# Modern Arabic mathematical notation

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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.

 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

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. ٣٫١٤١٥٩٢٦٥٣٥٨3.14159265358, ١٬٠٠٠٬٠٠٠٬٠٠٠1,000,000,000. 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. ٢/٧2/7.

### Mirrored Latin symbols

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

Latin Arabic Persian x4 س٤‎ [a] س۴‎ [b]
• ^a نهــــاnūn-hāʾ-ʾalif is derived from the first three letters of Arabic نهايةnihāya "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 3√x ٣‭√‬س‎[c] 3‭√‬س‎
• ^c مجــــ

مجموعmaǧmūʿ means "sum" in Arabic language.

However, in Iran, usually Latin symbols are used.

## Examples

### Mathematical letters

Latin Arabic Notes
${\displaystyle a}$ ا From the Arabic letter اʾalif; a and اʾalif are the first letters of the Latin alphabet and the Arabic alphabet's ʾabjadī sequence respectively
${\displaystyle b}$ ٮ A dotless بbāʾ; b and بbāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively
${\displaystyle c}$ حــــ From the initial form of حḥāʾ, or that of a dotless جjīm; c and جjīm are the third letters of the Latin alphabet and the ʾabjadī sequence respectively
${\displaystyle d}$ د From the Arabic letter دdāl; d and دdāl are the fourth letters of the Latin alphabet and the ʾabjadī sequence respectively
${\displaystyle x}$ س From the Arabic letter سsīn. It is contested that the usage of Latin x in maths is derived from the first letter شšīn (without its dots) of the Arabic word شيءšayʾ(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]
${\displaystyle y}$ ص From the Arabic letter صṣād
${\displaystyle z}$ ع From the Arabic letter عʿayn

### Mathematical constants and units

Description Latin Arabic Notes
Euler's number ${\displaystyle e}$ ھ Initial form of the Arabic letter هhāʾ. Both Latin letter e and Arabic letter هhāʾ are descendants of Phoenician letter .
imaginary unit ${\displaystyle i}$ ت From تtāʾ, which is in turn derived from the first letter of the second word of وحدة تخيليةwaḥdaẗun taḫīliyya "imaginary unit"
pi ${\displaystyle \pi }$ ط From طṭāʾ; also ${\displaystyle \pi }$ in some regions
radius ${\displaystyle r}$ نٯ From نnūn followed by a dotless قqāf, which is in turn derived from نصف القطرnuṣfu l-quṭr "radius"
kilogram kg كجم From كجمkāf-jīm-mīm. In some regions alternative symbols like ( كغ kāf-ġayn) or ( كلغ kāf-lām-ġayn) are used. All three abbreviations are derived from كيلوغرامkīlūġrām "kilogram" and its variant spellings.
gram g جم From جمjīm-mīm, which is in turn derived from جرامjrām, a variant spelling of غرامġrām "gram"
meter m م From مmīm, which is in turn derived from مترmitr "meter"
centimeter cm سم From سمsīn-mīm, which is in turn derived from سنتيمتر‎ "centimeter"
millimeter mm مم From ممmīm-mīm, which is in turn derived from مليمترmillīmitr "millimeter"
kilometer km كم From كمkāf-mīm; also ( كلم kāf-lām-mīm) in some regions; both are derived from كيلومترkīlūmitr "kilometer".
second s ث From ثṯāʾ, which is in turn derived from ثانيةṯāniya "second"
minute min د From دdālʾ, which is in turn derived from دقيقةdaqīqa "minute"; also ( ٯ, i.e. dotless قqāf) in some regions
hour h س From سsīnʾ, which is in turn derived from ساعةsāʿa "hour"
kilometer per hour km/h كم/س From the symbols for kilometer and hour
degree Celsius °C °س From سsīn, which is in turn derived from the second word of درجة سيلسيوسdarajat sīlsīūs "degree Celsius"; also ( °م) from مmīmʾ, which is in turn derived from the first letter of the third word of درجة حرارة مئوية‎ "degree centigrade"
degree Fahrenheit °F °ف From فfāʾ, which is in turn derived from the second word of درجة فهرنهايتdarajat fahranhāyt "degree Fahrenheit"
millimeters of mercury mmHg مم‌ز From مم‌زmīm-mīm zayn, which is in turn derived from the initial letters of the words مليمتر زئبق‎ "millimeters of mercury"
Ångström Å أْ From أْʾalif 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 ${\displaystyle \mathbb }$ ط From طṭāʾ, which is in turn derived from the first letter of the second word of عدد طبيعيʿadadun ṭabīʿiyyun "natural number"
Integers ${\displaystyle \mathbb }$ ص From صṣād, which is in turn derived from the first letter of the second word of عدد صحيحʿadadun ṣaḥīḥun "integer"
Rational numbers ${\displaystyle \mathbb }$ ن From نnūn, which is in turn derived from the first letter of نسبةnisba "ratio"
Real numbers ${\displaystyle \mathbb }$ ح From حḥāʾ, which is in turn derived from the first letter of the second word of عدد حقيقيʿadadun ḥaqīqiyyun "real number"
Imaginary numbers ${\displaystyle \mathbb }$ ت From تtāʾ, which is in turn derived from the first letter of the second word of عدد تخيليʿadadun taḫīliyyun "imaginary number"
Complex numbers ${\displaystyle \mathbb }$ م From مmīm, which is in turn derived from the first letter of the second word of عدد مركبʿadadun markabun "complex number"
Empty set ${\displaystyle \varnothing }$ ${\displaystyle \varnothing }$
Is an element of ${\displaystyle \in }$ ${\displaystyle \ni }$ A mirrored ∈
Subset ${\displaystyle \subset }$ ${\displaystyle \supset }$ A mirrored ⊂
Superset ${\displaystyle \supset }$ ${\displaystyle \subset }$ A mirrored ⊃
Universal set ${\displaystyle \mathbf }$ ش From شšīn, which is in turn derived from the first letter of the second word of مجموعة شاملةmajmūʿatun šāmila "universal set"

### Arithmetic and algebra

Description Latin Arabic Notes
Percent % ٪ e.g. 100% "٪١٠٠‎"
Permille ؉ ؊ is an Arabic equivalent of the per ten thousand sign ‱.
Is proportional to ${\displaystyle \propto }$ A mirrored ∝
n th root ${\displaystyle {\sqrt[]{\,\,\,}}}$ ں‭√‬ ں‎ is a dotless نnūn while is a mirrored radical sign √
Logarithm ${\displaystyle \log }$ لو From لوlām-wāw, which is in turn derived from لوغاريتم lūġārītm "logarithm"
Logarithm to base b ${\displaystyle \log _}$ لوٮ
Natural logarithm ${\displaystyle \ln }$ لوھ From the symbols of logarithm and Euler's number
Summation ${\displaystyle \sum }$ مجــــ مجـــmīm-medial form of jīm is derived from the first two letters of مجموعmajmūʿ "sum"; also (, a mirrored summation sign ∑) in some regions
Product ${\displaystyle \prod }$ جــــذ From جذjīm-ḏāl. The Arabic word for "product" is جداء jadāʾun. Also ${\displaystyle \prod }$ in some regions.
Factorial ${\displaystyle n!}$ ں Also ( ں!) in some regions
Permutations ${\displaystyle ^\mathbf _}$ ںلر Also ( ل(ں، ر)) is used in some regions as ${\displaystyle \mathbf (n,r)}$
Combinations ${\displaystyle ^\mathbf _}$ ںٯك Also ( ٯ(ں، ك)) is used in some regions as ${\displaystyle \mathbf (n,k)}$ and (ں
ك
) as the binomial coefficient ${\displaystyle n \choose k}$

### Trigonometric and hyperbolic functions

#### Trigonometric functions

Description Latin Arabic Notes
Sine ${\displaystyle \sin }$ حا from حاءḥāʾ (i.e. dotless جjīm)-ʾalif; also ( جب jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "sine" is جيبjayb
Cosine ${\displaystyle \cos }$ حتا from حتاḥāʾ (i.e. dotless جjīm)-tāʾ-ʾalif; also ( تجب tāʾ-jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "cosine" is جيب تمام
Tangent ${\displaystyle \tan }$ طا from طاṭāʾ (i.e. dotless ظẓāʾ)-ʾalif; also ( ظل ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "tangent" is ظلẓill
Cotangent ${\displaystyle \cot }$ طتا from طتاṭāʾ (i.e. dotless ظẓāʾ)-tāʾ-ʾalif; also ( تظل tāʾ-ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "cotangent" is ظل تمام
Secant ${\displaystyle \sec }$ ٯا from ٯا‎ dotless قqāf-ʾalif; Arabic for "secant" is أو قاطع
Cosecant ${\displaystyle \csc }$ ٯتا from ٯتا‎ dotless قqāf-tāʾ-ʾalif; Arabic for "cosecant" is أو قاطع تمام

#### Hyperbolic functions

The letter ( 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 ${\displaystyle \operatorname }$ is added to the end of trigonometric functions in Latin-based notation.

 Description Latin Arabic Hyperbolic sine Hyperbolic cosine Hyperbolic tangent Hyperbolic cotangent Hyperbolic secant Hyperbolic cosecant ${\displaystyle \sinh }$ ${\displaystyle \cosh }$ ${\displaystyle \tanh }$ ${\displaystyle \coth }$ ${\displaystyle \operatorname }$ ${\displaystyle \operatorname }$ 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 ${\displaystyle -1}$ in Latin-based notation.

 Description Latin Arabic Inverse sine Inverse cosine Inverse tangent Inverse cotangent Inverse secant Inverse cosecant ${\displaystyle \sin ^{-1}}$ ${\displaystyle \cos ^{-1}}$ ${\displaystyle \tan ^{-1}}$ ${\displaystyle \cot ^{-1}}$ ${\displaystyle \sec ^{-1}}$ ${\displaystyle \csc ^{-1}}$ حا−١‎ حتا−١‎ طا−١‎ طتا−١‎ ٯا−١‎ ٯتا−١‎

#### Inverse hyperbolic functions

 Description Latin Arabic Inverse hyperbolic sine Inverse hyperbolic cosine Inverse hyperbolic tangent Inverse hyperbolic cotangent Inverse hyperbolic secant Inverse hyperbolic cosecant ${\displaystyle \sinh ^{-1}}$ ${\displaystyle \cosh ^{-1}}$ ${\displaystyle \tanh ^{-1}}$ ${\displaystyle \coth ^{-1}}$ ${\displaystyle \operatorname ^{-1}}$ ${\displaystyle \operatorname ^{-1}}$ حاز−١‎ حتاز−١‎ طاز−١‎ طتاز−١‎ ٯاز−١‎ ٯتاز−١‎

### Calculus

Description Latin Arabic Notes
Limit ${\displaystyle \lim }$ نهــــا نهــــاnūn-hāʾ-ʾalif is derived from the first three letters of Arabic نهايةnihāya "limit"
function ${\displaystyle \mathbf (x)}$ د(س) دdāl is derived from the first letter of دالة‎ "function". Also called تابع‎, تا‎ for short, in some regions.
derivatives ${\displaystyle \mathbf (x),{\dfrac },{\dfrac y}}},{\dfrac {\partial }{\partial }}}$ د‵(س)، د‌ص/ د‌س ، د٢ص/ د‌س٢ ، ص/س ‵ is a mirrored prime ′ while ، is an Arabic comma. The signs should be mirrored: .
Integrals ${\displaystyle \int {},\iint {},\iiint {},\oint {}}$ ، ، ، Mirrored ∫, ∬, ∭ and ∮

### Complex analysis

Latin Arabic
${\displaystyle z=x+iy=r(\cos {\varphi }+i\sin {\varphi })=re^=r\angle {\varphi }}$
ع = س + ت ص = ل(حتا ى + ت حا ى) = ل ھت‌ى = لى