Ginti

Set me hae natural numbers, the integers (ℤ), the rational numbers (ℚ), the real numbers (ℝ), aur the complex numbers (ℂ)

Ginti, Mathematics ke ek concept hae, jiske kaam me laae ke giina aur naapa jaae hae. Ginti khaatir 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 ke kaam me laawa jaae hae. Ginti ke sab se sahaj namuna hae natural numbers 1, 2, 3, 4, 5, .....^[1] Ek-ek number ke koi bhasa me number sabd, nai to number symbols, jiske numerals bola jaawe hae, me likha jaae sake hae; jaise ki, "paanch" ek number sabd aur "5" iske numeral hae. Kaaheki khaali thorraa symbol ke yaad karaa jaae sake hae , basic numerals ke numeral system me arrange karaa jaae hae, jon ek number ke represent kare ke organized way hae. Sab se jaada kaam me laawe waala numeral system Hindu–Arabic numeral system hae, jisme koi non-negative integer ke ten fundamental numeric symbols adj. and n. "numeral, adj. and n." OED Online. Oxford University Press. Archived from the original on 2022-07-30. Retrieved 2017-05-16.</ref> Naape aur giine ke alaawa , ginti ke labels (jaise telephone numbers), order kare ke khaatir (as with serial numbers), and for codes (as with ISBNs). Sadharan kaam me numeral ke the number se ditinguish nai karaa jaae sake hae.

Mathematics me, number ke idea ke kuchh sau saal me barrhae ke zero (0) ke saamil karaa gais hae,^[2] negative numbers,^[3] rational number jaise one half $\left({\tfrac {1}{2}}\right)$ , real number jaise square root of 2 $\left({\sqrt {2}}\right)$ aur $π$ ,^[4] aur complex numbers^[5] jon real numbers ke extend kare hae square root of $-1$ (and its combinations with real numbers by adding or subtracting its multiples).^[3] Numbers se [Calculation]] arithmetical operations se karaa jaawe hae, jisme se sab se jaana maana hae addition, subtraction, multiplication, division, aur exponentiation. Iske adhyan aur kaam ke arithmetic bola jaawe hae, ek sabd jiske number theory ke khaatir kaam me bhi laawa jaawe hae, jon number ke properties ke adhyan hae.

Itihaas

Ginti ke sab se pahile kaam me laana

Haddi aur duusra chij ke paawa gais hae, jisme mark kaata gais hae, jiske biswas karaa jaawe hae ki tally marks hae.^[6] Kuchh itihaas ke parrhe waale ii suggest karis hae ki Lebombo bone (dated about 43,000 years ago) aur Ishango bone (dated about 22,000 to 30,000 years ago) sab se puraana arithmetic chij hae, lekin ii interpretation ke dispute karaa jaawe hae.^[7]^[8] Ii tally marks ke saait time ke jiine ke khaatir kaam me laawa gais rahaa, jaise ketna roj bitaa hae, chandarma ke cycles nai to quantities, jaise ketna jaanwar hae ke gine ke khaatir.^[9] Ek perceptual system for quantity jiske numeracy ke underly kare ke socha jaawe hae, jaise phylogenetic distribution jon suggest kare hae ki ii bhasa se pahile rahaa. ^[10]^[7]

Ek tallying system me place value (as in modern decimal notation) ke koi concept nai rahe hae, jon isse brraa ginti likhe nai de hae. Lekin tallying systems ke pahila rakam ke abstract numeral system maana jaawe hae.

Sab se puraana unambiguous ginti, archaeological record me Mesopotamian base 60 system hae(c. 3400 BC);^[11] jisme place value 3rd millennium BCE me aais rahaa.^[12] Sab se puraana, jaana jaawe waala base 10 system 3100 BC me Egypt me suruu karaa gais rahaa.^[13]

Numerals

Numbers, numerals se different hae. Numerals uu chinh hae jisse ginti ke likha jaawe hae. Egyptian log pahila ciphered numeral system ke banae rahin, aur Greeks iske baad rahin jon gine waala number ke Ionian aur Doric alphabets me mal kare rahin.^[14] Roman numerals, ek system hae Roman alphabet ke letters ke kaam me laawe hae, Europe me sab se jaada kaam me laawa jaawat rahaa, Hindu–Arabic numeral system ke late 14th century me dunia bhar me faile se pahile, aur Hindu–Arabic numeral system abhi sab se jaada kaam me laawe waala system hae. ^[15] Ii system ke khaas gun ii hae ki isme zero ke khaatir chinh hae, jiske puraana Indian mathematicians lagbhag 500 AD me develop kare rahin.^[15]

Zero

Pahila dafe jab zero ke kaam me laawa gais rahaa, AD 628 me rahaa, aur ii Brāhmasphuṭasiddhānta me likha hae, t Indian mathematician Brahmagupta ke khaas kaam. Uud 0 ke ek number ke ruup me kaam me laais rahaa aue iske operations ke discuss karis rahaa, jismedivision by zero bhi hae. Ii time talak (the 7th century), ii concept Cambodia pahunch gais rahaa, Khmer numerals me,^[16] aur iske baare me documentation dekhae hae ki idea baad me s China aur Islamic dunia talak faila rahaa.

Brahmagupta ke Brāhmasphuṭasiddhānta pahila book hae jon zero ke ek number ke ruup me batae hae, tab Brahmagupta ke jaada kar ke pahila jan maana jaawe hae jon concept of zero ke formulate karis rahaa. Uu zero ke kaam me laae ke niyam diis hae, negative aur positive numbers ke saathe, jaise "zero ke jab ek positive number se jorra jaawe hae tab answer ek positive number rahe hae, aur jab negative number ke zero se jorraa jaawe hae tab answe ek negative number" rahe hae. Brāhmasphuṭasiddhānta sabse puraana book hae jon zero ke ek number ke ruup me kaam me laawe hae.

↑ "number, n." (in en-GB). OED Online (Oxford University Press). http://www.oed.com/view/Entry/129082. Retrieved 2017-05-16.
↑ Matson, John. "The Origin of Zero", Scientific American. (in en)
↑ ^3.0 ^3.1 Hodgkin, Luke (2005-06-02) (in en). A History of Mathematics: From Mesopotamia to Modernity. OUP Oxford. pp. 85–88. ISBN 978-0-19-152383-0. https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88. Retrieved 2017-05-16.
↑ Mathematics across cultures : the history of non-western mathematics. Dordrecht: Kluwer Academic. 2000. pp. 410–411. ISBN 1-4020-0260-2.
↑ Descartes, René (1954). La Géométrie: The Geometry of René Descartes with a facsimile of the first edition. Dover Publications. ISBN 0-486-60068-8. https://archive.org/details/geometryofrenede00rend. Retrieved 20 April 2011.
↑ Marshack, Alexander (1971). The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation (1st ed.). New York: McGraw-Hill. ISBN 0-07-040535-2. OCLC 257105. https://books.google.com/books?id=vbQ9AAAAIAAJ.
↑ ^7.0 ^7.1 (Burgin 2022, pp. 2–3)
↑ (Thiam & Rochon 2019, p. 164)
↑ Ore, Øystein (1988). Number theory and its history. New York: Dover. ISBN 0-486-65620-9. OCLC 17413345. https://books.google.com/books?id=Sl_6BPp7S0AC.
↑ Coolidge, Frederick L.; Overmann, Karenleigh A. (2012). "Numerosity, Abstraction, and the Emergence of Symbolic Thinking". Current Anthropology 53 (2): 204–225. doi:10.1086/664818. https://osf.io/utn53/.
↑ Schmandt-Besserat, Denise (1992). Before Writing: From Counting to Cuneiform (2 vols). University of Texas Press.
↑ Robson, Eleanor (2008). Mathematics in Ancient Iraq: A Social History. Princeton University Press.
↑ "Egyptian Mathematical Papyri – Mathematicians of the African Diaspora". Math.buffalo.edu. Archived from the original on 2015-04-07. Retrieved 2012-01-30.
↑ Chrisomalis, Stephen (2003-09-01). "The Egyptian origin of the Greek alphabetic numerals". Antiquity 77 (297): 485–96. doi:10.1017/S0003598X00092541. ISSN 0003-598X.
↑ ^15.0 ^15.1 Bulliet, Richard; Crossley, Pamela; Headrick, Daniel; Hirsch, Steven; Johnson, Lyman (2010). The Earth and Its Peoples: A Global History, Volume 1. Cengage Learning. p. 192. ISBN 978-1-4390-8474-8. https://books.google.com/books?id=dOxl71w-jHEC&pg=PA192. Retrieved 2017-05-16. "Indian mathematicians invented the concept of zero and developed the "Arabic" numerals and system of place-value notation used in most parts of the world today"
↑ Aczel, Amir D. (2015-05-07). "My Quest to Find the First Zero". TIME (in English). Archived from the original on 2025-04-20. Retrieved 2025-02-15.

[1] "number, n." (in en-GB). OED Online (Oxford University Press). http://www.oed.com/view/Entry/129082. Retrieved 2017-05-16.

[2] Matson, John. "The Origin of Zero", Scientific American. (in en)

[:0-3] 3.0 ^3.1 Hodgkin, Luke (2005-06-02) (in en). A History of Mathematics: From Mesopotamia to Modernity. OUP Oxford. pp. 85–88. ISBN 978-0-19-152383-0. https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88. Retrieved 2017-05-16.

[4] Mathematics across cultures : the history of non-western mathematics. Dordrecht: Kluwer Academic. 2000. pp. 410–411. ISBN 1-4020-0260-2.

[5] Descartes, René (1954). La Géométrie: The Geometry of René Descartes with a facsimile of the first edition. Dover Publications. ISBN 0-486-60068-8. https://archive.org/details/geometryofrenede00rend. Retrieved 20 April 2011.

[6] Marshack, Alexander (1971). The roots of civilization; the cognitive beginnings of man's first art, symbol, and notation (1st ed.). New York: McGraw-Hill. ISBN 0-07-040535-2. OCLC 257105. https://books.google.com/books?id=vbQ9AAAAIAAJ.

[auto-7] 7.0 ^7.1 (Burgin 2022, pp. 2–3)

[8] (Thiam & Rochon 2019, p. 164)

[Ore-9] Ore, Øystein (1988). Number theory and its history. New York: Dover. ISBN 0-486-65620-9. OCLC 17413345. https://books.google.com/books?id=Sl_6BPp7S0AC.

[10] Coolidge, Frederick L.; Overmann, Karenleigh A. (2012). "Numerosity, Abstraction, and the Emergence of Symbolic Thinking". Current Anthropology 53 (2): 204–225. doi:10.1086/664818. https://osf.io/utn53/.

[11] Schmandt-Besserat, Denise (1992). Before Writing: From Counting to Cuneiform (2 vols). University of Texas Press.

[12] Robson, Eleanor (2008). Mathematics in Ancient Iraq: A Social History. Princeton University Press.

[13] "Egyptian Mathematical Papyri – Mathematicians of the African Diaspora". Math.buffalo.edu. Archived from the original on 2015-04-07. Retrieved 2012-01-30.

[14] Chrisomalis, Stephen (2003-09-01). "The Egyptian origin of the Greek alphabetic numerals". Antiquity 77 (297): 485–96. doi:10.1017/S0003598X00092541. ISSN 0003-598X.

[Cengage_Learning2-15] 15.0 ^15.1 Bulliet, Richard; Crossley, Pamela; Headrick, Daniel; Hirsch, Steven; Johnson, Lyman (2010). The Earth and Its Peoples: A Global History, Volume 1. Cengage Learning. p. 192. ISBN 978-1-4390-8474-8. https://books.google.com/books?id=dOxl71w-jHEC&pg=PA192. Retrieved 2017-05-16. "Indian mathematicians invented the concept of zero and developed the "Arabic" numerals and system of place-value notation used in most parts of the world today"

[16] Aczel, Amir D. (2015-05-07). "My Quest to Find the First Zero". TIME (in English). Archived from the original on 2025-04-20. Retrieved 2025-02-15.

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