View source for Modern Arabic mathematical notation

{{Short description|Mathematical notation based on the Arabic script}}
'''Modern Arabic mathematical notation''' is a [[Mathematical notation|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 '''{{lang|ar|نق}}''' ({{IPA-ar|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 [[Social:Tertiary education|tertiary education]], most regions use the [[Mathematical notation|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. [[Place:Egypt|Egypt]] and [[Place:Syria|Syria]])
*"Eastern Arabic-Indic numerals" are used in [[Social:Persian language|Persian]] and [[Social:Urdu|Urdu]] speaking regions (e.g. [[Religion:Iran|Iran]], {{wipe|Pakistan}}, {{wipe|India}})

{|class="wikitable center" style="line-height:normal" 
|- 
| European<br /><small>(descended from Western Arabic)</small>
|0 ||1 || 2 || 3 || 4
|5 || 6 || 7 || 8 || 9
|- style="font-size:160%"
| style="font-size:63%"|Arabic-Indic <small>(Eastern Arabic)</small><!-- U+0660 thru U+0669 -->
| {{lang|ar|٠|italic=no}} || {{lang|ar|١|italic=no}} || {{lang|ar|٢|italic=no}} || {{lang|ar|٣|italic=no}} || {{lang|ar|٤|italic=no}}
| {{lang|ar|٥|italic=no}} || {{lang|ar|٦|italic=no}} || {{lang|ar|٧|italic=no}} || {{lang|ar|٨|italic=no}} || {{lang|ar|٩|italic=no}}
|- style="font-size:160%"
| style="font-size:63%"| [[Social:Persian alphabet|Perso-Arabic variant]]<!-- U+06F0 thru U+06F9 -->
| {{lang|fa|۰|italic=no}} || {{lang|fa|۱|italic=no}} || {{lang|fa|۲|italic=no}} || {{lang|fa|۳|italic=no}} || {{lang|fa|۴|italic=no}}
| {{lang|fa|۵|italic=no}} || {{lang|fa|۶|italic=no}} || {{lang|fa|۷|italic=no}} || {{lang|fa|۸|italic=no}} || {{lang|fa|۹|italic=no}}
|- style="font-size:160%"
| style="font-size:63%"| Urdu variant
| {{Urdu numeral|0|23}} 
| {{Urdu numeral|1|23}} 
| {{Urdu numeral|2|23}} 
| {{Urdu numeral|3|23}} 
| {{Urdu numeral|4|23}}
| {{Urdu numeral|5|23}}
| {{Urdu numeral|6|23}}
| {{Urdu numeral|7|23}}
| {{Urdu numeral|8|23}}
| {{Urdu numeral|9|23}}
|}
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. {{lang|ar| ٣٫١٤١٥٩٢٦٥٣٥٨ }} ''3.14159265358'', {{lang|ar| ١٬٠٠٠٬٠٠٠٬٠٠٠ }} ''1,000,000,000''. Negative signs are written to the left of magnitudes, e.g. {{lang|ar| ٣− }} ''−3''. In-line fractions are written with the numerator and denominator on the left and right of the fraction slash respectively, e.g. {{lang|ar| ٢/٧ }} ''2/7''.

===Mirrored Latin symbols===

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

{| class="wikitable"
|-
| colspan=3 |[[Image:Difference between arabic and persian maths limit.PNG|500px|Arabic mathematical limit in different forms]]
|-
! style="width: 150px;" | Latin
! style="width: 150px;" | Arabic
! style="width: 150px;" | Persian
|-
| {{math| {{underset| ''x''→∞ | lim }} ''x''<sup>4</sup> }}
| {{math| {{lang|rtl=yes|ar-Zmth|س<sup>٤</sup> {{underset| س←∞&rlm; |نهــــــــــــا}} }} }}{{ref|a|[a]}}
| {{math| {{lang|rtl=yes|fa-Zmth|س<sup>۴</sup> {{underset| س←∞&rlm; | حــــــــــــد }} }} }}{{ref|b|[b]}}
|}
{{refbegin}}
* {{note|a|a}} {{lang|ar-Zmth|نهــــا}} ''nūn''-''hāʾ''-''ʾalif'' is derived from the first three letters of [[Social:Arabic language|Arabic]] {{lang|ar|نهاية}}  ''nihāya'' "limit".
* {{note|b|b}} {{lang|fa|حد}} ''ḥadd'' is [[Social:Persian language|Persian]] for "limit".
{{refend}}

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

{| class="wikitable"
|-
| colspan=3 |[[Image:Difference between arabic mathematical sum forms.PNG|500px|Arabic mathematical sum in different forms]]
|-
! style="width: 150px;" | Latin
! style="width: 150px;" | Arabic
! style="width: 150px;" | Mirrored Latin
|-
| {{math| {{underoverset| ∑ | ''n'' | ''x''{{=}}0 }} {{radic|''x''|3}} }}
| {{math|1={{lang|rtl=yes|ar-Zmth|2=<sup style="margin-left: -0.5em; vertical-align: 0.8em;">٣</sup><span style="{{mirrorH}}">&#x202d;√&#x202c;</span>{{overline|س}} {{underoverset| مجــــــــــــ | ں | س{{=}}٠ }} }}}}{{ref|c|[c]}}
| {{math|1={{lang|rtl=yes|ar-Zmth|2=&#x202a;<span style="{{mirrorH}}">√</span><sup style="margin-left: -0.5em; vertical-align: 0.8em;">3</sup>&#x202c;{{overline|س}} {{underoverset|1=<span style="{{mirrorH}}">&#x202d;∑&#x202c;</span> |2=ں |3= س{{=}}0 }} }}}}
|}
{{refbegin}}
* {{note|c|c}} {{lang|ar|مجــــ}} is derived from [[Social:Arabic language|Arabic]] {{lang|ar|مجموع}} ''maǧmūʿ'' "sum".
{{refend}}

However, in Iran, usually Latin symbols are used.

==Examples==

===Mathematical letters===

{| class="wikitable"
|-
! Latin
! colspan=2 | Arabic
! Notes
|-
| <math>a</math>
| [[Image:Arabic mathematical alif.PNG|12px]] ||  {{math| {{lang|ar|ا}} }} || From the Arabic letter {{lang|ar|ا}} ''ʾalif''; ''a'' and {{lang|ar|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
|-
| <math>b</math>
| [[Image:Arabic mathematical beh.PNG|20px]] ||  {{math| {{lang|ar|ٮ}} }} || A dotless {{lang|ar|ب}} ''bāʾ''; ''b'' and {{lang|ar|ب}} ''bāʾ'' are the second letters of the Latin alphabet and the ʾabjadī sequence respectively
|-
| <math>c</math>
| [[Image:Arabic mathematical geem.PNG|25px]] ||  {{math| {{lang|ar|حــــ}} }} || From the initial form of {{lang|ar|ح}} ''ḥāʾ'', or that of a dotless {{lang|ar|ج}} ''jīm''; ''c'' and {{lang|ar|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
|-
| <math>d</math>
|[[Image:Arabic mathematical dal.PNG|15px]] ||  {{math| {{lang|ar|د}} }} || From the Arabic letter {{lang|ar|د}} ''dāl''; ''d'' and {{lang|ar|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
|-
| <math>x</math>
|[[Image:Arabic mathematical seen.PNG|25px]] ||  {{math| {{lang|ar|س}} }} || From the Arabic letter {{lang|ar|س}} ''sīn''. It is contested that the usage of Latin ''x'' in maths is derived from the first letter {{lang|ar|ش}} ''šīn'' (without its dots) of the Arabic word {{lang|ar|شيء}} ''šayʾ(un)'' {{IPA-ar|ʃajʔ(un)|}}, meaning ''thing''.<ref>{{cite web|last=Moore|first=Terry|title=Why is X the Unknown|url=http://www.ted.com/talks/terry_moore_why_is_x_the_unknown.html|publisher=Ted Talk|access-date=2012-10-11|archive-date=2014-02-22|archive-url=https://web.archive.org/web/20140222163108/http://www.ted.com/talks/terry_moore_why_is_x_the_unknown.html|url-status=dead}}</ref>  (''X'' was used in old [[Social:Spanish language|Spanish]] for the sound /ʃ/). However, according to others there is no historical evidence for this.<ref>{{cite book|last=Cajori|first=Florian|title=A History of Mathematical Notation|year=1993|url=https://archive.org/details/historyofmathema00cajo_0|url-access=registration|publisher=Courier Dover Publications|accessdate=11 October 2012|pages=[https://archive.org/details/historyofmathema00cajo_0/page/382 382]–383|isbn=9780486677668|quote=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.'}}</ref><ref>{{cite book|title=Oxford Dictionary, 2nd Edition|quote=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.}}</ref>
|-
| <math>y</math>
| [[Image:Arabic mathematical sad.PNG|20px]] ||  {{math| {{lang|ar|ص}} }} || From the Arabic letter {{lang|ar|ص}} ''ṣād''
|-
| <math>z</math>
|[[Image:Arabic mathematical ain.PNG|15px]] ||  {{math| {{lang|ar|ع}} }} || From the Arabic letter {{lang|ar|ع}} ''ʿayn''
|-
|}

===Mathematical constants and units===
{| class="wikitable"
|-
! Description
! Latin
! colspan=2 | Arabic
! Notes
|-
| Euler's number
| <math>e</math>
|[[Image:Arabic mathematical heh.PNG|15px]]  || {{math| {{lang|ar| ھ }} }}
| Initial form of the Arabic letter {{lang|ar| ه }} ''hāʾ''. Both Latin letter ''e'' and Arabic letter {{lang|ar| ه }} ''hāʾ'' are descendants of [[Phoenician alphabet|Phoenician letter]] [[File:Phoenician_he.svg|12px|]] ''hē''. 
|-
| [[Imaginary unit|imaginary unit]]
| <math>i</math>
| [[Image:Arabic mathematical teh.PNG|20px]]  || {{math| {{lang|ar| ت }} }}
| From {{lang|ar| ت }} ''tāʾ'', which is in turn derived from the first letter of the second word of {{lang|ar|  وحدة تخيلية }} ''waḥdaẗun taḫīliyya'' "imaginary unit"
|-
| [[Pi|pi]]
| <math>\pi</math>
|[[Image:Arabic mathematical tah.PNG|20px]]  || {{math| {{lang|ar| ط }} }}
| From {{lang|ar| ط }} ''ṭāʾ''; also <math>\pi</math> in some regions
|-
| [[Radius|radius]]
| <math>r</math>
|[[Image:Arabic mathematical radius.PNG|20px]]  || {{math| {{lang|ar| نٯ }} }}
| From {{lang|ar| ن }} ''nūn'' followed by a dotless {{lang|ar| ق }} ''qāf'', which is in turn derived from {{lang|ar| نصف القطر }} ''nuṣfu l-quṭr'' "radius"
|-
| [[Kilogram|kilogram]]
| kg
|[[Image:Arabic kilogram.PNG|25px]]  || {{math| {{lang|ar| كجم }} }}
| From {{lang|ar| كجم }} ''kāf''-''jīm''-''mīm''. In some regions alternative symbols like [[Image:Arabic alternative kilogram 2.PNG|20px]] ({{math|{{lang|ar| كغ }}}} ''kāf''-''ġayn'') or  [[Image:Arabic alternative kilogram 1.PNG|25px]] ({{math|{{lang|ar| كلغ }}}} ''kāf''-''lām''-''ġayn'') are used. All three abbreviations are derived from {{lang|ar| كيلوغرام }} ''kīlūġrām'' "kilogram" and its variant spellings.
|-
| [[Gram|gram]]
| g
|[[Image:Arabic gram.PNG|20px]]  || {{math|{{lang|ar| جم }}}}
| From {{lang|ar| جم }} ''jīm''-''mīm'', which is in turn derived from {{lang|ar| جرام }} ''jrām'', a variant spelling of {{lang|ar| غرام }} ''ġrām'' "gram"
|-
| meter
| m
| [[Image:Arabic mathematical meem.PNG|12px]]  || {{math| {{lang|ar| م }} }}
| From {{lang|ar| م }} ''mīm'', which is in turn derived from {{lang|ar| متر }} ''mitr'' "meter"
|-
| centimeter
| cm
| [[Image:Arabic cm.PNG|20px]]  || {{math| {{lang|ar| سم }} }}
| From {{lang|ar| سم }} ''sīn''-''mīm'', which is in turn derived from {{lang|ar| سنتيمتر }} "centimeter"
|-
| millimeter
| mm
| [[Image:Arabic mm.PNG|20px]]  || {{math| {{lang|ar| مم }} }}
| From {{lang|ar| مم }} ''mīm''-''mīm'', which is in turn derived from {{lang|ar| مليمتر }} ''millīmitr'' "millimeter"
|-
| kilometer
| km
| [[Image:Arabic Km.PNG|25px]]  || {{math| {{lang|ar| كم }} }}
| From {{lang|ar| كم }} ''kāf''-''mīm''; also [[Image:Arabic alternative km.PNG|30px]] ({{math|{{lang|ar| كلم }}}} ''kāf''-''lām''-''mīm'') in some regions; both are derived from {{lang|ar|كيلومتر}} ''kīlūmitr'' "kilometer".
|-
| second
| s
| [[Image:Arabic mathematical theh.PNG|20px]]  || {{math| {{lang|ar| ث }} }}
| From {{lang|ar| ث }} ''ṯāʾ'', which is in turn derived from {{lang|ar| ثانية }} ''ṯāniya'' "second"
|-
| minute
| min
| [[Image:Arabic mathematical Dal large.PNG|10px]]  || {{bigmath| {{lang|ar| د }} }}
| From {{lang|ar| د }} ''dālʾ'', which is in turn derived from {{lang|ar| دقيقة }} ''daqīqa'' "minute"; also [[Image:Arabic mathematical qaf.PNG|20px]] ({{math|{{lang|ar| ٯ }}}}, i.e. dotless {{lang|ar| ق }} ''qāf'') in some regions
|-
| hour
| h
|[[Image:Arabic mathematical seen.PNG|25px]]  || {{math| {{lang|ar| س }} }}
| From {{lang|ar| س }} ''sīnʾ'', which is in turn derived from {{lang|ar| ساعة }} ''sāʿa'' "hour"
|-
| kilometer per hour
| km/h
| [[Image:Arabic kmph.PNG|40px]]  || {{math|{{lang|ar| كم/س }}}}
| From the symbols for kilometer and hour
|-
| [[Physics:Degree Celsius|degree Celsius]]
| °C
| [[Image:Arabic celsius degree.PNG|30px]]  || {{math| {{lang|ar| °س }} }}
| From {{lang|ar| س }} ''sīn'', which is in turn derived from the second word of {{lang|ar| درجة سيلسيوس }} ''darajat sīlsīūs'' "degree Celsius"; also [[Image:Arabic centegrade degree.PNG|20px]] ({{math|{{lang|ar| °م }}}}) from {{lang|ar| م }} ''mīmʾ'', which is in turn derived from the first letter of the third word of {{lang|ar| درجة حرارة مئوية }}  "degree centigrade"
|-
| [[Physics:Degree Fahrenheit|degree Fahrenheit]]
| °F
|[[Image:Arabic fahrenheit degree.PNG|25px]]  || {{math| {{lang|ar| °ف }} }}
| From {{lang|ar| ف }} ''fāʾ'', which is in turn derived from the second word of {{lang|ar| درجة فهرنهايت }} ''darajat fahranhāyt'' "degree Fahrenheit"
|- 
| [[Torr|millimeters of mercury]]
| mmHg
| [[Image:Arabic mmHg.PNG|35px]]  || {{math| {{lang|ar| مم&zwnj;ز }} }}
| From {{lang|ar| مم&zwnj;ز }} ''mīm''-''mīm'' ''zayn'', which is in turn derived from the initial letters of the words {{lang|ar| مليمتر زئبق }} "millimeters of mercury"
|- 
| [[Ångström]]
| Å
| [[Image:Arabic angestrom.PNG|17px]]  || {{math| {{lang|ar| أْ }} }}
|  From {{lang|ar| أْ }} ''ʾalif'' with ''hamzah'' and ring above, which is in turn derived from the first letter of "Ångström", variously spelled {{lang|ar| أنغستروم }} or {{lang|ar| أنجستروم }}
|-
 
|}

===[[Set (mathematics)|Sets]] and number systems===

{| class="wikitable"
|-
! Description
! Latin
! colspan=2 | Arabic
! Notes
|-
| Natural numbers
| <math>\mathbb{N}</math>
| [[Image:Arabic mathematical tah large.PNG|22px]] || {{bigmath| {{lang|ar| ط }} }}
| From {{lang|ar| ط }} ''ṭāʾ'', which is in turn derived from the first letter of the second word of {{lang|ar| عدد طبيعي }}''ʿadadun ṭabīʿiyyun'' "natural number"
|-
| Integers
| <math>\mathbb{Z}</math>
| [[Image:Arabic mathematical Sad large.PNG|30px]] || {{bigmath| {{lang|ar| ص }} }}
| From {{lang|ar| ص }} ''ṣād'', which is in turn derived from the first letter of the second word of {{lang|ar| عدد صحيح }} ''ʿadadun ṣaḥīḥun'' "integer"
|-
| Rational numbers
| <math>\mathbb{Q}</math>
| [[Image:Arabic mathematical noon large.PNG|22px]] || {{bigmath| {{lang|ar| ن }} }}
| From {{lang|ar| ن }} ''nūn'', which is in turn derived from the first letter of {{lang|ar| نسبة }} ''nisba'' "ratio"
|-
| Real numbers
| <math>\mathbb{R}</math>
| [[Image:Arabic mathematical hah large.PNG|22px]] || {{bigmath| {{lang|ar| ح }} }}
| From {{lang|ar| ح }} ''ḥāʾ'', which is in turn derived from the first letter of the second word of {{lang|ar| عدد حقيقي }} ''ʿadadun ḥaqīqiyyun'' "real number"
|-
| Imaginary numbers
| <math>\mathbb{I}</math>
| [[Image:Arabic mathematical teh large.PNG|25px]] || {{bigmath| {{lang|ar| ت }} }}
| From {{lang|ar| ت }} ''tāʾ'', which is in turn derived from the first letter of the second word of {{lang|ar| عدد تخيلي }} ''ʿadadun taḫīliyyun'' "imaginary number"
|-
| Complex numbers
| <math>\mathbb{C}</math>
| [[Image:Arabic mathematical meem large.PNG|22px]] || {{bigmath| {{lang|ar| م }} }}
| From {{lang|ar| م }} ''mīm'', which is in turn derived from the first letter of the second word of {{lang|ar| عدد مركب }} ''ʿadadun murakkabun'' "complex number"
|-
| [[Empty set]]
| <math>\varnothing</math>
| <math>\varnothing</math> || ∅
| 
|-
| Is an [[Element (mathematics)|element]] of
| <math>\in</math>
| <math>\ni</math> || <span style="{{mirrorH}}">∈</span>
| A mirrored ∈
|-
| [[Subset]]
| <math>\subset</math>
| <math>\supset</math> || <span style="{{mirrorH}}">⊂</span>
| A mirrored ⊂
|-
| Superset
| <math>\supset</math>
| <math>\subset</math> || <span style="{{mirrorH}}">⊃</span>
| A mirrored ⊃
|-
| [[Universal set]]
| <math>\mathbf{S}</math>
| [[Image:Arabic mathematical sheen large.PNG|25px]] || {{bigmath| {{lang|ar| ش }} }}
| From {{lang|ar| ش }} ''šīn'', which is in turn derived from the first letter of the second word of {{lang|ar| مجموعة شاملة }} ''majmūʿatun šāmila'' "universal set"
|-
|}

===[[Arithmetic]] and [[Algebra|algebra]]===
{| class="wikitable"
|-
! Description
! Latin
! colspan=2 | Arabic
! Notes

|- 
| Percent
| %
| [[Image:Arabic percent.PNG|25px]] || {{math| {{lang|ar| ٪ }} }}
| e.g. ''100%'' "{{lang|ar| ٪١٠٠ }}"
|- 
| Permille
| ‰ 
| [[Image:Arabic permille.PNG|25px]] || {{math| {{lang|ar| ؉ }} }}
| {{math| {{lang|ar| ؊ }} }} is an Arabic equivalent of the [[Finance:Basis point|per ten thousand sign]] ‱.
|-
| [[Proportionality (mathematics)|Is proportional to]]
| <math>\propto</math>
| [[Image:Arabic prop.PNG|25px]] || <span style="{{mirrorH}}">∝</span>
| A mirrored ∝
|-  
| [[Nth root|n <sup>th</sup> root]]
| <math> \sqrt[n]{\,\,\,}</math>
| [[Image:Arabic mathematical nth root.PNG|50px]] || {{math| {{lang|ar|2=<sup style="margin-left: -0.5em; vertical-align: 0.8em;">ں</sup><span style="{{mirrorH}}">&#8237;√&#8236;</span>{{overline|&ensp;}}&rlm;}} }}
| {{lang|ar| ں }} is a dotless 	{{lang|ar| ن }} ''nūn'' while <span style="{{mirrorH}}">√</span> is a mirrored radical sign √
|-
| [[Logarithm]]
| <math>\log</math>
| [[Image:Arabic mathematical log.PNG|25px]] || {{math| {{lang|ar| لو }} }}
| From {{lang|ar| لو }} ''lām''-''wāw'', which is in turn derived from لوغاريتم {{transl|ar|lūġārītm}} "logarithm"
|- 
| [[Logarithm|Logarithm to base b]]
| <math>\log_b</math>
| [[Image:Arabic mathematical log b.PNG|40px]] || {{math| {{lang|ar| لو<sub>ٮ</sub> }}}}
|
|- 
| [[Natural logarithm]]
| <math>\ln</math>
| [[Image:Arabic mathematical ln.PNG|40px]] || {{math| {{lang|ar| لو<sub>ھ</sub> }}}}
| From the symbols of logarithm and Euler's number
|- 
| [[Summation]]
| <math>\sum</math>
| [[Image:Arabic mathematical sum.PNG|100px]] || {{math| {{lang|ar| مجــــ }} }}
| {{lang|ar|مجـــ}} ''mīm''-medial form of ''jīm'' is derived from the first two letters of {{lang|ar| مجموع }} ''majmūʿ''  "sum"; also [[Image:Arabic mathematical mirrored sum.PNG|20px]] (<span style="{{mirrorH}}">∑</span>, a mirrored summation sign ∑) in some regions
|-
| [[Product (mathematics)|Product]]
| <math>\prod</math>
|[[Image:Arabic mathematical product.PNG|100px]] || {{math| {{lang|ar| جــــذ  }}  }}
| From {{lang|ar| جذ }} ''jīm''-''ḏāl''. The Arabic word for "product" is جداء ''jadāʾun''. Also <math>\prod</math> in some regions.
|-
| [[Factorial]]
| <math>n!</math>
| [[Image:Arabic mathematical factorial.PNG|30px]]
| {{math| {{lang|ar|2=<span style="border-bottom: 1px #000 solid; border-right: 1px #000 solid; display: inline; line-height: 1em; padding: 0 0.25em;">ں</span>}} }}
| Also [[Image:Arabic mathematical fact.PNG|30px]] ({{math| {{lang|ar|ں!}} }}) in some regions
|-
| Permutations
| <math>^n\mathbf{P}_r</math>
| [[Image:Arabic mathematical nPr.PNG|50px]] || {{math| {{lang|ar|<sup>ں</sup>ل<sub>ر</sub>}} }}
| Also [[Image:Arabic mathematical P(n,r).PNG|75px]] ({{math| {{lang|ar| ل(ں، ر) }} }}) is used in some regions as <math>\mathbf{P}(n,r)</math>
|-
| Combinations
| <math>^n\mathbf{C}_k</math>
| [[Image:Arabic mathematical nCk.PNG|50px]] || {{math| {{lang|ar|<sup>ں</sup>ٯ<sub>ك</sub>}} }}
| Also [[Image:Arabic mathematical C(n,k).PNG|75px]] ({{math| {{lang|ar| ٯ(ں، ك) }} }}) is used in some regions as <math>\mathbf{C}(n,k) </math> and 40px  ({{math|{{ldelim|a=round|b=1}}{{su|lh=1.8|va=middle|p=ں|b=ك}}{{rdelim|a=round|b=1}}}} ) as the [[Binomial coefficient|binomial coefficient]] <math> n \choose k</math>
|-
|}

===Trigonometric and hyperbolic functions===

====[[Trigonometric functions]]====
{| class="wikitable"
|-
! Description
! Latin
! colspan=2 | Arabic
! Notes
|-
| Sine
| <math>\sin</math>
| [[Image:Arabic mathematical sin.PNG|20px]] || {{math| {{lang|ar| حا }} }}
| from {{lang|ar|حاء}} ''ḥāʾ'' (i.e. dotless {{lang|ar|ج}} ''jīm'')-''ʾalif''; also [[Image:Arabic mathematical sins.PNG|30px]] ({{math|{{lang|ar|جب}}}} ''jīm''-''bāʾ'') is used in some regions (e.g. Syria); Arabic for "sine" is {{lang|ar| جيب }} ''jayb''
|-
| Cosine
| <math>\cos</math>
| [[Image:Arabic mathematical cos.PNG|25px]] || {{math| {{lang|ar| حتا }} }}
| from {{lang|ar| حتا }} ''ḥāʾ'' (i.e. dotless {{lang|ar| ج }} ''jīm'')-''tāʾ''-''ʾalif''; also [[Image:Arabic mathematical coss.PNG|30px]] ({{math|{{lang|ar|تجب}}}} ''tāʾ''-''jīm''-''bāʾ'') is used in some regions (e.g. Syria); Arabic for "cosine" is {{lang|ar| جيب تمام }}
|-
| Tangent
| <math>\tan</math>
| [[Image:Arabic mathematical tan.PNG|25px]] || {{math| {{lang|ar| طا }} }}
| from {{lang|ar| طا }} ''ṭāʾ'' (i.e. dotless {{lang|ar| ظ }} ''ẓāʾ'')-''ʾalif''; also [[Image:Arabic mathematical tans.PNG|30px]] ({{math|{{lang|ar|ظل}}}} ''ẓāʾ''-''lām'') is used in some regions (e.g. Syria); Arabic for "tangent" is {{lang|ar| ظل }} ''ẓill''
|-
| Cotangent
| <math>\cot</math>
| [[Image:Arabic mathematical cot.PNG|25px]] || {{math| {{lang|ar| طتا }} }}
| from {{lang|ar| طتا }} ''ṭāʾ'' (i.e. dotless {{lang|ar| ظ }} ''ẓāʾ'')-''tāʾ''-''ʾalif''; also [[Image:Arabic mathematical cots.PNG|30px]] ({{math|{{lang|ar|تظل}}}} ''tāʾ''-''ẓāʾ''-''lām'') is used in some regions (e.g. Syria); Arabic for "cotangent" is {{lang|ar| ظل تمام }}
|-
| Secant
| <math>\sec</math>
| [[Image:Arabic mathematical sec.PNG|18px]] || {{math| {{lang|ar| ٯا }} }}
| from {{lang|ar| ٯا }} dotless {{lang|ar| ق }} ''qāf''-''ʾalif''; Arabic for "secant" is {{lang|ar| قاطع }}
|-
| Cosecant
| <math>\csc</math>
| [[Image:Arabic mathematical csc.PNG|25px]] || {{math| {{lang|ar| ٯتا }} }}
| from {{lang|ar| ٯتا }} dotless {{lang|ar| ق }} ''qāf''-''tāʾ''-''ʾalif''; Arabic for "cosecant" is {{lang|ar| قاطع تمام }}
|}

====[[Hyperbolic functions]]====
The letter [[Image:Arabic mathematical zain.PNG|20px]] ({{math|{{lang|ar|ز}}}} ''zayn'', from the first letter of the second word of {{lang|ar|دالة زائدية}} "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to the way <math>\operatorname{h}</math> is added to the end of trigonometric functions in Latin-based notation.
{| class="wikitable"
| colspan=7 | [[Image:Arabic mathematical hf.PNG|500px|Arabic hyperbolic functions]]
|-
! style="font-size:80%; width:70px;" | Description
| style="font-size:80%; width:60px;" | Hyperbolic sine
| style="font-size:80%; width:60px;" | Hyperbolic cosine
| style="font-size:80%; width:60px;" | Hyperbolic tangent
| style="font-size:80%; width:60px;" | Hyperbolic cotangent
| style="font-size:80%; width:60px;" | Hyperbolic secant
| style="font-size:80%; width:60px;" | Hyperbolic cosecant
|-
! style="font-size:80%; width:70px;" | Latin
| <math>\sinh</math> || <math>\cosh</math> || <math>\tanh</math> || <math>\coth</math> || <math>\operatorname{sech}</math> || <math>\operatorname{csch}</math>
|-
! style="font-size:80%; width:70px;" | Arabic
| {{math| {{lang|ar| حاز }} }}
| {{math| {{lang|ar| حتاز }} }}
| {{math| {{lang|ar| طاز }} }}
| {{math| {{lang|ar| طتاز }} }}
| {{math| {{lang|ar| ٯاز }} }}
| {{math| {{lang|ar| ٯتاز }} }}
|}

====[[Inverse trigonometric functions]]====
For inverse trigonometric functions, the superscript {{math| {{lang|ar| −١ }} }} in Arabic notation is similar in usage to the superscript <math>-1</math> in Latin-based notation.
{| class="wikitable"
| colspan=7 | [[Image:Arabic mathematical inverse tf.PNG|500px|Arabic inverse trigonometric functions]]
|-
! style="font-size:80%; width:70px;" | Description
| style="font-size:80%; width:60px;" | Inverse sine
| style="font-size:80%; width:60px;" | Inverse cosine
| style="font-size:80%; width:60px;" | Inverse tangent
| style="font-size:80%; width:60px;" | Inverse cotangent
| style="font-size:80%; width:60px;" | Inverse secant
| style="font-size:80%; width:60px;" | Inverse cosecant
|-
! style="font-size:80%; width:70px;" | Latin
|  <math>\sin^{-1}</math> || <math>\cos^{-1}</math> || <math>\tan^{-1}</math> || <math>\cot^{-1}</math> || <math>\sec^{-1}</math> || <math>\csc^{-1}</math>
|-
! style="font-size:80%; width:70px;" | Arabic
| {{math| {{lang|ar| حا<sup>−١</sup> }} }}
| {{math| {{lang|ar| حتا<sup>−١</sup> }} }}
| {{math| {{lang|ar| طا<sup>−١</sup> }} }}
| {{math| {{lang|ar| طتا<sup>−١</sup> }} }}
| {{math| {{lang|ar| ٯا<sup>−١</sup> }} }}
| {{math| {{lang|ar| ٯتا<sup>−١</sup> }} }}
|}

====Inverse hyperbolic functions====
{| class="wikitable"
| colspan=7 | [[Image:Arabic mathematical inverse hf.PNG|500px|Arabic inverse hyperbolic functions]]
|-
! style="font-size:80%; width:70px;" | Description
| style="font-size:80%; width:60px;" | Inverse hyperbolic sine
| style="font-size:80%; width:60px;" | Inverse hyperbolic cosine
| style="font-size:80%; width:60px;" | Inverse hyperbolic tangent
| style="font-size:80%; width:60px;" | Inverse hyperbolic cotangent
| style="font-size:80%; width:60px;" | Inverse hyperbolic secant
| style="font-size:80%; width:60px;" | Inverse hyperbolic cosecant
|-
! style="font-size:80%; width:70px;" | Latin
| <math>\sinh^{-1}</math> || <math>\cosh^{-1}</math> || <math>\tanh^{-1}</math> || <math>\coth^{-1}</math> || <math>\operatorname{sech}^{-1}</math> || <math>\operatorname{csch}^{-1}</math>
|-
! style="font-size:80%; width:70px;" | Arabic
| {{math| {{lang|ar| حاز<sup>−١</sup> }} }}
| {{math| {{lang|ar| حتاز<sup>−١</sup> }} }}
| {{math| {{lang|ar| طاز<sup>−١</sup> }} }}
| {{math| {{lang|ar| طتاز<sup>−١</sup> }} }}
| {{math| {{lang|ar| ٯاز<sup>−١</sup> }} }}
| {{math| {{lang|ar| ٯتاز<sup>−١</sup> }} }}
|}

===[[Calculus]]===

{| class="wikitable"
|-
! Description
! Latin
! colspan=2 | Arabic
! Notes
|-
| [[Limit (mathematics)|Limit]]
| <math>\lim</math>
| [[Image:Arabic mathematical limit.PNG|75px]] || {{math| {{lang|ar-Zmth| نهــــا }} }}
| {{lang|ar-Zmth| نهــــا }} ''nūn''-''hāʾ''-''ʾalif'' is derived from the first three letters of Arabic {{lang|ar| نهاية }} ''nihāya'' "limit"
|-
| [[Function (mathematics)|Function]]
| <math>\mathbf{f}(x)</math>
| [[Image:Arabic mathematical f(x).PNG|45px]] || {{math| {{lang|ar-Zmth| د(س) }} }}
| {{lang|ar-Zmth| د }} ''dāl'' is derived from the first letter of  {{lang|ar| دالة }} "function". Also called {{lang|ar| تابع}}, {{lang|ar| تا }} for short, in some regions.
|-
| Derivatives
| <math>\mathbf{f'}(x), \dfrac{dy}{dx} , \dfrac{d^2y}{dx^2} , \dfrac{\partial {y}}{\partial{x}}</math>
| [[Image:Arabic mathematical derivatives.PNG|250px]] || {{math| {{lang|ar-Zmth|{{sfrac|1= ص<span style="{{mirrorH}}">∂</span> |2=س<span style="{{mirrorH}}">∂</span> }} ،{{sfrac| د<sup>٢</sup>ص | د&zwnj;س<sup>٢</sup> }} ،{{sfrac| د&zwnj;ص | د&zwnj;س }} ،(س)‵د}} }}
| ‵ is a mirrored [[Prime (symbol)|prime]] ′ while ، is an Arabic comma. The {{math|∂}} signs should be mirrored: <span style="{{mirrorH}}">∂</span>.
|-
| Integrals
| <math>\int{} , \iint{} ,\iiint{}, \oint{} </math>
| [[Image:Arabic mathematical integrals.PNG|200px]] || &#x202a;<span style="{{mirrorH}}">∮</span> ،<span style="{{mirrorH}}">∭</span> ،<span style="{{mirrorH}}">∬</span> ،<span style="{{mirrorH}}">∫</span>&#x202c;
| Mirrored ∫, ∬, ∭ and ∮
|-
|}

===Complex analysis===
{| class="wikitable"
|-
! Latin
! Arabic
|-
| rowspan=2 | <math>z = x + iy = r(\cos{\varphi}+i \sin{\varphi})= r e^{i\varphi} = r\angle{\varphi}</math>
| [[Image:Arabic mathematical complex analysis.PNG|400px]]
|-
| {{math| {{lang|ar-Zmth|2= ع {{=}} س + ت ص  {{=}} ل(حتا ى + ت حا ى) {{=}} ل ھ<sup>ت&zwnj;ى</sup> {{=}} ل<span style="{{mirrorH}}">∠</span>ى}} }}
|-
|}

==See also==
*[[Mathematical notation]]
*[[Arabic Mathematical Alphabetic Symbols]]

==References==
{{Reflist}}

==External links==
* [https://www.ima.umn.edu/materials/2006-2007/SW12.8-9.06/2289/MathArabIMAe.pdf Multilingual mathematical e-document processing]
* [http://www.w3.org/TR/arabic-math/ Arabic mathematical notation] - W3C Interest Group Note.
* [http://www.wiris.com/ar_sa/editor/demo/ Arabic math editor] - by [[Software:WIRIS|WIRIS]].

[[Category:Mathematical notation|Arabic, modern]]

{{Sourceattribution|Modern Arabic mathematical notation}}