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Roots of algebraic equations

Cubic equation

The roots of 3rd order algebraic equation x^3 + bx^2+cx+d=0 are given by the Tartaglia-Cardano formulas:

  • x_0 = X_0 + X_{+} + X_{-},
  • x_1 = X_0 + \omega X_{+} + \omega^2 X_{-},
  • x_2 = X_0 + \omega^2 X_{+} + \omega X_{-},


  • \omega:\equiv e^{i\frac{2\pi}{3}} = \frac{-1+i\sqrt{3}}2,
  • X_0 :\equiv -\frac{b}3,
  • X_{\pm} :\equiv \left\{ - \left(\frac{27 d -9bc+2b^3}{54}\right) \pm \sqrt{\left(\frac{27 d -9bc+2b^3}{54}\right)^2 - \left( \frac{b^2-3c}{9}\right)^3} \right\}^{\frac13},
  • X_{+} X_{-} = \frac{b^2-3c}{9},
  • X_{+}^2 + X_{-}^2 = - \frac{27 d -9bc+2b^3}{54}.


Special case: reduced cubic equation (Cardano’s formula)

Change of variables x =  y -\frac{b}3 reduces the general third-order algebraic equation x^3 + bx^2+cx+d=0 to the reduced form y^3+p y + q = 0. This one can be solved using the Cardan’s ansatz

  • y = \sqrt[3]{A+B} + \sqrt[3]{A-B} for some not yet determined A and B.
  • This gives: y^3 = 2A+ 3y \sqrt[3]{A^2-B^2}
  • Comparing coefficients with the y^3 = -p y - q gives 2A=q and 3 \sqrt[3]{A^2-B^2} = p
  • Their solution is: A=\frac{q}2, B = \sqrt{\left(\frac{q}{2}\right)^2-\left(\frac{p}3\right)^3}.

So, finally, one root of the reduced equation y^3+p y + q = 0 is (the Cardano’s formula):

    y_1 =  \sqrt[3]{\frac{q}2+\sqrt{\left(\frac{q}{2}\right)^2-\left(\frac{p}3\right)^3}} + \sqrt[3]{\frac{q}2-\sqrt{\left(\frac{q}{2}\right)^2-\left(\frac{p}3\right)^3}}

The remaining two roots are solutions of the quadratic equation y^2 + y_1 y + y_1^2+p=0:

    y_{2,3} = \frac12(-y_1 \pm \sqrt{-3 y^2_1 -4 p}).

Quartic equation

Special case: y^4+y=x

The solution is y = \frac12 \epsilon \sqrt{A} +  \frac12 \epsilon' \sqrt{B} for \epsilon, \epsilon' = \pm 1, where B = -A - \frac{2\epsilon}{\sqrt{A}}, and A is any of three roots of the cubic equation A^3+4xA-1=0. For example, we may take the Cardan’s solution: A_1 = \sqrt[3]{a+b} +  \sqrt[3]{a-b} where a=\frac12 and b = \frac1{2\sqrt{3}}\sqrt{27+256 x^3}. Putting all elements together,

    y = \frac12 \epsilon \sqrt{\sqrt[3]{\frac12+\frac1{2\sqrt{3}}\sqrt{27+256 x^3}} +  \sqrt[3]{\frac12-\frac1{2\sqrt{3}}\sqrt{27+256 x^3}}} +  \frac12 \epsilon' \sqrt{- \sqrt[3]{\frac12+\frac1{2\sqrt{3}}\sqrt{27+256 x^3}} -  \sqrt[3]{\frac12-\frac1{2\sqrt{3}}\sqrt{27+256 x^3}} -\frac{2\epsilon}{\sqrt{\sqrt[3]{\frac12+\frac1{2\sqrt{3}}\sqrt{27+256 x^3}} +  \sqrt[3]{\frac12-\frac1{2\sqrt{3}}\sqrt{27+256 x^3}}}}}

Derivation: Start from the ansatz y = \frac12 \epsilon \sqrt{A} +  \frac12 \epsilon' \sqrt{B} for yet unknown A & B (Q: How? Why? ;-). Then

  • y^2 = \frac14 (A+B) + \frac{\epsilon\epsilon'}2 \sqrt{AB},
  • y^4 = \frac1{16} (A+B)^2  + \frac{\epsilon\epsilon'}4 (A+B) \sqrt{AB} + \frac14 AB,
  • Require that A+B = \frac{\alpha}{\sqrt{A}} (Again: How? Why?)
  • Then y^4 = \frac{\alpha^2}{16 A} -\frac{A^2}4  + \frac{\epsilon\epsilon'\alpha}4  \sqrt{B} + \frac{\alpha}4 \sqrt{A},
  • and y^4 + y = \frac{\alpha^2}{16 A} -\frac{A^2}4  + \left(\frac{\epsilon\epsilon'\alpha}4+\frac{\epsilon'}2\right)  \sqrt{B} + \left(\frac{\alpha}4 +\frac{\epsilon}2\right)\sqrt{A}.
  • Setting \alpha=-2\epsilon simplifies that equation by eliminating roots: y^4 + y = \frac{\alpha^2}{16 A} -\frac{A^2}4 = x
  • which is equivalent to the cubic equation A^3+4xA-1=0 solvable via Cardano’s procedure.

Example: A simple (and trivial) example is for x=0, when y^4+y=0 has roots \{y_1=0\} \cup \{y_{n}=e^{i\frac{\pi}3(n-1)} | n=2,3,4\}. Here b=a=\frac12, A_1=1, B_1=-1-2\epsilon and y_{\epsilon,\epsilon'}(A_1) = \frac{\epsilon+\epsilon' \sqrt{-1-2\epsilon}}2, i.e.

  • y_{+,+} = \frac{1+i\sqrt{3}}2 \equiv y_2
  • y_{+,-} = \frac{1-i\sqrt{3}}2 \equiv y_3
  • y_{-,+} = 0 \equiv y_1
  • y_{-,-} = -1 \equiv y_4


Quintic equation


Some continued fractions

Filed under: mathematics — Tags: — sandokan65 @ 21:01
  • \sqrt{2} = 1+ \frac1{2+}\frac1{2+}\frac1{2+}\cdots = [1;2,2,2,\cdots] = [1;\dot{2}].
  • \sqrt{3} = [1;\dot{1},\dot{2}].
  • \sqrt{5} = [2; \dot{4}].
  • \sqrt{7} = [2; \dot{1},1,1,\dot{4}].
  • e = [2;1,2,1,1,4,1,1,6,1,1,8,\cdots].
  • \frac\pi4 = \frac1{1+} \frac{1^2}{2+} \frac{3^2}{2+} \frac{5^2}{2+} \frac{7^2}{2+} \frac{9^2}{2+} \cdots.
  • [3;\dot{3}] = \frac{3+\sqrt{13}}2 = 3.302,77\cdots.
  • [a;\dot{b}] = a+\frac{\sqrt{b^2+4}-b}2.
  • [4;\dot{6}] = 1+\sqrt{10}.
  • [a;\dot{b},\dot{c}] = ab + \frac{\sqrt{bc(bc+4)}-bc}2.

Theorem: A simple infinite continued fraction [a_0; a_1, a_2, \cdots] converges IFF \sum_n a_n diverges.

Every real number x has the unique simple continued fraction expression. That expression is:

  • finite IFF x \in {\Bbb Q},
  • infinite and repeating IFF x \not\in {\Bbb Q} but algebraic,
  • infinite and non-repeating IFF x is transcendental.

The 50 Most Brilliant Atheists of All Time

Filed under: atheism, critical thinking — sandokan65 @ 20:31

The Brainz.org has an article titled “The 50 Most Brilliant Atheists of All Time” – http://brainz.org/50-most-brilliant-atheists-all-time/.
According to them, following people were some kind of atheists (secularists, atheists, positivists, etc):

  • 1. Democritus – an ancient Greek philosopher
  • 2. Diagoras of Melos – 5th century b.c.e. poet and sophist from Melos known as Diagoras the Atheist.
  • 3. Epicurus – Born in 341 b.c.e. in Athens, Epicurus established the school of philosophy known as Epicureanism
  • 4. Theodorus the Atheist – lived around 300 b.c.e
  • 5. Andrew Carnegie [1835-1919]
  • 6. Ivan Petrovich Pavlov [1849-1936] – a Russian physiologist, psychologist and physician, won the Nobel Prize in medicine in 1904 for research on the digestive system.
  • 7. Sigmund Freud (Sigismund Schlomo Freud) [1856-1939] – an Austrian psychiatrist founded the psychoanalytic school of psychology.
  • 8. Clarence Seward Darrow [1857-1938] – an American lawyer, a leading member of the ACLU and a notable defense attorney.
  • 9. Richard Georg Strauss [1864-1949] – a brilliant German composer who began writing music at the age of six and continued almost until his death.
  • 10. Bertrand Arthur William Russell [1872-1970], 3rd Earl of Russell – a British philosopher, logician, mathematician, historian, pacifist and social activist, awarded the Nobel Prize for literature in 1950.
  • 11. Jawaharlal Nehru [1889-1964] – Prime Minister of an India from 1947 to 1964.
  • 12. Linus Carl Pauling [1901-1994] – one of only 4 individuals ever to have won solo Nobel Prizes in separate and unrelated fields – for chemistry in 1954, and the Nobel Peace Prize for his tireless campaign against atmospheric nuclear bomb testing in 1962.
  • 13. Paul Adrien Maurice Dirac [1902-1984] – a British theoretical physicist who contributed to the early development of quantum mechanics and quantum electrodynamics [QED].
  • 14. Ayn Rand [1905-1982] – Best known for her sweeping intellectual masterpiece Atlas Shrugged, the fiction mystery allowed her to fully develop her philosophy of objectivism.
  • 15. Katherine Houghton Hepburn [1907-2003] – an acclaimed actress in film, television and stage for 73 years of her long life.
  • 16. Jacques Lucien Monod [1910-1976] – awarded the Nobel Prize in Physiology or Medicine in 1965.
  • 17. Padma Vibhushan Subrahmanyan Chandresekhar [1910-1995] – Awarded the Nobel Prize in Physics in 1983 for his important contributions to knowledge about the evolution of stars
  • 18. Alan Mathison Turing [1912-1954] – a mathematician, logician, computer scientist and cryptanalyst from England.
  • 19. Francis Harry Compton Crick [1916-2004] – best known as the co-discoverer of the structure of DNA.
  • 20. Claude Elwood Shannon [1916-2001] – an electronic engineer and mathematician known as “the father of information theory.”
  • 21. Richard Phillips Feynman [1918-1988] – won the Nobel Prize in 1965 for QED
  • 22. Avram Noam Chomsky [b. 1928] – one of the most notable American philosophers of any age. Professor emeritus of linguistics at MIT, and is considered a father of modern linguistics. Also a prolific writer, he has also become famous for being an outspoken political dissident, anarchist, humanist freethinker and libertarian socialist.
  • 23. James Dewey Watson [b. 1928] received the Nobel Prize in physiology or medicine in 1962 as co-discoverer along with Francis Crick and Maurice Wilkins of the molecular structure of DNA.
  • 24. Peter Ware Higgs [b. 1929] – a theoretical physicist and emeritus professor at the University of Edinburgh in Scotland.
  • 25. Warren Edward Buffett [b. 1930] – an American businessman and CEO of Berkshire Hathaway
  • 26. John Rogers Searle [b. 1932] – an American philosopher whose contributions to the philosophy of mind, philosophy of language and social philosophy made him an influential member and spokesperson for the Free Speech Movement in Berkeley during the late 1960s and early ’70s.
  • 27. Steven Weinberg [b. 1933] – an American physicist best known for his work on unification of electromagnetism and the weak force, for which he shared the Nobel Prize in physics in 1979.
  • 28. Carl Edward Sagan [1934-1996] – an American astronomer, astrochemist, and successful popularizer of science.
  • 29. David Takayoshi Suzuki [b. 1936] – a Canadian zoologist, geneticist, science broadcaster and entironmental activist.
  • 30. George Denis Patrick Carlin [1937-2008] – one of the most popular and controversial comedians during his lifetime, having won five Grammy awards for his comedy albums.
  • 31. Bruce Jun Fan Lee [1940- 1973] – an American born Chinese martial artist, philosopher, instructor and actor, the founder of the Jeet Kune Do combat form.
  • 32. Leonard Susskind [b. 1940] – an American physicist specializing in string theory and quantum field theory.
  • 33. Stephen Jay Gould [1941-2002] – a paleontologist, evolutionary biologist and historian of science
  • 34. Clinton Richard Dawkins [b. 1941] – the most prominent scientific atheist in the world today
  • 35. Daniel Clement Dennett [b. 1942] – an American philosopher specializing in the philosophies of mind, science and biology.
  • 36. Stephen William Hawking [b. 1942] – the Lucasian Professor of Mathematics at Cambridge
  • 37. Sir Michael Philip “Mick” Jagger [b. 1943] – singer of Rolling Stones
  • 38. Richard Erskine Frere Leakey [b. 1944] – discovered Australopithecus boisei.
  • 39. David Jon Gilmour [b. 1946] – member of rock group Pink Floyd
  • 40. Brian Eno (Brian Peter George St. John le Baptiste de la Salle Eno) [b. 1948] – an English musician, composer, record producer, music theorist and singer best known as the father of ambient music.
  • 41. David Sloan Wilson [b. 1949] – SUNY Distinguished Professor of Biology and Anthropology at Binghamton University in New York, a prolific popular science writer, and a promoter of evolution by group and multi-level selection.
  • 42. Stephen Gary “Woz” Wozniak [b. 1950] – founder of the Apple computer company with Steve Jobs.
  • 43. Douglas Noel Adams [1952-2001] – an English writer, dramatist and musician, best known for his Hitchhiker’s Guide to the Galaxy series.
  • 44. Steven Arthur Pinker [b. 1954] – an experimental psychologist and cognitive scientist best known for his advocacy of evolutionary psychology and the computational theory of mind.
  • 45. PZ (Paul Zachary) Myers [b. 1957] – an evolutionary developmental biologist and professor of biology at the University of Minnesota, Morris
  • 46. Jodie Foster (Alicia Christian Foster) [b. 1962] – an American film actor
  • 47. Stephen Russell Davies [b. 1963] – a Welsh writer and producer of the modern version of the popular science fiction television series Doctor Who.
  • 48. David John Chalmers [b. 1966] – an Australian philosopher.
  • 49. Sean M. Carroll [b. 1966] – is a theoretical cosmologist at Caltech.
  • 50. Mark Elliot Zuckerberg [b. 1984] – founder of Facebook.

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