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[edit]

  • 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[edit]

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[edit]

There are three numeral systems used in right to left mathematical notation.

Table of numerals
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. 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.

Mirrored Latin symbols[edit]

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

Arabic mathematical limit in different forms
Latin Arabic Persian
lim x→∞ x4 ‏نهـــــــا ‏س‏←‏∞‏ ‏س٤ [a] ‏حـــــــد ‏س‏←‏∞‏ ‏س۴ [b]
  • ^a نهـــا 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 symbols are used in Arabic mathematical notation (especially in western Arabic regions):

Arabic mathematical sum in different forms
Latin Arabic Mirrored Latin
n x=0 3x ‏ں ‏مجـــ ‏س‏=‏٠‏٣س[c] ‏ں‏س‏=‏0‏3س
  • ^c مجـــ mīm-medial form of ǧīm is derived from the first two letters of Arabic مجموع maǧmūʿ "sum".

However, in Iran, usually Latin symbols are used.

Examples[edit]

Mathematical letters[edit]

Latin Arabic Notes
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
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
Arabic mathematical geem.PNG حـ From the initial form of ح ḥāʾ, or that of a dotless ج ǧīm; c and ج ǧīm are the third letters of the Latin alphabet and the ʾabjadī sequence respectively
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
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.[1] (X was used in old Spanish for the sound /ʃ/). However, according to others there is no historical evidence for this.[2][3]
Arabic mathematical sad.PNG ص From the Arabic letter ص ṣād
Arabic mathematical ain.PNG ع From the Arabic letter ع ʿayn

Mathematical constants and units[edit]

Description Latin Arabic Notes
Euler's number 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 Phoenician he.svg .
imaginary unit 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 Arabic mathematical tah.PNG ط From ط ṭāʾ; also in some regions
radius 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 Arabic kilogram.PNG كجم From كجم kāf-ǧīm-mīm. In some regions alternative symbols like Arabic alternative kilogram 2.PNG ( كغ kāf-ġayn) or Arabic alternative kilogram 1.PNG ( كغم kāf-ġayn-mīm) are used. All three abbreviations are derived from كيلوغرام kīlūġrām "kilogram" and its variant spellings.
gram g Arabic gram.PNG جم From جم ǧīm-mīm, which is in turn derived from جرام ǧrām, a variant spelling of غرام ġrām "gram"
meter m Arabic mathematical meem.PNG م From م mīm, which is in turn derived from متر mitr "meter"
centimeter cm Arabic cm.PNG سم From سم sīn-mīm, which is in turn derived from سنتيمتر "centimeter"
millimeter mm Arabic mm.PNG مم From مم mīm-mīm, which is in turn derived from مليمتر millīmitr "millimeter"
kilometer km Arabic Km.PNG كم From كم kāf-mīm; also Arabic alternative km.PNG ( كلم kāf-lām-mīm) in some regions; both are derived from كيلومتر kīlūmitr "kilometer".
second s Arabic mathematical theh.PNG ث From ث ṯāʾ, which is in turn derived from ثانية ṯāniya "second"
minute min Arabic mathematical Dal large.PNG د From د dālʾ, which is in turn derived from دقيقة daqīqa "minute"; also Arabic mathematical qaf.PNG ( ٯ , i.e. dotless ق qāf) in some regions
hour h Arabic mathematical seen.PNG س From س sīnʾ, which is in turn derived from ساعة sāʿa "hour"
kilometer per hour km/h Arabic kmph.PNG كم/س From the symbols for kilometer and hour
degree Celsius °C 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 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 Arabic fahrenheit degree.PNG ف° From ف fāʾ, which is in turn derived from the second word of درجة فهرنهايت darajat fahranhāyt "degree Fahrenheit"
millimeters of mercury mmHg Arabic mmHg.PNG مم ز From مم ز mīm-mīm zayn, which is in turn derived from the initial letters of the words مليمتر زئبق "millimeters of mercury"
Ångström Å Arabic angestrom.PNG أ̊ From أ̊ 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[edit]

Description Latin Arabic Notes
Natural numbers Arabic mathematical tah large.PNG ط From ط ṭāʾ, which is in turn derived from the first letter of the second word of عدد طبيعي ʿadadun ṭabīʿiyyun "natural number"
Integers 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 Arabic mathematical noon large.PNG ن From ن nūn, which is in turn derived from the first letter of نسبة nisba "ratio"
Real numbers 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 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 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 markabun "complex number"
Empty set
Is an element of A mirrored ∈
Subset A mirrored ⊂
Superset A mirrored ⊃
Universal set 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 maǧmūʿaẗun šāmila "universal set"

Arithmetic and algebra[edit]

Description Latin Arabic Notes
Percent  % Arabic percent.PNG ٪ e.g. 100% " ٪١٠٠ "
Permille Arabic permille.PNG ؉ ؊ is an Arabic equivalent of the per ten thousand sign ‱.
Is proportional to Arabic prop.PNG A mirrored ∝
n th root Arabic mathematical nth root.PNG ں  ں is a dotless ن nūn while is a mirrored radical sign √
Logarithm Arabic mathematical log.PNG لو From لو lām-nūn, which is in turn derived from لوغاريتم lūġārītum "logarithm"
Logarithm to base b Arabic mathematical log b.PNG لوٮ
Natural logarithm Arabic mathematical ln.PNG لوھ From the symbols of logarithm and Euler's number
Summation Arabic mathematical sum.PNG مجـــ مجـــ mīm-medial form of ǧīm is derived from the first two letters of مجموع maǧmūʿ "sum"; also Arabic mathematical mirrored sum.PNG (, a mirrored summation sign ∑) in some regions
Product Arabic mathematical product.PNG جـــذ From جذ ǧīm-ḏāl. The Arabic word for "product" is جداء ǧadāʾun. Also (, a mirrored product sign ∏) in some regions.
Factorial Arabic mathematical factorial.PNG ں| Also Arabic mathematical fact.PNG ( ) in some regions
Permutations Arabic mathematical nPr.PNG ںلر Also Arabic mathematical P(n,r).PNG ( ل(‏ں،ر)‏ ) is used in some regions as
Combinations Arabic mathematical nCk.PNG ںٯك Also Arabic mathematical C(n,k).PNG ( ٯ(‏ں،ك)‏ ) is used in some regions as and Arabic mathematical b(n,k).PNG ( (ں
ك
) ) as the binomial coefficient

Trigonometric and hyperbolic functions[edit]

Trigonometric functions[edit]

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

Hyperbolic functions[edit]

The letter 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 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
Arabic حاز حتاز طاز طتاز ٯاز ٯتاز

Inverse trigonometric functions[edit]

For inverse trigonometric functions, the superscript −١ in Arabic notation is similar in usage to the superscript in Latin-based notation.

Arabic inverse trigonometric functions
Description Inverse sine Inverse cosine Inverse tangent Inverse cotangent Inverse secant Inverse cosecant
Latin
Arabic حا‏−١ حتا‏−١ طا‏−١ طتا‏−١ ٯا‏−١ ٯتا‏−١

Inverse hyperbolic functions[edit]

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 حاز‏−١ حتاز‏−١ طاز‏−١ طتاز‏−١ ٯاز‏−١ ٯتاز‏−١

Calculus[edit]

Description Latin Arabic Notes
Limit Arabic mathematical limit.PNG نهـــا نهـــا nūn-hāʾ-ʾalif is derived from the first three letters of Arabic نهاية nihāya "limit"
function Arabic mathematical f(x).PNG د(‏س)‏ د dāl is derived from the first letter of دالة "function"
derivatives Arabic mathematical derivatives.PNG د‏‵‏(‏س)‏، ‏د‌ص‏/ د‌س‏ ‏، ‏د‏٢‏ص‏/ د‌س‏٢ ‏،‏ ‏∂‏ص‏/ ‏∂‏س‏ ‵ is a mirrored prime ′ while ، is an Arabic comma. The signs should be mirrored: .
Integrals Arabic mathematical integrals.PNG ‏، ‏‏‏، ‏‏‏، ‏‏ Mirrored ∫, ∬, ∭ and ∮

Complex analysis[edit]

Latin Arabic
Arabic mathematical complex analysis.PNG
ع =‏ س +‏ ت‏ ص =‏ ل(‏ حتا‏ ى‏ +‏ ت‏ حا‏ ى‏ )‏ =‏ ل‌ھ ت‌ى‏ =‏ لى

See also[edit]

References[edit]

  1. ^ Moore, Terry. "Why is X the Unknown". Ted Talk. 
  2. ^ Cajori, Florian. 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. 

External links[edit]