا nūn-hāʾ-ʾalif is derived from the first three letters of Arabic نهاية nihāya "limit"
Function	$𝐟 (x)$	File:Arabic mathematical f(x).PNG	$د(س)$	د dāl is derived from the first letter of دالة "function". Also called تابع, تا for short, in some regions.
Derivatives	$𝐟^{'} (x), \frac{d y}{d x}, \frac{d^{2} y}{d x^{2}}, \frac{\partial y}{\partial x}$	File:Arabic mathematical derivatives.PNG	$.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠ص∂/س∂⁠ ،⁠د٢ص/ د‌س٢ ⁠ ،⁠د‌ص/ د‌س ⁠ ،(س)‵د$	‵ is a mirrored prime ′ while ، is an Arabic comma. The $\partial$ signs should be mirrored: ∂.
Integrals	$\int, \iint, ∭, \oint$	File:Arabic mathematical integrals.PNG	‪∮ ،∭ ،∬ ،∫‬	Mirrored ∫, ∬, ∭ and ∮

Complex analysis

Latin/Greek	Arabic
$z = x + i y = r (\cos φ + i \sin φ) = r e^{i φ} = r ∠ φ$	File:Arabic mathematical complex analysis.PNG
$z = x + i y = r (\cos φ + i \sin φ) = r e^{i φ} = r ∠ φ$	$ع = س + ت ص = ل(حتا ى + ت حا ى) = ل ھ ت‌ى = ل ∠ o$

References

^ Oaks (2012). "Algebraic symbolism in medieval Arabic algebra" Philosophia, 87 27--83.
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

External links

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

[1] Oaks (2012). "Algebraic symbolism in medieval Arabic algebra" Philosophia, 87 27--83.

[2] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[3] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[4] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

[1]

[2]

[3]

[4]

Modern Arabic mathematical notation

Contents

Features

Variations

Numeral systems

Symbols

Examples

Mathematical letters

Mathematical constants and units

Sets and number systems

Arithmetic and algebra

Trigonometric and hyperbolic functions

Trigonometric functions

Hyperbolic functions

Inverse trigonometric functions

Inverse hyperbolic functions

Calculus

Complex analysis

See also

References

External links

Navigation menu

Modern Arabic mathematical notation

Features

Variations

Numeral systems

Symbols

Examples

Mathematical letters

Mathematical constants and units

Sets and number systems

Arithmetic and algebra

Trigonometric and hyperbolic functions

Trigonometric functions

Hyperbolic functions

Inverse trigonometric functions

Inverse hyperbolic functions

Calculus

Complex analysis

See also

References

External links

Navigation menu

Search