Matematika

Ti Wikipédia, énsiklopédi bébas

Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantosanna diantos kanggo narjamahkeun.

Matematika sacara umum dihartikeun salaku ulikan pola struktur, parobahan, jeung ruang; jéntréna, urang bisa nyebut ulikan ngeunaan 'gambar jeung angka'. Tina jihat formal, disebut salaku panalungtikan ngeunaan struktur teu nyata nu ditangtukeun sacara aksiomatis migunakeun logika jeung lambang matematis; sawangan séjén dijéntrékeun dina Filosofi matematik. Matematika meunang ditingali salaku basa simpel tina basa tulisan jeun ucapan, kalawan aturan jeun tatabahasa anu jelas katut pasti, for the purpose of describing and exploring physical and conceptual relationships.

Struktur husus nu ditalungtik ku para ahli matematika mun dicukcruk sasakalana sok kapanggih dina élmu alam, pangmindengna fisika, tapi matematikawan ogé nangtukeun jeung nalungtik struktur pikeun alesan-alesan nu murni pikeun dunya matematik, sabab struktur éta nyadiakeun, misalna, generalisasi panghiji pikeun sababaraha subwidang, atawa pakakas pikeun itung-itungan biasa. Pamungkas, loba matematikawan ngulik wewengkon ukur pikeun alesan éstétik, némbongkeun matematik salaku hiji wujud seni batan salaku hiji élmu terapan atawa praktis. Sababaraha matematikawan mikaresep gelar "the Queen of Sciences" pikeun widangna.

Matematika kadang diringkes jadi math (na American English) atawa maths (na British English). Mun ceuk barudak sakola urang, matematik téh sok disebut ogé maté.

Daptar eusi 1 Ihtisar jeung sajarah matematika 2 Jejer-jejer na matematik 2.1 Kuantitas 2.2 Parobahan 2.3 Struktur 2.4 Space 2.5 Matematik Diskrit 2.6 Matematik terapan 2.7 Famous theorems and conjectures 2.8 Important theorems 2.9 Foundations and methods 2.10 Sajarah jeung jagat matematikawan 2.11 Matematik jeung widang séjénna 2.12 Mathematical coincidences 3 Pakakas matematis 4 Quotes 5 Mathematics is not... 6 Bibliografi 7 Tumbu kaluar

[édit] Ihtisar jeung sajarah matematika

Pikeun leuwih lengkep, tempo artikel sajarah matematik.

Kecap "matematik" datangna tina Basa Yunani μάθημα (máthema) nu hartina "élmu, pangaweruh, atawa diajar"; μαθηματικός (mathematikós) nu hartina "fond of learning".

Disiplin utama dina matematik nyelengceng tina kabutuh nyieun rupa-rupa itungan dina widang bilintik/usaha, pikeun ngukur taneuh jeung pikeun ngira-ngira kajadian-kajadian astronomis. Tilu pangabutuh ieu sacara kasar bisa dipatalikeun ka rupa-rupa bagbagan matematik nu lega kana ulikan struktur, spasi (rohangan), jeung parobahan.

Ulikan ngeunaan struktur dimimitian ku wilangan, mimiti nu geus pada mikawanoh wilangan natural jeung wilangan buleud sarta operasi aritmatikna, nu dicatetkeun dina aljabar dasar. Sipat wilangan nu leuwih jero diulik dina tiori wilangan. Panalungtikan ngeunaan métode-métode pikeun ngudar/meupeuskeun persamaan ngawujud jadi widang aljabar abstrak, nu, di antara nu séjén, ngulik rings jeung fields, struktur nu ngajabarkeun sifat-sifat nu dipibanda ku angka-anka anu geus umum. The physically important concept of vectors, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.

Ulikan ngeunaan rohangan dimimitian ku géometri, kahiji géométri Euclid jeung trigonométri dina rohangan tilu diménsi, tapi kadieunakeun dijieun leuwih umum ku ulikan non-Euclidean geometries nu ngabogaan pangaruh nu utama dina general relativity. Sababaraha masalah klasik ngeunaan ruler and compass constructions ahirna bisa dijawab ku Galois theory. Widang modern ngeunaan differential geometry jeung algebraic geometry ngalegakeun geometri ka arah anu rada beda: geometri differensial nekenkeun konsep fungsi, fiber bundles, derivatives, smoothness jeung arah, sedengkeun aljabar geometri naliti wangun geometri anu dijieun tina jawaban sasaruaan (persamaan) sakumpulan polynomial. Group theory naliti konsep simetri sacara abstrak jeung mere kaitan antra ulikan rohangan jeung ulikan struktur. Topology ngaitkeun ulikan rohangan jeung ulikan parobahan ku alatan nekenkeun kana konsep continuity.

Bisa ngarti jeung ngajelaskeun parobahan dina kuantitas nu ka ukur mangrupakeun salah sahiji tema elmu alam. Kalkulus mangrupakeun salah sahiji alat nu utama pikeun ngajelaskeun eta perkara. Konsep nu utama pikeun nerangkeun parobahan variabel nyaeta ku konsep fungsi. Loba masalah anu bisa diterangkeun sacara alami ku kaitan antara kuantitas jeung laju parobahannana, metoda pikeun ngajawab hal ieu di ulik dina widang differential equations. Wilangan anu dipake pikeun nerangkeun kasinambungan kuantitas nyeta wilangan real numbers, ulikan nu taliti ngeunaan sifat wilangan real jeung fungsi nu ngabogaan niley real disebut real analysis. Ku sababaraha alesan, wilangan real perlu dilegakeun ka complex numbernu di ulik dina widang complex analysis. Functional analysis nekenkeun ulikanna kana(typically infinite-dimensional) rohangan fungsi, nu mere dadasar pikeun quantum mechanics diantaran nu sejenna. Loba kajadian di alam nu bisa dijelaskeun ku dynamical systems jeung chaos theory ngurus sistim anu kalakuanna mengpar tina kalakuan nu galib.

Ku perluna ngajentrekeun jeung naliti dadasar matematik, widang tiori set, logika matematik jeung tiori model dikembangkeun.

Nalikakomputer mimiti katimu, sababaraha konsep tioritis anu utama diwangun ku matematikawan, nu ngalahirkeun widang tiori itungan, tiori itungan komplek, tiori informasi jeung tiori informasi algoritma. Loba pamasalahan ieu nu ayeuna di taliti dina widang sain komputer tioritis. Matematik Diskrit nyaeta ngaran anu galib pikeun widang matematika anu kapake dina elmu komputer. Salah sahiji widang anu penting dina matematika terapan nyaeta statistik, nu ngagunakeun tiori kamungkinan pikeun jadi alat nu mampuh nerangkeun, nganalisis jeung nyawang kajadian-kajadian nu bakal tumiba. Elmu ieu dipake ampir ku sakabeh elmu alam. analisis angka naliti metode anu efisien mecahkeun(meupeuskeun???) rupa-rupa masalah matematika sacara numerik ngagunakeun komputer dimana kasalahan ngitung oge dipertimabangkeun.

[édit] Jejer-jejer na matematik

Di handap ieu béréndélan subwidang jeung jejer-jejer nu ngagambarkeun salasahiji sawangan organisasional matematik.

[édit] Kuantitas

Sacara umum, jejer jeung pamendak némbongkeun ukuran-ukuran éksplisit ukuran wilangan atawa sét, atawa cara-cara pikeun manggihan pangukuran-pangukuran nu sarupa.

Wilangan -- Wilangan natural -- Pi -- Integers -- Wilangan rasional -- Wilangan real -- Wilangan kompléks -- Wilangan hiperkompléks -- Quaternions -- Octonions -- Sedenions -- Hyperreal numbers -- Surreal numbers -- Ordinal numbers -- Cardinal numbers -- p-adic numbers -- Integer sequences -- Konstanta matematiks -- Number names -- Infinity -- Base

[édit] Parobahan

Jejer-jejer di handap méré jalan pikeun ngukur parobahan dina rumus matematis jeung parobahan antarwilangan.

Aritatik -- Kalkulus -- Kalkulus véktor -- Analisis -- Differential equations -- Sistem dinamis jeung chaos theory -- Béréndélan rumus

[édit] Struktur

Rangkadak dahan matematik nu aya di handap nangtukeun ukuran jeung simétri wilangan, sarta rupa-rupa wangun.

Aljabar abstrak -- Téori wilangan -- Géométri aljabar -- Group theory -- Monoids -- Analisis -- Topologi -- Aljabar liniér -- Téori grafik -- Aljabar universal -- Téori kategori -- Order theory

[édit] Space

These topics tend to quantify a more visual approach to mathematics than others.

Topology -- Geometry -- Trigonometry -- Algebraic geometry -- Differential geometry -- Differential topology -- Algebraic topology -- Linear algebra -- Fractal geometry

[édit] Matematik Diskrit

Such topics deal with branches of mathematics with objects that can only take on specific, separated values.

Combinatorics -- Naive set theory -- Probability -- Theory of computation -- Finite mathematics -- Cryptography -- Graph theory -- Game theory

[édit] Matematik terapan

Widang-widang di handap nerapkeun pangaweruh matematik dina masalah-masalah kahirupan nyata.

Mékanik -- Analisis numeris -- Optimization -- Probability -- Statistik -- Financial mathematics

[édit] Famous theorems and conjectures

These theorems have interested mathematicians and non-mathematicians alike.

Fermat's last theorem -- Goldbach's conjecture -- Twin Prime Conjecture -- Gödel's incompleteness theorems -- Poincaré conjecture -- Cantor's diagonal argument -- -- Four color theorem -- Zorn's lemma -- Euler's identity -- Scholz Conjecture -- Church-Turing thesis

[édit] Important theorems

These are theorems that have changed the face of mathematics throughout history.

Riemann hypothesis -- Continuum hypothesis -- P=NP -- Pythagorean theorem -- Central limit theorem -- Fundamental theorem of calculus -- Fundamental theorem of algebra -- Fundamental theorem of arithmetic --Fundamental theorem of projective geometry -- classification theorems of surfaces -- Gauss-Bonnet theorem

[édit] Foundations and methods

Such topics are approaches to mathematics, and influence the way mathematicians study their subject.

Philosophy of mathematics -- Mathematical intuitionism -- Mathematical constructivism -- Foundations of mathematics -- Set theory -- Symbolic logic -- Model theory -- Category theory -- Theorem-proving -- Logic -- Reverse Mathematics -- Table of mathematical symbols

[édit] Sajarah jeung jagat matematikawan

Sajarah matematik -- Timeline of mathematics -- Matematikawan -- Fields medal -- Abel Prize -- Millennium Prize Problems (Clay Math Prize) -- International Mathematical Union -- Mathematics competitions -- Lateral thinking

[édit] Matematik jeung widang séjénna

Matematik jeung arsitéktur -- Matematik jeung atikan -- Mathematics of musical scales

[édit] Mathematical coincidences

List of mathematical coincidences

[édit] Pakakas matematis

Heubeul:

• Abacus
• Napier's bones, Slide Rule
• Jidar jeung Kompas
• Mental calculation

Anyar:

• Kalkulator jeung komputer
• Programming languages
• Computer algebra systems (listing)
• Internet shorthand notation
• software analisis statistis
  • SPSS
  • SAS

[édit] Quotes

Referring to the axiomatic method, where certain properties of an (otherwise unknown) structure are assumed and consequences thereof are then logically derived, Bertrand Russell said:

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

This may explain why John Von Neumann once said:

In mathematics you don't understand things. You just get used to them.

About the beauty of Mathematics, Bertrand Russell said in Study of Mathematics:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

Elucidating the symmetry between the creative and logical aspects of mathematics, W.S. Anglin observed, in Mathematics and History:

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

[édit] Mathematics is not...

• Numerology

[édit] Bibliografi

• Courant, R. and H. Robbins, What Is Mathematics? (1941);
• Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
• Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. Ihtisar matematik énsiklopédis nu dipedar maké basa nu jéntré tur basajan.
• Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. Vérsi tarjamah énsiklopédi Matematik Soviet nu dilegaan dina sapuluh jilid, karya nu panglengkepna tur pangmundelna. Ogé aya dina rupa CD-ROM.
• Kline, M., Mathematical Thought from Ancient to Modern Times (1973);

[édit] Tumbu kaluar

• Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics.
• Planet Math. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL license, allowing article exchange with Wikipedia. Uses TeX markup.
• Weisstein, Eric et al.: World of Mathematics. An online encyclopedia of mathematics, focusing on classical mathematics.
• Stefanov, Alexandre: Textbooks in Mathematics. A list of free online textbooks and lecture notes in mathematics.
• A mathematical thesaurus maintained by the NRICH project at the University of Cambridge (UK), Connecting Mathematics
• Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. A huge collection of articles on various math topics with more than 400 illustrated with Java applets.
• Mathforge. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
• Metamath. A site and a language, that formalize math from its foundations.