Mathematics

Mathematics (jiske chhota kar ke "maths" nai to "math" bola jaae hae), ginti (number), dhaancha (shape) aur namuuna (pattern) ke adhyan hae. Ii sabd Greek μάθημα (máthema) se aais hae, jiske matlab vigyan, gyaan, nai to sikhnaa hae.

Ii papyrus rhind se puraana Egypt me mathematics ke jaankari mile hae

Mathematics me adhyan karaa jaae hae:

• Ginti: jisme chij ke kaise gina jaae, bhi hae.
• Dhaancha (structure):chij ke kaise bandobast (organize) karaa gais hae, lekin ii bhi ki ii kaise rahaa hoi. Iske jaada kar ke algebra bola jaae hae.
• Place: jahaan chij hae, ii kaise arrange karaa gais, jisme iske dhaancha ke arrangement bhi hae. Iske jaada kar ke geometry bola jaae hae.
• Change: kaise chij ke biich me antar (difference) hae. Iske jaada kae ke Mathematical analysis bola jaae hae.

Applied math, asli dunia me problem ke solve kare me kaamil (useful) hae. Jon log business, vigyan, engineering aur construction me kaam kare hae, mathematics ke kaam me laae hae.^[1][2]

Mathematics me samasya ke suljhaana (problem solving)

Mathematics, logic ke kaam me laae ke samasya ke sulghae hae.Ek khaas aujaar jiske mathematician log kaam me laae hae, deduction. Deduction me puraana sachchaai ke kaam me laae ke nawaa sachchaai ke khoja jaae hae. Deduction ke kaam me laana, uu chij hae jon mathematics ke duusra rakam ke vigyanik soch (scientific thinking) se different hae, kaaheki usme experiment nai to interview pe nirbhar rahaa jaae hae.^[3]

Mathematician log logic aur reasoning ke kaam me laae ke general niyam (rule) banae hae, jon mathematics ke khaatir jaruri hae. Ii niyam uu jaankari ke nikaal de hai jon jaruri nai hae, tab ek niyam dher haalaat me kaam me laawa jaae sake hae. General niyam ke paae se, mathematicians dher samasya ke suljhaae sake hae, kaaheki ii niyam duusra samaya me bhi kaam me laawa jaae sake hae.^[4] Ii niyam ke theorem bola jaae hae (agar iske saabit kar dewa gais hae), nai to conjecture bola jaae hae.^[5] Jaada mathematicians non-logical aur creative reasoning ke kaam me laae ke logical proof ke paae hae.^[6]

Areas of study in mathematics

Ginti

Mathematics me ginti aur quantities ke adhyan karaa jaae hae. Ii vigyan ke ek hissa hae jisme logic of shape, quantity, aur arrangement hae. Niche dewa gais areas ke mathematics ke dher field me adhyan karaa jaae hae, jisme set theory aur mathematical logic hae. Number theory ke adhyan jaada kar ke integer ke structure aur behavior ke adhyan kare hae .

${\displaystyle 1,2,3,\ldots }$	${\displaystyle \ldots ,-1,0,1,\ldots }$	${\displaystyle {\frac {1}{2}},{\frac {2}{3}},0.125,\ldots }$	${\displaystyle \pi ,\tau ,e,{\sqrt {2}},\ldots }$	${\displaystyle 1+i,2e^{i\pi /3},\ldots }$
Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers
${\displaystyle 0,1,\ldots ,\omega ,\omega +1,\ldots ,2\omega ,\ldots }$	${\displaystyle \aleph _{0},\aleph _{1},\ldots }$	${\displaystyle +,-,\times ,/}$	${\displaystyle <,\leq ,=,\geq ,>}$	${\displaystyle f(x)={\sqrt {x}}}$
Ordinal numbers	Cardinal numbers	Arithmetic operations	Arithmetic relations	Functions, see also special functions

Structure

Structural mathematics objects' and constructions' ke shape aur integrity ke adhyan kare hae. Isme algebra aur calculus hae.

Number theory	Abstract algebra	Linear algebra	Order theory	Graph theory

Shape

Mathematics ke kuchh hussa chij ke dhaancha ke adhyan kare hae, Ii sab jaada kar ke geometry ke adhuan ke hissa hae.

Topology	Geometry	Trigonometry	Differential geometry	Fractal geometry

Change

Some areas of mathematics study the way things change. Most of these areas are part of the study of analysis.

Calculus	Vector calculus	Analysis
Differential equations	Dynamical systems	Chaos theory

Mathematics ke itihaas

Puraana itihaas

Babylonian=

Egyptian

Greek

Roman

Chinese

Indian

Jon akchhar ke gine ke khaatir Bakhshali manuscript me kaam me laawa gais rahaa. Ii 2nd century BC aur 2nd century AD ke biich ke hae.

Indian numerals ke patthar aur tamba (copper) me ke inscriptions^[7]

Puraana Brahmi numerals, India ke ek hissa me

Indian subcontinent ke sab se puraana civilization Indus Valley civilization hae (mature second phase: 2600 to 1900 BC) jon Indus Naddi ke basin me rahaa. Iske city geometric shape me banaa rahaa, lekin ii civilization ke koi mathematical document abhi talak nai bachaa hae.^[8]

India ke sab se puraana mathematical records, jon abhi talak bachaa hae, Sulba Sutras hae (jiske 8th century BC se 2nd century AD talak ke bataawa jaae hae),^[9] Ii time, dharmik kitaab me murti ke kaise banaawa jaae sake ke niyam me dher shape, jaise squares, rectangles, parallelograms hae.^[10] Egypt ke rakam, mandir banae me jaada dhyan dena, ii dekhae hae ki mathematics, dharmik ritual se suruu bhais hae.^[9] Sulba Sutras ii batae hae ki ek circle, jiske area ek square ke baraabar hae, ke kaise banaawa jaae sake hae, jon π ke dher approxinations dekhae hae. ^[11][12] π ke approximate hae 4 x (13/15)² (3.0044...), 25/8 (3.125), 900/289 (3.11418685...), 1156/361 (3.202216...), aur 339/108 (3.1389). Iske alaawa, uu log 2 ke square root ke decimal places talak work our kare rahin, Pythagorean triples ke suchi banae rahin, aur Pythagorean theorem ke bhi batae rahin.^[12] Ii sab chij Babylonian mathematics me bhi hae, aur ii Mesopotamian influence dekhae hae.^[9] Ii nai jaana jaae hae ki Sulba Sutra ke ketnaa asar baad ke Indian mathematicians pe rahaa. China ke rakam, Indian mathematics ke development contunuous nai rahaa; kabhi-kabhi sher pragati ke baad dher din talak koi nawaa chij nai bhais rahaa.^[9]

Panini (c. 5th century BC) me Sanskrit grammar ke niyam banais rahaa.^[13] Uske notation, modern mathematical notation ke rakam rahaa, aur metarules, transformations, aur recursion ke kaam me laais rahaa.^[14] Pingala (lagbhag 3rd–1st centuries BC) aapan thesis prosody me ek aujaar ke kaam me laais rahaa jon binary numeral system ke rakam rahaa.^[15][16] Uske combinatorics of meters, sangeet me, binomial theorem ke rakam rahaa. Pingala ke kaam me Fibonacci number (jiske mātrāmeru bola gais rahaa) ke basic ideas of hae.^[17]

India ke aglaa khaas mathematical document Sulba Sutras hae, jon Siddhantas hae, jisme astronomical treatises 4th aur 5th centuries AD (Gupta period) ke hae jon Greece ke asar dekhae hae.^[18] Isme trigonometric relations hae jon half-chord pe based hae, jaise modern trigonometry me, aur full chord nai, jaise Ptolemaic trigonometry me rahaa.^[19] Kuchh ranslation me galti ke kaaran, sabd "sine" aur "cosine" Sanskrit "jiya" aur "kojiya" se aae hae.^[19]

Yuktibhāṣā me sine rule ke niyam

Lagbhag 500 AD me, Aryabhata Aryabhatiya likhis rahaa, ek patraa volume, jon verse me loha gias rahaa, jiske astronomy aur mathematical mensuration ke rules ke samjhae ke khaatir likha gis rahaa.^[20] Aryabhatiya me decimal place-value system ke pahila dafe dekha jaae hae.

7th century me, Brahmagupta, Brahmagupta theorem banais, josme uu Brahmagupta's identity aur Brahmagupta's formula ke diis, aur pahila dafe Brahma-sphuta-siddhanta me, uu zeroke , as both a placeholder aur decimal digit ke ruup me samjhais, aur Hindu–Arabic numeral system ke bhi sanjhais.^[21] Ii text ke translation ke (c. 770) se Islamic mathematicians ke numeral system introduce karaa gais rahaa, jon iske Arabic numerals ke ruup me apnae liin. Islamic scholars, number system ke ii gyan ke Europe me 12th century me pahunchain, aur iske puraana munber system ke jagha kaam me laae jaae lagaa gais. Jab ki ii number system ke dher rakam likha jaae hae, ii sab Brahmi numeral se aais hae. India ke ek darjan script ke aapan number likhe ke dhang hae. 10th century me, jab Halayudha , Pingala ke kaam ke baare me likhis tab isme Fibonacci sequence,^[22] aur Pascal's triangle bhi rahaa,^[23] aur ii matrix kaise bane hae ke bhi samjhais rahaa.

12th century me, Bhāskara II,^[24] jon south India me rahat raha, jaana jaae waala mathematics ke branches ke baare me dher chij likhis rahaa. Uske kaam me mathematical objects equivalent or approximately equivalent to infinitesimals, the mean value theorem aur sine function ke derivative hae.^[25][26] 14th century me, Narayana Pandita Ganita Kaumudi ke likh ke khalaas karis.^[27]

14th century bhi, Madhava of Sangamagrama, jon Kerala School of Mathematics ke suruu karis rahaa, Madhava–Leibniz series ke banais jiske uu transformed series se suruu karis rahaa, jiske pahila 21 terms ke kaam me laae ke π ke value 3.14159265359 paawa gais hae. Madhava the Madhava-Gregory series ke bhi paais jiske uu kaam me laae ke arctangent ke calculate karis, Madhava-Newton power series ke kaam me laae ke sine aur cosine aur the Taylor approximation sine aur cosine functions ke calculate karis.^[28] 16th century me, Jyesthadeva Kerala School ke banaawa gais theorem ke Yukti-bhāṣā me likhis.^[29][30] Ii bhi arue karaa gais hae ki kuchh calculus ke idea, jaise infinite series aur taylor series aur kuchh trigonometry functions, ke Europe talak 16th century me^[31] Jesuit missionaries aur traders log pahunchaain jon puraana Muziris port me uu time me rahin aur ii rakam se European me analysis aur calculus ke development me madat karin.^[32]