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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.

CONTENTS

* 1 Features

* 2 Variations

* 2.1 Numeral systems * 2.2 Mirrored Latin symbols

* 3 Examples

* 3.1 Mathematical letters * 3.2 Mathematical constants and units * 3.3 Sets and number systems * 3.4 Arithmetic and algebra

* 3.5 Trigonometric and hyperbolic functions

* 3.5.1 Trigonometric functions
Trigonometric functions
* 3.5.2 Hyperbolic functions
Hyperbolic functions
* 3.5.3 Inverse trigonometric functions
Inverse trigonometric functions
* 3.5.4 Inverse hyperbolic functions
Inverse hyperbolic functions

* 3.6 Calculus * 3.7 Complex analysis

* 4 See also * 5 References * 6 External links

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: ), 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.

* " Western Arabic numerals " (sometimes called European) are used in western Arabic regions (e.g. Morocco
Morocco
) * " Eastern Arabic numerals " are used in middle and eastern Arabic regions (e.g. Egypt
Egypt
and Syria
Syria
) * "Eastern Arabic-Indic numerals" are used in Persian and Urdu speaking regions (e.g. Iran
Iran
, Pakistan
Pakistan
, India
India
)

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
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. ٣٫١٤١٥٩٢٦٥٣٥٨‎ 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

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

LATIN ARABIC PERSIAN

lim x→∞ x4 نهــــــــــــا س←∞ س٤‎ حــــــــــــد س←∞ س۴‎

* ^A نهــــا‎ 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):

LATIN ARABIC MIRRORED LATIN

n ∑ x=0 3√x ں مجــــــــــــ س=٠ ٣‭√‬س‎ ں‭∑‬س=0 3‭√‬س‎

* ^C مجــــ‎ 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

MATHEMATICAL LETTERS

LATIN ARABIC NOTES

a {displaystyle a}

ا‎ From the Arabic letter ا‎ ʾalif; a and ا‎ ʾalif are the first letters of the Latin alphabet and the Arabic alphabet
Arabic alphabet
's ʾabjadī sequence respectively

b {displaystyle b}

ٮ‎ A dotless ب‎ bāʾ; b and ب‎ bāʾ are the second letters of the Latin alphabet and the ʾabjadī sequence respectively

c {displaystyle c}

حــــ‎ 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

d {displaystyle d}

د‎ From the Arabic letter د‎ dāl; d and د‎ dāl are the fourth letters of the Latin alphabet and the ʾabjadī sequence respectively

x {displaystyle x}

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) , meaning thing. (X was used in old Spanish for the sound /ʃ/ ). However, according to others there is no historical evidence for this.

y {displaystyle y}

ص‎ From the Arabic letter ص‎ ṣād

z {displaystyle z}

ع‎ From the Arabic letter ع‎ ʿayn

MATHEMATICAL CONSTANTS AND UNITS

DESCRIPTION LATIN ARABIC NOTES

Euler\'s number e {displaystyle e}

ھ‎ Initial form of the Arabic letter ه‎ hāʾ. Both Latin letter e and Arabic letter ه‎ hāʾ are descendants of Phoenician letter hē.

imaginary unit i {displaystyle i}

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 {displaystyle pi }

ط‎ From ط‎ ṭāʾ; also {displaystyle pi } in some regions

radius r {displaystyle r}

نٯ‎ From ن‎ nūn followed by a dotless ق‎ qāf, which is in turn derived from نصف القطر‎ nuṣfu l-quṭr "radius"

kilogram kg

كجم‎ From كجم‎ kāf-ǧīm-mīm. In some regions alternative symbols like ( كغ‎ kāf-ġayn) or ( كغم‎ kāf-ġayn-mīm) are used. All three abbreviations are derived from كيلوغرام‎ kīlūġrām "kilogram" and its variant spellings.

gram g

جم‎ From جم‎ ǧīm-mīm, which is in turn derived from جرام‎ ǧrām, a variant spelling of غرام‎ ġrām "gram"

meter m

م‎ From م‎ mīm, which is in turn derived from متر‎ mitr "meter"

centimeter cm

سم‎ From سم‎ sīn-mīm, which is in turn derived from سنتيمتر‎ "centimeter"

millimeter mm

مم‎ From مم‎ mīm-mīm, which is in turn derived from مليمتر‎ millīmitr "millimeter"

kilometer km

كم‎ From كم‎ kāf-mīm; also ( كلم‎ kāf-lām-mīm) in some regions; both are derived from كيلومتر‎ kīlūmitr "kilometer".

second s

ث‎ From ث‎ ṯāʾ, which is in turn derived from ثانية‎ ṯāniya "second"

minute min

د‎ From د‎ dālʾ, which is in turn derived from دقيقة‎ daqīqa "minute"; also ( ٯ‎ , i.e. dotless ق‎ qāf) in some regions

hour h

س‎ From س‎ sīnʾ, which is in turn derived from ساعة‎ sāʿa "hour"

kilometer per hour km/h

كم/س‎ From the symbols for kilometer and hour

degree Celsius °C

°س‎ From س‎ sīn, which is in turn derived from the second word of درجة سيلسيوس‎ darajat sīlsīūs "degree Celsius"; also ( °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

°ف‎ From ف‎ fāʾ, which is in turn derived from the second word of درجة فهرنهايت‎ darajat fahranhāyt "degree Fahrenheit"

millimeters of mercury mmHg

مم‌ز‎ From مم‌ز‎ mīm-mīm zayn, which is in turn derived from the initial letters of the words مليمتر زئبق‎ "millimeters of mercury"

Ångström
Ångström
Å

أْ‎ From أْ‎ 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

DESCRIPTION LATIN ARABIC NOTES

Natural numbers
Natural numbers
N {displaystyle mathbb {N} }

ط‎ From ط‎ ṭāʾ, which is in turn derived from the first letter of the second word of عدد طبيعي‎ʿadadun ṭabīʿiyyun "natural number"

Integers
Integers
Z {displaystyle mathbb {Z} }

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
Rational numbers
Q {displaystyle mathbb {Q} }

ن‎ From ن‎ nūn, which is in turn derived from the first letter of نسبة‎ nisba "ratio"

Real numbers
Real numbers
R {displaystyle mathbb {R} }

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
Imaginary numbers
I {displaystyle mathbb {I} }

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 C {displaystyle mathbb {C} }

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
Empty set
{displaystyle varnothing } {displaystyle varnothing } ∅

Is an element of {displaystyle in } {displaystyle ni } ∈ A mirrored ∈

Subset {displaystyle subset } {displaystyle supset } ⊂ A mirrored ⊂

Superset
Superset
{displaystyle supset } {displaystyle subset } ⊃ A mirrored ⊃

Universal set S {displaystyle mathbf {S} }

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

DESCRIPTION LATIN ARABIC NOTES

Percent
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 n {displaystyle {sqrt{,,,}}}

ں‭√‬ ‎ ں‎ is a dotless ن‎ nūn while √ is a mirrored radical sign √

Logarithm log {displaystyle log }

لو‎ From لو‎ lām-wāw, which is in turn derived from لوغاريتم lūġārītum "logarithm"

Logarithm to base b log b {displaystyle log _{b}}

لوٮ‎

Natural logarithm ln {displaystyle ln }

لوھ‎ From the symbols of logarithm and Euler's number

Summation {displaystyle sum }

مجــــ‎ مجـــ‎ mīm-medial form of ǧīm is derived from the first two letters of مجموع‎ maǧmūʿ "sum"; also (∑, a mirrored summation sign ∑) in some regions

Product {displaystyle prod }

جــــذ‎ From جذ‎ ǧīm-ḏāl. The Arabic word for "product" is جداء ǧadāʾun. Also {displaystyle prod } in some regions.

Factorial n ! {displaystyle n!}

ں‎ Also ( ں!‎ ) in some regions

Permutations
Permutations
n P r {displaystyle ^{n}mathbf {P} _{r}}

ںلر‎ Also ( ل(ں،ر)‎ ) is used in some regions as P ( n , r ) {displaystyle mathbf {P} (n,r)}

Combinations
Combinations
n C k {displaystyle ^{n}mathbf {C} _{k}}

ںٯك‎ Also ( ٯ(ں،ك)‎ ) is used in some regions as C ( n , k ) {displaystyle mathbf {C} (n,k)} and ( ⎛⎝ں ك⎞⎠ ) as the binomial coefficient ( n k ) {displaystyle n choose k}

TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS

Trigonometric Functions

DESCRIPTION LATIN ARABIC NOTES

Sine sin {displaystyle sin }

حا‎ from حتا‎ ḥāʾ (i.e. dotless ج‎ ǧīm)-tāʾ-ʾalif; also ( جب‎ ǧīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "sine" is جيب‎ ǧayb

Cosine cos {displaystyle cos }

حتا‎ from حتا‎ ḥāʾ (i.e. dotless ج‎ ǧīm)-tāʾ-ʾalif; also ( تجب‎ tāʾ-ǧīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "cosine" is جيب تمام‎

Tangent tan {displaystyle tan }

طا‎ from طا‎ ṭāʾ (i.e. dotless ظ‎ ẓāʾ)-tāʾ-ʾalif; also ( ظل‎ ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "tangent" is ظل‎ ẓill

Cotangent cot {displaystyle cot }

طتا‎ from طتا‎ ṭāʾ (i.e. dotless ظ‎ ẓāʾ)-tāʾ-ʾalif; also ( تظل‎ tāʾ-ẓāʾ-lām) is used in some regions (e.g. Syria); Arabic for "cotangent" is ظل تمام‎

Secant sec {displaystyle sec }

ٯا‎ from ٯا‎ dotless ق‎ qāf-ʾalif; Arabic for "secant" is أو قاطع‎

Cosecant csc {displaystyle csc }

ٯتا‎ from ٯتا‎ dotless ق‎ qāf-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 h {displaystyle operatorname {h} } 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 sinh {displaystyle sinh } cosh {displaystyle cosh } tanh {displaystyle tanh } coth {displaystyle coth } sech {displaystyle operatorname {sech} } csch {displaystyle operatorname {csch} }

ARABIC 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 1 {displaystyle -1} in Latin-based notation.

DESCRIPTION Inverse sine Inverse cosine Inverse tangent Inverse cotangent Inverse secant Inverse cosecant

LATIN sin 1 {displaystyle sin ^{-1}} cos 1 {displaystyle cos ^{-1}} tan 1 {displaystyle tan ^{-1}} cot 1 {displaystyle cot ^{-1}} sec 1 {displaystyle sec ^{-1}} csc 1 {displaystyle csc ^{-1}}

ARABIC حا−١‎ حتا−١‎ طا−١‎ طتا−١‎ ٯا−١‎ ٯتا−١‎

Inverse Hyperbolic Functions

DESCRIPTION Inverse hyperbolic sine Inverse hyperbolic cosine Inverse hyperbolic tangent Inverse hyperbolic cotangent Inverse hyperbolic secant Inverse hyperbolic cosecant

LATIN sinh 1 {displaystyle sinh ^{-1}} cosh 1 {displaystyle cosh ^{-1}} tanh 1 {displaystyle tanh ^{-1}} coth 1 {displaystyle coth ^{-1}} sech 1 {displaystyle operatorname {sech} ^{-1}} csch 1 {displaystyle operatorname {csch} ^{-1}}

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

CALCULUS

DESCRIPTION LATIN ARABIC NOTES

Limit lim {displaystyle lim }

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

function f ( x ) {displaystyle mathbf {f} (x)}

د(س)‎ د‎ dāl is derived from the first letter of دالة‎ "function"

derivatives f ( x ) , d y d x , d 2 y d x 2 , y x {displaystyle mathbf {f'} (x),{dfrac {dy}{dx}},{dfrac {d^{2}y}{dx^{2}}},{dfrac {partial {y}}{partial {x}}}}

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

z = x + i y = r ( cos + i sin ) = r e i = r {displaystyle z=x+iy=r(cos {varphi }+isin {varphi })=re^{ivarphi }=rangle {varphi }}

ع = س + ت ص = ل(حتا ى + ت حا ى) = ل ھت‌ى = ل∠ى‎

SEE ALSO

* Mathematical notation * Arabic Mathematical Alphabetic Symbols

REFERENCES

* ^ Moore, Terry. "Why is X the Unknown". Ted Talk. * ^ 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.' * ^ 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

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

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