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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 3.5.2 Hyperbolic functions 3.5.3 Inverse trigonometric functions 3.5.4 Inverse hyperbolic functions

3.6 Calculus 3.7 Complex analysis

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: [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 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) "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 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

Tamil 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 [a]

حــــــــــــد س←∞ س۴‎ [b]

^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[c]

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'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 jīm; c and ج‎ jīm 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) [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]

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

π

displaystyle pi

ط‎

From ط‎ ṭāʾ; also

π

displaystyle pi

in some regions

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

gram

g

جم‎

From جم‎ jīm-mīm, which is in turn derived from جرام‎ jrām, 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

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

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

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

Q

displaystyle mathbb Q

ن‎

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

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

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

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

S

displaystyle mathbf S

ش‎

From ش‎ šīn, which is in turn derived from the first letter of the second word of مجموعة شاملة‎ majmūʿatun šāmila "universal set"

Arithmetic and algebra

Description

Latin

Arabic

Notes

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[ n ] ,,,

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 lūġārītm "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 jīm is derived from the first two letters of مجموع‎ majmūʿ "sum"; also (∑, a mirrored summation sign ∑) in some regions

Product

displaystyle prod

جــــذ‎

From جذ‎ jīm-ḏāl. The Arabic word for "product" is جداء jadāʾun. Also

displaystyle prod

in some regions.

Factorial

n !

displaystyle n!

ں‎

Also ( ں!‎ ) in some regions

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

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 (  ⎛⎝ںك⎞⎠   ) 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 حاء‎ ḥāʾ (i.e. dotless ج‎ jīm)-ʾalif; also ( جب‎ jīm-bāʾ) is used in some regions (e.g. Syria); Arabic for "sine" is جيب‎ jayb

Cosine

cos

displaystyle cos

حتا‎

from حتا‎ ḥāʾ (i.e. dotless ج‎ jīm)-tāʾ-ʾalif; also ( تجب‎ tāʾ-jīm-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)-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". Also called تابع‎, تا‎ for short, in some regions.

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 ⁡

φ

+ i sin ⁡

φ

) = r

e

i φ

= r ∠

φ

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