Modern Arabic mathematical notation

The designation modern Arabic mathematical notation is used for a mathematical notation based on the Arabic script, 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 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)

Mirrored Latin symbols

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

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

However, in Iran, usually Latin symbols are used.

Examples

Mathematical letters

Latin	Arabic
$a$
$b$
$c$
$d$
$x$
$y$
$z$

• It is contested that the usage of Latin x in maths is derived from the first letter of the Arabic word شيء [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]

Mathematical constants and units

Description	Latin	Arabic	Notes
Euler's number	$e$
imaginary unit	$i$
pi	$\pi$		also $\pi$ in some regions
radius	$r$
kilogram	kg		In some regions alternative symbols like or are used
gram	g
meter	m
centimeter	cm
millimeter	mm
kilometer	km		also in some regions
second	s
minute	min		also in some regions
hour	h
kilometer per hour	km/h
degree Celsius	°C		also
degree Fahrenheit	°F
millimeters of mercury	mmHg
Ångström	Å

Sets and number systems

Description	Latin	Arabic
Natural numbers	$\mathbb{N}$
Integers	$\mathbb{Z}$
Rational numbers	$\mathbb{Q}$
Real numbers	$\mathbb{R}$
Imaginary numbers	$\mathbb{I}$
Complex numbers	$\mathbb{C}$
Empty set	$\varnothing$	$\varnothing$
Is an element of	$\in$	$\ni$
Subset	$\subset$	$\supset$
Superset	$\supset$	$\subset$
Universal set	$\mathbf{S}$

Arithmetic and algebra

Description	Latin	Arabic	Notes
Percent	%
Permille	‰
Is proportional to	$\propto$
n ^th root	$\sqrt[n]{\,\,\,}$
Logarithm	$\log$
Logarithm to base b	$\log_b$
Natural logarithm	$\ln$
Summation	$\sum$		also in some regions
Product	$\prod$		also $\prod$ in some regions
factorial	$n!$		also in some regions
permutations	$^n\mathbf{P}_r$		also is used in some regions as $\mathbf{P}(n,r)$
Combinations	$^n\mathbf{C}_k$		also is used in some regions as $\mathbf{C}(n,k)$ and as the binomial coefficient $n \choose k$

Trigonometric and hyperbolic functions

Trigonometric functions

Latin	Arabic	Notes
$\sin$		also is used in some regions (e.g. : Syria)
$\cos$		also is used in some regions (e.g. : Syria)
$\tan$		also is used in some regions (e.g. : Syria)
$\cot$		also is used in some regions (e.g. : Syria)
$\sec$
$\csc$

Hyperbolic functions

The letter is added to the end of trigonometric functions to express hyperbolic functions (the same way h is used in Latin notation).

Inverse trigonometric functions

The notation $sin^{-1}$ is the one used in Arabic notation for the inverse functions like:

Inverse hyperbolic functions

Calculus

Description	Latin	Arabic	Notes
Limit	$\lim$
function	$\mathbf{f}(x)$
derivatives	$\mathbf{f'}(x), \dfrac{dy}{dx} , \dfrac{d^2y}{dx^2} , \dfrac{\partial {y}}{\partial{x}}$
Integrals	$\int{} , \iint{} ,\iiint{}, \oint{}$

Complex analysis

Latin	Arabic
$z = x + iy = r(\cos{\varphi}+i \sin{\varphi})= r e^{i\varphi} = r\angle{\varphi}$

See also

• Mathematical notation

External links

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

References

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.