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ⓘ Matematika



                                               

Produk (matematika)

Dina matematik, produk nyaéta hasil multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no béaring on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative. Several products are considered in mathematics: the cartesian product of sets, the product of rings ...

                                               

Fungsi karakteristik

Sababaraha ahli matematika maké istilah fungsi karakteristik sarua jeung "fungsi indikator". The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at éach point in A and 0 at éach point that is in B but not in A. In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any variabel acak with the distribution in question: φ X t = E ⁡ e i t X = ∫ Ω e i t x d F x = ∫ − ∞ ∞ f x e i t x d x. {\displaystyle \varphi _{X}t=\operatorname {E} \lefte^{itX}\right=\int _{\O ...

Matematika
                                     

ⓘ Matematika

Matématika nyaéta élmu pangaweruh anu museurkeun dirina dina konsép-konsép sarupaning kuantitas, struktur, rohang, katut parobahan, sarta mangrupa widang akademik anu maluruhna. Benjamin Peirce nyebutkeun yén matématika téh "élmu nu ngahasilkeun kacindekan nu diperlukeun". Praktisi matématika séjénna nyebutkeun yén matématika téh élmu ngeunaan pola, sarta matématikawan téh tukang néang atawa nalungtik pola-pola anu aya dina wilangan, rohang, sains, komputer, gambaran abstrak atawa di mana waé ayana. Matématikawan ngéksplorasi konsép-konsép éta pikeun ngarumuskeun konjéktur-konjéktur atawa téori-téori anyar sarta mengkuhkeun bener-henteuna ku cara déduksi nu kukuh tina pilihan aksioma tur définisi nu jelas tur cocog.

Ku cara abstraksi jeung nalar logis, matématika kabangun tina prosés ngitung, ngukur, sarta studi bentuk en: shape sacara sitematis jeung usikna banda-banda fisis. Pangaweruh tur pamakéan matématika dasar geus lila jadi hal anu inhéren sarta ngahiji dina kahirupan, boh kahirupan saurang atawa kelompok. Prosés nyampurnakeun idé-idé dasar katémbong dina téks-téks matématis nu asalna ti Mesir kuna, Mésopotamia, India kuna, Cina kuna, sarta Yunani kuna. Argumén nu kukuh kasampak dina tulisan Euclid Elements. Matématika terus mekar sanajan rada reup-reupan en: fitful nepikeun ka jaman Rénésans dina abad 16, harita inovasi matématika pinanggih jeung timuan-timuan sains, nu nyababkeun panalungtikan jadi ngagancangan, nerus nepikeun ka kiwari.

Kiwari, matématika dipaké di rupa-rupa widang sakuliah dunya, élmu alam, rékayasa, ubar, sarta élmu sosial contona ékonomi. Matématika terapan dina widang-widang kasebut, bisa ngilhamkeun timuan-timuan matématis anyar sarta sakapeung nyababkeun mekarna widang anu anyar pisan. Matématikawan ogé aya nu museurkeun usahana dina matématika murni anu awalna tanpa mikirkeun terapkeuneunana, tapi kahareupnakeun sok aya waé anu bisa diterapkeun.

                                     

1. Étimologi

Kecap matématika Yunani: μαθηματικά or mathēmatiká asalna téh tina basa Yunani kuna μάθημα máthēma, anu hartina diajar, nalungtik, sains, anu saterusna boga harti nu ngaheureutan sarta leuwih téhnis "studi matématis", saprak di jaman klasik kénéh. Kecap sipatna nyaéta μαθηματικός mathēmatikós, anu hartina hal anu pakait jeung diajar, atawa studious, anu sigana nerus miboga harti matématis en: mathematical. Harti husus lainna μαθηματικὴ τέχνη mathēmatikḗ tékhnē, dina basa Latin ars mathematica, nu hartina seni matématis.

Dina basa Inggris, mathematics mangrupa noun kecap barang, nu mindeng disingget jadi math di Amérika Kalér atawa maths. Mun ceuk barudak sakola urang, matématika téh sok disebut ogé maté.

                                     

2. Ihtisar jeung sajarah matematika

Pikeun leuwih lengkep, tempo artikel sajarah matematik.

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 mibanda pangaruh nu utama dina general relativity. Sababaraha masalah klasik ngeunaan ruler and compass constructions ahirna bisa dijawab ku Galois theory. Widang modérn ngeunaan differential geometry jeung algebraic geometry ngalegakeun géometri ka arah anu rada beda: géometri differensial nekenkeun konsep fungsi, fiber bundles, derivatives, smoothness jeung arah, sedengkeun aljabar géometri naliti wangun géometri anu dijieun tina jawaban sasaruaan persamaan sakumpulan polynomial. Group theory naliti konsep simetri sacara abstrak jeung méré 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 mangrupa salah sahiji tema elmu alam. Kalkulus mangrupa salah sahiji alat nu utama pikeun ngajelaskeun éta perkara. Konsep nu utama pikeun nerangkeun parobahan variabel nyaéta ku konsep fungsi. Loba masalah anu bisa diterangkeun sacara alami ku kaitan antara kuantitas jeung laju parobahannana, métodeu pikeun ngajawab hal ieu di ulik dina widang differential equations. Wilangan anu dipaké pikeun nerangkeun kasinambungan kuantitas nyeta wilangan real numbers, ulikan nu taliti ngeunaan sifat wilangan réal jeung fungsi nu mibanda niley réal disebut real analysis. Ku sababaraha alesan, wilangan réal perlu dilegakeun ka complex numbernu di ulik dina widang complex analysis. Functional analysis nekenkeun ulikanna kanatypically infinite-dimensional rohangan fungsi, nu méré dadasar pikeun quantum mechanics di antaran nu séjénna. 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 nyaéta ngaran anu galib pikeun widang matematika anu kapake dina elmu komputer. Salah sahiji widang anu penting dina matematika terapan nyaéta statistik, nu ngagunakeun tiori kamungkinan pikeun jadi alat nu mampuh nerangkeun, nganalisis jeung nyawang kajadian-kajadian nu bakal tumiba. Elmu ieu dipaké ampir ku sakabéh elmu alam. analisis angka nalitimétodeuanu efisien mecahkeunmeupeuskeun rupa-rupa masalah matematika sacara numerik ngagunakeun komputer di mana kasalahan ngitung ogé dipertimabangkeun.

                                     

3. Jejer-jejer na matematik

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

Matematik jeung widang séjénna

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

3.1. Jejer-jejer na matematik 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
                                     

3.2. Jejer-jejer na matematik 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
                                     

3.3. Jejer-jejer na matematik 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
                                     

3.4. Jejer-jejer na matematik Matematik Diskrit

Such topics déal 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
                                     

3.5. Jejer-jejer na matematik Matematik terapan

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

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

3.6. Jejer-jejer na matematik Famous theorems and conjectures

These théorems have interested mathematicians and non-mathematicians alike.

Fermats last theorem -- Goldbachs conjecture -- Twin Prime Conjecture -- Gödels incompleteness theorems -- Poincaré conjecture -- Cantors diagonal argument -- -- Four color theorem -- Zorns lemma -- Eulers identity -- Scholz Conjecture -- Church-Turing thesis
                                     

3.7. Jejer-jejer na matematik Important theorems

These are théorems 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
                                     

3.8. Jejer-jejer na matematik 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
                                     

3.9. Jejer-jejer na matematik 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
                                     

3.10. Jejer-jejer na matematik Matematik jeung widang séjénna

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

4. Pakakas matematis

Heubeul:

  • Napiers bones, Slide Rule
  • Jidar jeung Kompas
  • Mental calculation
  • Abacus

Anyar:

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

5. Quotes

Referring to the axiomatic method, where certain properties of an otherwise unknown structure are assumed and consequences theréof 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 dont understand things. You just get used to them.

About the béauty 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 créative 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.


                                     

6. Bibliografi

  • Kline, M., Mathematical Thought from Ancient to Modern Times 1973;
  • 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.
  • 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.
                                     

7. Tumbu kaluar

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

Interval

Watesan interval dipaké dina sabaraha hal anu pakait jeung matematika jeung dina tembang: Baca: interval waktu interval tembang interval matematika

Ékstrapolasi
                                               

Ékstrapolasi

Dina matematika, ékstrapolasi téh hiji tipe interpolasi. Nalika hiji tabulasi fungsi diinterpolasikeun teu dina rentang anu geus ditangtukeun, nyaéta di luareun rentangna, mangka disebut ékstrapolasi. Ékstrapolasi sacara kasar mah bisa baé dianggap bener, tapi hasilna mah justru mindeng teu valid.

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