Modern Arabic Mathematical Notation
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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 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 (Arabic pronunciation: [n?q]), which is written using the two letters n?n and q?f. 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) ?‎ ?‎ ?‎ ?‎ ?‎ ?‎ ?‎ ?‎ ?‎ ?‎ 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. 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.

### Mirrored Latin symbols

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

Latin Arabic Persian x4 ‎ [a] ‎ [b]
• ^a ?n?n-h-?alif is derived from the first three letters of Arabic nih?ya "limit".
• ^b ?add is Persian for "limit".

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

Latin Arabic ?‭?‬?‎[c] 3‭?‬?‎

ma?m means "sum" in Arabic.

However, in Iran, usually Latin symbols are used.

## Examples

### Mathematical letters

Latin Arabic Notes
${\displaystyle a}$ ? From the Arabic letter ??alif; a and ??alif are the first letters of the Latin alphabet and the Arabic alphabet's ?abjad? sequence respectively
${\displaystyle b}$ ? A dotless ?b; b and ?b are the second letters of the Latin alphabet and the ?abjad? sequence respectively
${\displaystyle c}$ From the initial form of ?, or that of a dotless ?j?m; c and ?j?m are the third letters of the Latin alphabet and the ?abjad? sequence respectively
${\displaystyle d}$ ? From the Arabic letter ?d?l; d and ?d?l are the fourth letters of the Latin alphabet and the ?abjad? sequence respectively
${\displaystyle x}$ ? From the Arabic letter ?s?n. It is contested that the usage of Latin x in maths is derived from the first letter ?n (without its dots) of the Arabic word ?ay?(un) [?aj?(un)], meaning thing.[1] (X was used in old Spanish for the sound /?/). However, according to others there is no historical evidence for this.[2][3]
${\displaystyle y}$ ? From the Arabic letter ?d
${\displaystyle z}$ ? From the Arabic letter ??ayn

### Mathematical constants and units

Description Latin Arabic Notes
Euler's number ${\displaystyle e}$ ? Initial form of the Arabic letter ?h. Both Latin letter e and Arabic letter ?h are descendants of Phoenician letter h?.
imaginary unit ${\displaystyle i}$ ? From ?t, which is in turn derived from the first letter of the second word of ? wa?da?un taliyya "imaginary unit"
pi ${\displaystyle \pi }$ ? From ?; also ${\displaystyle \pi }$ in some regions
radius ${\displaystyle r}$ From ?n?n followed by a dotless ?q?f, which is in turn derived from nu?fu l-qu?r "radius"
kilogram kg From k?f-j?m-m?m. In some regions alternative symbols like ( k?f-?ayn) or ( k?f-l?m-?ayn) are used. All three abbreviations are derived from k?lr?m "kilogram" and its variant spellings.
gram g From j?m-m?m, which is in turn derived from ?jr?m, a variant spelling of ??r?m "gram"
meter m ? From ?m?m, which is in turn derived from mitr "meter"
centimeter cm From s?n-m?m, which is in turn derived from ?‎ "centimeter"
millimeter mm From m?m-m?m, which is in turn derived from mill?mitr "millimeter"
kilometer km From k?f-m?m; also ( k?f-l?m-m?m) in some regions; both are derived from ?k?l?mitr "kilometer".
second s ? From ?, which is in turn derived from niya "second"
minute min ? From ?d?l?, which is in turn derived from daq?qa "minute"; also ( ?, i.e. dotless ?q?f) in some regions
hour h ? From ?s?n?, which is in turn derived from ?sa "hour"
kilometer per hour km/h /? From the symbols for kilometer and hour
degree Celsius °C °? From ?s?n, which is in turn derived from the second word of ? ?darajat s?lss "degree Celsius"; also ( °?) from ?m?m?, which is in turn derived from the first letter of the third word of ? ‎ "degree centigrade"
degree Fahrenheit °F °? From ?f, which is in turn derived from the second word of ? darajat fahranh?yt "degree Fahrenheit"
millimeters of mercury mmHg ‌? From ‌?m?m-m?m zayn, which is in turn derived from the initial letters of the words ?‎ "millimeters of mercury"
Ångström Å From ?alif 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 ${\displaystyle \mathbb {N} }$ ? From ?, which is in turn derived from the first letter of the second word of ?adadun ?abiyyun "natural number"
Integers ${\displaystyle \mathbb {Z} }$ ? From ?d, which is in turn derived from the first letter of the second word of ??adadun ?aun "integer"
Rational numbers ${\displaystyle \mathbb {Q} }$ ? From ?n?n, which is in turn derived from the first letter of ?nisba "ratio"
Real numbers ${\displaystyle \mathbb {R} }$ ? From ?, which is in turn derived from the first letter of the second word of ?adadun ?aq?qiyyun "real number"
Imaginary numbers ${\displaystyle \mathbb {I} }$ ? From ?t, which is in turn derived from the first letter of the second word of ?adadun taliyyun "imaginary number"
Complex numbers ${\displaystyle \mathbb {C} }$ ? From ?m?m, which is in turn derived from the first letter of the second word of ??adadun 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 ${\displaystyle \mathbf {S} }$ ? From ?n, which is in turn derived from the first letter of the second word of majmatun mila "universal set"

### Arithmetic and algebra

Description Latin Arabic Notes
Percent % ? e.g. 100% "?‎"
Permille ? ? ? is an Arabic equivalent of the per ten thousand sign ?.
Is proportional to ${\displaystyle \propto }$ ? A mirrored ?
n th root ${\displaystyle {\sqrt[{n}]{\,\,\,}}}$ ?‭?‬ ?‎ is a dotless ?n?n while ? is a mirrored radical sign ?
Logarithm ${\displaystyle \log }$ From l?m-w?w, which is in turn derived from lr?tm "logarithm"
Logarithm to base b ${\displaystyle \log _{b}}$ ?
Natural logarithm ${\displaystyle \ln }$ ? From the symbols of logarithm and Euler's number
Summation ${\displaystyle \sum }$ m?m-medial form of j?m is derived from the first two letters of majm "sum"; also (?, a mirrored summation sign ?) in some regions
Product ${\displaystyle \prod }$ From j?m-l. The Arabic word for "product" is ? jadun. Also ${\displaystyle \prod }$ in some regions.
Factorial ${\displaystyle n!}$ ? Also ( ?!) in some regions
Permutations ${\displaystyle ^{n}\mathbf {P} _{r}}$ ??? Also ( ?( ?)) is used in some regions as ${\displaystyle \mathbf {P} (n,r)}$
Combinations ${\displaystyle ^{n}\mathbf {C} _{k}}$ ??? Also ( ?( ?)) is used in some regions as ${\displaystyle \mathbf {C} (n,k)}$ and (?
?
) as the binomial coefficient ${\displaystyle n \choose k}$

### Trigonometric and hyperbolic functions

#### Trigonometric functions

Description Latin Arabic Notes
Sine ${\displaystyle \sin }$ from (i.e. dotless ?j?m)-?alif; also ( j?m-b) is used in some regions (e.g. Syria); Arabic for "sine" is jayb
Cosine ${\displaystyle \cos }$ from (i.e. dotless ?j?m)-t-?alif; also ( t-j?m-b) is used in some regions (e.g. Syria); Arabic for "cosine" is ?
Tangent ${\displaystyle \tan }$ from (i.e. dotless ?)-?alif; also ( -l?m) is used in some regions (e.g. Syria); Arabic for "tangent" is ?ill
Cotangent ${\displaystyle \cot }$ from (i.e. dotless ?)-t-?alif; also ( t--l?m) is used in some regions (e.g. Syria); Arabic for "cotangent" is ?
Secant ${\displaystyle \sec }$ from ‎ dotless ?q?f-?alif; Arabic for "secant" is ?
Cosecant ${\displaystyle \csc }$ from ‎ dotless ?q?f-t-?alif; 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 ${\displaystyle \operatorname {h} }$ 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 ${\displaystyle \sinh }$ ${\displaystyle \cosh }$ ${\displaystyle \tanh }$ ${\displaystyle \coth }$ ${\displaystyle \operatorname {sech} }$ ${\displaystyle \operatorname {csch} }$ ‎ ?‎ ‎ ?‎ ‎ ?‎

#### Inverse trigonometric functions

For inverse trigonometric functions, the superscript -? 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 ${\displaystyle \sin ^{-1}}$ ${\displaystyle \cos ^{-1}}$ ${\displaystyle \tan ^{-1}}$ ${\displaystyle \cot ^{-1}}$ ${\displaystyle \sec ^{-1}}$ ${\displaystyle \csc ^{-1}}$ -?‎ -?‎ -?‎ -?‎ -?‎ -?‎

#### Inverse hyperbolic functions

 Description Latin Arabic Inverse hyperbolic sine Inverse hyperbolic cosine Inverse hyperbolic tangent Inverse hyperbolic cotangent Inverse hyperbolic secant Inverse hyperbolic cosecant ${\displaystyle \sinh ^{-1}}$ ${\displaystyle \cosh ^{-1}}$ ${\displaystyle \tanh ^{-1}}$ ${\displaystyle \coth ^{-1}}$ ${\displaystyle \operatorname {sech} ^{-1}}$ ${\displaystyle \operatorname {csch} ^{-1}}$ -?‎ ?-?‎ -?‎ ?-?‎ -?‎ ?-?‎

### Calculus

Description Latin Arabic Notes
Limit ${\displaystyle \lim }$ ? ?n?n-h-?alif is derived from the first three letters of Arabic nih?ya "limit"
function ${\displaystyle \mathbf {f} (x)}$ ?(?) ?d?l is derived from the first letter of ?‎ "function". Also called ?‎, ‎ for short, in some regions.
derivatives ${\displaystyle \mathbf {f'} (x),{\dfrac {dy}{dx}},{\dfrac {d^{2}y}{dx^{2}}},{\dfrac {\partial {y}}{\partial {x}}}}$ ?(?)? ?‌?/ ?‌? ? ???/ ?‌?? ? ??/??  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
${\displaystyle z=x+iy=r(\cos {\varphi }+i\sin {\varphi })=re^{i\varphi }=r\angle {\varphi }}$
? = ? + ? ? = ?( ? + ? ?) = ? ??‌? = ???

## References

1. ^ Moore, Terry. "Why is X the Unknown". Ted Talk.
2. ^ Cajori, Florian (1993). A History of Mathematical Notation. Courier Dover Publications. pp. 382-383. Retrieved 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.