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 45 6 7 8 9Arabic-Indic (Eastern Arabic) Perso-Arabic variant 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: 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.Oaks (2012). "Algebraic symbolism in medieval Arabic algebra" Philosophia, 87 27--83. 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:

Latin Arabic Persian

nūn-hāʾ-ʾalif is derived from the first three letters of Arabic nihāya "limit". 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

is derived from Arabic maǧmūʿ "sum".

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

Examples

Mathematical letters

Latin Arabic Notes 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 A dotless bāʾ; b and bāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively 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 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 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. From the Arabic letter ṣād From the Arabic letter ʿayn

Mathematical constants and units Description Latin Arabic Notes Euler's number Initial form of the Arabic letter hāʾ. Both Latin letter e and Arabic letter hāʾ are descendants of Phoenician letter hē. imaginary unit From tāʾ, which is in turn derived from the first letter of the second word of waḥdaẗun taḫīliyya "imaginary unit" pi From ṭāʾ; also in some regions radius From nūn followed by a dotless qāf, which is in turn derived from nuṣfu l-quṭr "radius" kilogram kg From kāf-jīm-mīm. In some regions alternative symbols like ( kāf-ġayn) or ( kāf-lām-ġayn) are used. All three abbreviations are derived from kīlūġrām "kilogram" and its variant spellings. gram g From jīm-mīm, which is in turn derived from jrām, a variant spelling of ġrām "gram" metre m From mīm, which is in turn derived from mitr "metre" centimetre cm From sīn-mīm, which is in turn derived from "centimetre" millimetre mm From mīm-mīm, which is in turn derived from millīmitr "millimetre" kilometre km From kāf-mīm; also ( kāf-lām-mīm) in some regions; both are derived from kīlūmitr "kilometre". second s From ṯāʾ, which is in turn derived from ṯāniya "second" minute min From dālʾ, which is in turn derived from daqīqa "minute"; also (, i.e. dotless qāf) in some regions hour h From sīnʾ, which is in turn derived from sāʿa "hour" kilometre per hour km/h From the symbols for kilometre and hour degree Celsius °C From sīn, which is in turn derived from the second word of darajat sīlsīūs "degree Celsius"; also () from mīmʾ, which is in turn derived from the first letter of the third word of "degree centigrade" degree Fahrenheit °F From fāʾ, which is in turn derived from the second word of darajat fahranhāyt "degree Fahrenheit" millimetres of mercury mmHg From mīm-mīm zayn, which is in turn derived from the initial letters of the words "millimetres of mercury" Ångström Å 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 From ṭāʾ, which is in turn derived from the first letter of the second word of ʿadadun ṭabīʿiyyun "natural number" Integers From ṣād, which is in turn derived from the first letter of the second word of ʿadadun ṣaḥīḥun "integer" Rational numbers From nūn, which is in turn derived from the first letter of nisba "ratio" Real numbers From ḥāʾ, which is in turn derived from the first letter of the second word of ʿadadun ḥaqīqiyyun "real number" Imaginary numbers From tāʾ, which is in turn derived from the first letter of the second word of ʿadadun taḫīliyyun "imaginary number" Complex numbers From mīm, which is in turn derived from the first letter of the second word of ʿadadun murakkabun "complex number" Empty set ∅ Is an element of ∈ A mirrored ∈ Subset ⊂ A mirrored ⊂ Superset ⊃ A mirrored ⊃ Universal set 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 % e.g. 100% "" Permille ‰ is an Arabic equivalent of the per ten thousand sign ‱. Is proportional to ∝ A mirrored ∝ n th root is a dotless nūn while √ is a mirrored radical sign √ Logarithm From lām-wāw, which is in turn derived from لوغاريتم "logarithm" Logarithm to base b Natural logarithm From the symbols of logarithm and Euler's number Summation mīm-medial form of jīm is derived from the first two letters of majmūʿ "sum"; also (∑, a mirrored summation sign ∑) in some regions Product From jīm-ḏāl. The Arabic word for "product" is جداء jadāʾun. Also in some regions. Factorial Also () in some regions Permutations Also () is used in some regions as Combinations Also () is used in some regions as and ( ) as the binomial coefficient

Trigonometric and hyperbolic functions

Trigonometric functions Description Latin Arabic Notes Sine from ḥāʾ (i.e. dotless jīm)-ʾalif; also ( jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "sine" is jayb Cosine from ḥāʾ (i.e. dotless jīm)-tāʾ-ʾalif; also ( tāʾ-jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "cosine" is Tangent from ṭāʾ (i.e. dotless ẓāʾ)-ʾalif; also ( ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "tangent" is ẓill Cotangent from ṭāʾ (i.e. dotless ẓāʾ)-tāʾ-ʾalif; also ( tāʾ-ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "cotangent" is Secant from dotless qāf-ʾalif; Arabic for "secant" is Cosecant from dotless qāf-tāʾ-ʾalif; Arabic for "cosecant" is

Hyperbolic functions The letter ( 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 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 Arabic

Inverse trigonometric functions For inverse trigonometric functions, the superscript in Arabic notation is similar in usage to the superscript in Latin-based notation. Description Inverse sine Inverse cosine Inverse tangent Inverse cotangent Inverse secant Inverse cosecant Latin Arabic

Inverse hyperbolic functions Description Inverse hyperbolic sine Inverse hyperbolic cosine Inverse hyperbolic tangent Inverse hyperbolic cotangent Inverse hyperbolic secant Inverse hyperbolic cosecant Latin Arabic

Calculus

Description Latin Arabic Notes Limit nūn-hāʾ-ʾalif is derived from the first three letters of Arabic nihāya "limit" Function dāl is derived from the first letter of "function". Also called , for short, in some regions. Derivatives ‵ is a mirrored prime ′ while ، is an Arabic comma. The signs should be mirrored: ∂. Integrals ‪∮ ،∭ ،∬ ،∫‬ Mirrored ∫, ∬, ∭ and ∮

Complex analysis Latin/Greek Arabic

See also Mathematical notation Arabic Mathematical Alphabetic Symbols

References

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