Ik45/wikipedia_dataset_science_en_id huggingface dataset:
According to Mario Livio, Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. ...
Menurut Mario Livio: Sekian banyak cendekiawan matematika dari berbagai era, seperti Pythagoras dan Euclid dari Yunani kuno, sampai ahli matematika Italia abad pertengahan Leonardo da Pisa dan ahli astronomi Renaissance Johannes Kepler, hingga tokoh-tokoh ilmuwan seperti pakar fisika dari Oxford Roger Penrose, telah menghabiskan banyak waktu untuk memahami rasio sederhana ini dengan sifat-sifatnya.
Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal.
Ahli biologi, seniman, musisi, sejarawan, arsitek, psikolog, dan bahkan ahli mistik telah berdebat mengenai hakikat keserbaadaannya dan daya tariknya.
Pacioli also saw Catholic religious significance in the ratio, which led to his work's title.
Pacioli juga melihat signifikansi iman Katolik dalam rasio ini, hal inilah yang mendorongnya memberi judul karyanya seperti demikian.
A continued square root form for ⁠ φ ⁠ can be obtained from ⁠ φ 21 + φ1+\varphi } ⁠, yielding: φ1 + 1 + 1 + ⋯ ) .
{\varphi }}=\varphi .} Dikalikan dengan φ menghasilkan φ + 1φ 2 } yang dapat diatur menjadi φ 2 − φ − 10. ^{2}-\varphi -1=0.} Dengan menggunakan formula kuadrat menghasilkan hasil positif yakni sebagai berikut.
φ n − ( − φ ) − n 5φ n − ( 1 − φ ) n 51 5 [ ( 1 + 5 2 ) n − ( 1 − 5 2 ) n ] , -(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}={\frac {1}{\sqrt {5}}}\left[\left({1+{\sqrt {5}} \over 2}\right)^{n}-\left({1-{\sqrt {5}} \over 2}\right)^{n}\right],} L ( n )φ n + ( − φ )
φ− ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 1 ) 2 ( 1 )1 ± 1 + 4 21 ± 5 2 \varphi &={\frac {-(-1)\pm {\sqrt {(-1)^{2}-4(1)(-1)}}}{2(1)}}\\&={\frac {1\pm {\sqrt {1+4}}}{2}}\\&={\frac {1\pm {\sqrt {5}}}{2}}\end{aligned}}} Dari perhitungan persamaan kuadrat, dapat diambil nilai positifnya, yakni sebagai berikut.
Letting ⁠ s ⁠ be the semiperimeter of the triangle, ⁠ s1 2 ( a + b + c ) {2}}(a+b+c)} ⁠, the area ⁠ A ⁠ is As ( s − a ) ( s − b ) ( s − c ) .
Menurut rumus ini, luas (L) suatu segitiga dengan panjang sisi a, b dan c adalah Ls ( s − a ) ( s − b ) ( s − c ) }} di mana s adalah semiperimeter segitiga itu: sa + b + c 2 .
be the triangle with sides ⁠ a4 ⁠, ⁠ b13 ⁠, and ⁠ c15 ⁠. This triangle's semiperimeter is s1 2 ( a + b + c )
Menurut rumus ini, luas (L) suatu segitiga dengan panjang sisi a, b dan c adalah Ls ( s − a ) ( s − b ) ( s − c ) }} di mana s adalah semiperimeter segitiga itu: sa + b + c 2 .
Applying the law of cosines we get cos ⁡ γa 2 + b 2 − c 2 2 a b +b^{2}-c^{2}}{2ab}}} From this proof, we get the algebraic statement that sin ⁡ γ1 − cos 2 ⁡ γ4 a 2 b 2 − ( a 2 + b 2 − c 2 ) ) 2 2 a b .
Pertama sesuai dengan aturan kosinus: cos ⁡ Ca 2 + b 2 − c 2 2 a b +b^{2}-c^{2}}{2ab}}} Kedua sesuai dengan aturan trigonometri: sin ⁡ C1 − cos 2 ⁡ C4 a 2 b 2 − ( a 2 + b 2 − c 2 ) 2 2 a b .
\gamma }}={\frac {\sqrt {4a^{2}b^{2}-{\bigl (}a^{2}+b^{2}-c^{2}{\bigr )}{\vphantom {)}}^{2}}}{2ab}}.}
C}}={\frac {\sqrt {4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})^{2}}}{2ab}}.}
As a mystical symbol, it was very important to the secret worship of Pythagoreanism.
Sebagai simbol mistik, tetraktis sangat berpengaruh pada pemujaan rahasia Pythagoreanisme.
There were four seasons, and the number was also associated with planetary motions and music.
Terdapat empat musim, dan bilangan tersebut juga diasosiasikan dengan gerak planetarium dan musik.
The Pythagorean Sourcebook and Library Archived 2015-05-09 at the Wayback Machine.
The Pythagorean Sourcebook and Library Diarsipkan 2015-05-09 di Wayback Machine.. Phanes Press, 1987.
Khipu (pronounced [ˈkʰipu]) comes from Cusco Quechua, while many other Quechua varieties use the term kipu.
Khipu (diucapkan [ˈkʰipu]) adalah kata untuk "simpul tali" pada bahasa Quechua Cusco (bahasa asli Inka).
The Bakhshali manuscript is an ancient Indian mathematical text written on birch bark that was found in 1881 in the village of Bakhshali, Mardan (near Peshawar in present-day Pakistan, historical Gandhara).
Manuskrip Bakhshali adalah sebuah manuskrip matematis ditulis pada kulit kayu birch yang ditemukan dekat desa Bakhshali (dekat Mardan di Pakistan saat ini) pada tahun 1881.
It is perhaps "the oldest extant manuscript in Indian mathematics".
Manuskrip ini terkenal karena menjadi "manuskrip tertua yang ada dalam matematika India."
The first research on the manuscript was done by A. F. R. Hoernlé.
Penelitian pertama terhadap manuskrip tersebut dilakukan oleh A. F. R. Hoernlé.
It is kept at the Bodleian Library at the University of Oxford (MS. Sansk.
Manuskrip ini terdapat di Perpustakaan Bodleian di Universitas Oxford (MS. Sansk.
"A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics" (PDF).
"A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics" (PDF)
) Hoernle: On the Bakhshali Manuscript, 1887, archive.org "A Big Zero: Research uncovers the date of the Bakhshali Manuscript", YouTube video, University of Oxford Plofker, Kim, Agathe Keller, Takao Hayashi, Clemency Montelle, and Dominik Wujastyk. 2017.
Hoernle: On the Bakhshali Manuscript, 1887, archive.org "A Big Zero: Research uncovers the date of the Bakhshali Manuscript", YouTube video, University of Oxford
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century.
Matematika India muncul di anak benua India dari 1200 SM sampai akhir abad ke-18.
In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, Varāhamihira, and Madhava.
Pada zaman klasik dari matematika India (400 M sampai 1200 M), kontribusi menonjol dibuat oleh para cendekiawan seperti Aryabhata, Brahmagupta, Bhaskara II, dan Varāhamihira.
Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra.
Para matematikawan India membuat kontribusi awal untuk kajian konsep 0 (nol) sebagai sebuah bilangan, bilangan negatif, aritmetika, dan aljabar.
In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there.
Selain itu, trigonometri berkembang di India, dan definisi modern sinus dan kosinus juga dikembangkan disana.
These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.
Konsep matematika tersebut dibawa ke Timur Tengah, Tiongkok dan Eropa dan berujung pada pengembangan lanjutan yang sekarang membentuk fondasi-fondasi dari banyak bidang matematika.
Indian Mathematicians Index of Ancient Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2004.
Indian Mathematicians Index of Ancient Indian mathematics Diarsipkan 2019-10-22 di Wayback Machine., MacTutor History of Mathematics Archive, St Andrews University, 2004.
Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha (around 1500), and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship.
Hasil terpenting mereka, yaitu ekspansi deret untuk fungsi trigonometri, digambarkan dalam syair bahasa Sanskerta dalam sebuah buku karangan Neelakanta yang bernama Tantrasangraha
The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c. 1530), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.
Teorema-teorema yang ada dinyatakan tanpa bukti, tetapi bukti-bukti untuk deret sinus, kosinus, dan tangen terbalik diberikan satu abad kemudian dalam karya Yuktibhasa (k. 1500 - k. 1610), yang ditulis dalam bahasa Malayalam oleh Jyesthadeva, dan juga dalam komentar tentang Tantrasangraha.
Indian Mathematics: Redressing the balance, MacTutor History of Mathematics archive, 2002.
(Inggris)Matematika India: Menyesuaikan keseimbangan, Arsip Sejarah Matematika MacTutor, 2002.
Tantrasamgraha, or Tantrasangraha, (literally, A Compilation of the System) is an astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics.
Tantrasamgraha (juga ditransliterasikan sebagai Tantrasangraha) adalah sebuah penulisan astronomi berpengaruh yang ditulis oleh Nilakantha Somayaji, seorang astronom/matematikawan yang berasal dari aliran astronomi dan matematika Kerala.
Bulletin of the Astronomical Society of India. 26 (11–31): 11.
Bulletin of the Astronomical Society of India. 26 (11–31).
Bibcode:1998BASI...26...11R. Ranjan Roy, R. (December 1990).
Bibcode:1998BASI...26...11R. Ranjan Roy, R. (DDec. 1990).
"The discovery of the series formula for π by Leibniz, Gregory and Nilakantha" (PDF).
"The discovery of the series formula for π by Leibnitz, Gregory and Nilakantha" (PDF).
Mathematical Association of America: 291–306. doi:10.2307/2690896.
Mathematical Association of America: 291–306.
Numerous geometric patterns and symmetries can be observed in a Vedic square, some of which can be found in traditional Islamic art.
Berbagai pola dan simetri geometris dapat diamati dalam persegi Weda, beberapa di antaranya dapat ditemukan dalam seni Islam tradisional.
(the operator ∘ refers to the abstract "multiplication" between the elements of this monoid).
(operator ∘ mengacu pada "perkalian" abstrak antara elemen-elemen monoid ini).
If a , b are elements of ( ( Z / 9 Z ) × , { 1 , ∘ } ) /9\mathbb
Jika a , b adalah elemen dari ( ( Z / 9 Z ) × , { 1 , ∘ } ) {\displaystyle ((\mathbb {
{Z} )^{\times },\{1,\circ \})} then a ∘ b can be defined as ( a × b ) mod 9 } , where the element 9 is representative of the residue class of 0 rather than the traditional choice of 0.
Z} /9\mathbb {Z} )^{\times },\{1,\circ \})} , maka a ∘ b dapat didefinisikan sebagai ( a × b ) mod 9 }} , di mana elemen 9 merupakan representasi dari kelas sisa 0, bukan 0
This does not form a group because not every non-zero element has a corresponding inverse element; for example 6 ∘ 39 but there is no a ∈ { 1 , ⋯ , 9 } } such that 9 ∘ a6.
Struktur ini tidak membentuk suatu grup karena tidak setiap elemen bukan nol memiliki elemen invers; misalnya 6 ∘ 39 tetapi tidak ada a ∈ { 1 , ⋯ , 9 } } sedemikian hingga 9 ∘ a6.
The subset { 1 , 2 , 4 , 5 , 7 , 8 } } forms a cyclic group with 2 as one choice of generator - this is the group of multiplicative units in the ring Z / 9 Z /9\mathbb
Subset { 1 , 2 , 4 , 5 , 7 , 8 } } membentuk sebuah grup siklik dengan 2 sebagai salah satu pilihan pembangkit — ini merupakan grup unit multiplikatif dalam gelanggang Z / 9 Z /9\mathbb
Every column and row includes all six numbers - so this subset forms a Latin square.
Setiap kolom dan baris mencakup keenam angka tersebut — sehingga subset ini membentuk sebuah persegi Latin.
A Vedic cube is defined as the layout of each digital root in a three-dimensional multiplication table.
Sebuah persegi Weda didefinisikan sebagai susunan setiap akar digital dalam tabel perkalian tiga dimensi.
Using the calculation above, ( a × b ) m o d ( base − 1 ) }-1)}} .
Menggunakan perhitungan di atas, ( a × b ) mod ( basis − 1 ) }-1)}}} .
The images in this section are color-coded so that the digital root of 1 is dark and the digital root of (base-1) is light.
Gambar dalam bagian ini diberi kode warna sehingga akar digital 1 ditampilkan gelap dan akar digital (basis − 1) ditampilkan terang.
119–122, ISBN 0-521-53162-4 Ghannam, Talal (2012), The Mystery of Numbers: Revealed Through Their Digital Root, CreateSpace Publications, pp.
119–122, ISBN 0-521-53162-4 Ghannam, Talal (2012), The Mystery of Numbers: Revealed Through Their Digital Root, CreateSpace Publications, hlm.
68–73, ISBN 978-1-4776-7841-1 Teknomo, Kadi (2005), Digital Root: Vedic Square Chia-Yu, Lin (2016), Digital Root Patterns of Three-Dimensional Space, Recreational Mathematics Magazine, pp.
68–73, ISBN 978-1-4776-7841-1 Teknomo, Kadi (2005), Digital Root: Vedic Square Chia-Yu, Lin (2016), Digital Root Patterns of Three-Dimensional Space, Recreational Mathematics Magazine, hlm.
The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school.
Risalah ini merupakan konsolidasi dari berbagai penemuan dari Madhava dari Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, dan astronom-matematikawan lainnya dari Mazhab Kerala.
The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam.
Namun, risalah tersebut umumnya tidak diketahui luar Kerala, karena buku itu ditulis dalam bahasa lokal (bahasa Malayalam).
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
American Mathematical Society (AMS) adalah sebuah asosiasi matematikawan profesional yang dicurahkan untuk peminatan riset dan beasiswa matematika, dan menyajikan komunitas nasional dan mancanegara melalui publikasinyam pertemuannya, advokasinya dan program lainnya.
The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences.
Perhimpunan tersebut adalah salah satu dari empat bagian dari Joint Policy Board for Mathematics dan anggota dari Conference Board of the Mathematical Sciences.
Charles B. Morrey Jr. (1967–68) Oscar Zariski (1969–70) Nathan Jacobson (1971–72) Saunders Mac Lane (1973–74) Lipman Bers (1975–76) R. H. Bing (1977–78) Peter Lax (1979–80) Andrew Gleason (1981–82) Julia Robinson (1983–84) Irving Kaplansky (1985–86) George Mostow (1987–88) William Browder (1989–90) Michael Artin (1991–92) Ronald Graham (1993–94)
Charles B. Morrey, Jr. (1967–68) Oscar Zariski (1969–70) Nathan Jacobson (1971–72) Saunders Mac Lane (1973–74) Lipman Bers (1975–76) R. H. Bing (1977–78) Peter Lax (1979–80) Andrew Gleason (1981–82) Julia Robinson (1983–84) Irving Kaplansky (1985–86) George Mostow (1987–88) William Browder (1989–90) Michael Artin (1991–92) Ronald Graham (1993–94)
Computational mathematics is a field of study that focuses on the interaction of mathematical sciences, computer science, and algorithms.
Matematika komputasi meliputi penelitian matematis dalam matematika dan bidang ilmu pengetahuan lainnya di mana komputasi memilki peran penting, serta menekankan algoritma, metode numerik, dan komputasi simbolis.
A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful.
Matematika komputasi terapan terdiri dari menggunakan matematika untuk memungkinkan dan memperbaiki penggunaan komputasi komputer dalam matematika terapan.
Computational mathematics refers also to the use of computers for mathematics itself.
Matematika komputasi juga bisa berarti penggunaan komputer dalam matematika.
Computational mathematics emerged as a distinct part of applied mathematics by the early 1950s.
Matematika komputasi menjadi bagian tersendiri dari matematika terapan pada awal 1950-an.
Foundations of Computational Mathematics, a non-profit organization International Journal of Computer Discovered Mathematics
Foundations of Computational Mathematics, sebuah organisasi nirlaba International Journal of Computer Discovered Mathematics
In addition to mathematical analysis of cryptographic algorithms, cryptanalysis includes the study of side-channel attacks that do not target weaknesses in the cryptographic algorithms themselves, but instead exploit weaknesses in their implementation.
Selain analisis matematis algoritma kriptografi, analisis kriptografi juga meliputi ilmu tentang serangan saluran samping yang tidak menyerang kelemahan algoritma, tetapi menyerang kelemahan implementasinya.
Applied Cryptanalysis: Breaking Ciphers in the Real World.
ISBN 978-1-60750-844-1. Applied Cryptanalysis: Breaking Ciphers in the Real World.
A mathematical model is an abstract description of a concrete system using mathematical concepts and language.
Model matematika adalah deskripsi dari suatu sistem dengan menggunakan konsep dan bahasa matematika.
The process of developing a mathematical model is termed mathematical modeling.
Proses pengembangan model matematika disebut pemodelan matematika.
In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations.
Penggunaan model matematika untuk memecahkan masalah dalam bisnis atau operasi militer adalah bagian besar dari bidang riset operasi.
A model may help to characterize a system by studying the effects of different components, which may be used to make predictions about behavior or solve specific problems.
Sebuah model dapat membantu menjelaskan sebuah sistem dan mempelajari efek dari komponen yang berbeda, dan untuk membuat prediksi tentang perilaku.
ISBN 0-486-68131-9 Bender, E.A. [ 1978 ] ( 2000 ).
ISBN 0-486-68131-9ISBN 0-486-68131-9 Bender, E.A. [ 1978 ] ( 2000 ).
ISBN 0-486-41180-X Gary Chartrand (1977) Graphs as Mathematical Models, Prindle, Webber & Schmidt ISBN 0871502364 Dubois, G. (2018) "Modeling and Simulation", Taylor & Francis, CRC Press.
ISBN 0-486-41180-XISBN 0-486-41180-X Gary Chartrand (1977) Graphs as Mathematical Models, Prindle, Webber & Schmidt ISBN 0871502364 Dubois, G. (2018) "Modeling and Simulation", Taylor & Francis, CRC Press.
Geography, Environment, Sustainability 1(3), 67–80.
Geography, Environment, Sustainability 1(3), 67-80.
Bibcode:1980ConPh..21....3P. doi:10.1080/00107518008210938. An Introduction to Infectious Disease Modelling Archived February 22, 2016, at the Wayback Machine by Emilia Vynnycky and Richard G White.
Bibcode:1980ConPh..21....3P. doi:10.1080/00107518008210938. An Introduction to Infectious Disease Modelling by Emilia Vynnycky and Richard G White.
General reference Patrone, F. Introduction to modeling via differential equations, with critical remarks.
Referensi umum Patrone, F. Introduction to modeling via differential equations, with critical remarks.
Philosophical Frigg, R. and S. Hartmann, Models in Science, in: The Stanford Encyclopedia of Philosophy, (Spring 2006 Edition) Griffiths, E. C. (2010)
Filosofis Frigg, R. and S. Hartmann, Models in Science, in: The Stanford Encyclopedia of Philosophy, (Spring 2006 Edition) Griffiths, E. C. (2010)
Society for Mathematical Psychology
Psikologi matematika secara kasar
Underfitting occurs when a mathematical model cannot adequately capture the underlying structure of the data.
Underfitting terjadi karena model masih mempelajari struktur dari data.
Social choice theory is a branch of welfare economics that extends the theory of rational choice to collective decision-making.
Teori pemilihan sosial atau pemilihan sosial adalah sebuah susunan teoterikal untuk analisis mengkombinasikan opini individual, preferensi, pemahaman atau kemampunan untuk mencapai keputusan kolektif atau kesejahteraan sosial dalam beberapa kasus.
Kenneth Arrow's book Social Choice and Individual Values is often recognized as inaugurating the modern era of social choice theory.
Social Choice and Individual Values (1951) karya Kenneth Arrow dan teorema imposlibilitas Arrow di dalamnya umumnya dianggap sebagai dasar teori pemilihan sosial modern.
Handbook of Social Choice and Welfare, v. 1. Chapter-preview links. _____, ed. (2011).
Chapter-preview links. _____, ed., 2011, Handbook of Social Choice and Welfare, v. 2. Chapter-preview links.
Bossert, Walter and John A. Weymark (2008).
Walter Bossert and John A. Weymark, 2008.
"Social Choice (New Developments)," The New Palgrave Dictionary of Economics, 2nd Edition, London:
"social choice (new developments)," The New Palgrave Dictionary of Economics, 2nd Edition, Abstract.
Dryzek, John S. and Christian List (2003).
John S. Dryzek and Christian List, 2003.
"Social Choice Theory and Deliberative Democracy: A Reconciliation," British Journal of Political Science, 33(1), pp.
"Social Choice Theory and Deliberative Democracy: A Reconciliation," British Journal of Political Science, 33(1), pp. 1-28. 2002 PDF link.
Feldman, Allan M. and Roberto Serrano (2006).
Allan M. Feldman and Roberto Serrano, 2006.
"Interpersonal Utility Comparisons," The New Palgrave: A Dictionary of Economics, v. 2, London:
John C. Harsanyi, 1987, "interpersonal utility comparisons," The New Palgrave: A Dictionary of Economics, v. 2, pp.
"Fundamentals of social choice theory".
A primer in social choice theory.
ISBN 978-0-521-72213-1. Robbins, Lionel (1935).
ISBN 0-521-72213-6. Lionel Robbins, 1935, 2nd ed..
"Social Choice," The New Palgrave: A Dictionary of Economics, v. 4, London:
"social choice (new developments)," The New Palgrave Dictionary of Economics, 2nd Edition, Abstract.
The New Palgrave Dictionary of Economics, 2nd Edition, London: Palgrave Abstract.
The New Palgrave Dictionary of Economics, 2nd Edition, Abstract.
"Social Choice Theory".
A primer in social choice theory.
The force generated by the mass is proportional to the acceleration of the mass as given by Newton's second law of motion: Σ Fm am x ¨m d 2 x d t 2 .
Sesuai Hukum kedua Newton gaya yang ditimbulkan sebanding dengan percepatan massa: Σ Fm am x ¨m d 2 x d t 2{\displaystyle \Sigma \
A cos ⁡ ( 2 π f n t ) .
A cos ⁡ ( 2 π f n t ) t)\!}
The frequency in this case is called the "damped natural frequency", f d , },} and is related to the undamped natural frequency by the following formula:
Frekuensi dalam hal ini disebut "frekuensi alamiah teredam", fd, dan terhubung dengan frekuensi alamiah takredam lewat rumus berikut.
Formally, given a set S and an equivalence relation ∼ on S , the equivalence class of an element a in S is denoted [ a ] or, equivalently,
Secara formal, kelas ekuivalen didefinisikan sebagai berikut:Bila ada himpunan S dan relasi ekuivalen ∼ , kelas ekuivalen suatu elemen a dalam S adalah himpunan { x ∈ S ∣ x ∼ a } } elemen-elemen yang ekuivalen dengan a .