tag:blogger.com,1999:blog-3948816397620270756.post5922202313662688355..comments2023-12-04T07:52:16.028-08:00Comments on KARAKALPAKSTAN BLOG: The Origins of Algebra - al khorezmiVance Painterhttp://www.blogger.com/profile/13211389044326396600noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-3948816397620270756.post-90478360284309302802013-02-08T04:50:38.404-08:002013-02-08T04:50:38.404-08:00In the Fibonacci sequence, every number after 0 an...In the Fibonacci sequence, every number after 0 and 1 is the sum of the previous two numbers, so that the sequence runs: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10,946 and so on. The series appears in nature in many forms, including the spiral arrangements of sunflower seeds, pineapple fruitlets and pinecone scales; it appears in geometry, where, starting with the number 5, every other Fibonacci number is the length of the hypotenuse of a Pythagorean right triangle with integral sides; it recurs in mathematics, where the ratio between successive Fibonacci numbers approaches the classical “golden ratio” of 1:1.618033…. <br /><br />Like the Polish astronomer Copernicus and the Spanish physician Michael Servetus in the 16th century, Fibonacci, who was one of the founders of western mathematics, constructed a substantial portion of his pioneering scientific research on the foundations laid by his Arabic-speaking predecessors. Using Latin translations of Muhammad ibn Musa al-Khwarizmi’s treatises on algebra and algorithms, Fibonacci, also known as Leonard of Pisa, wrote the Liber abaci, the first widely available book on Arabic numerals and arithmetical problems, expanding Indian-based concepts that had arrived in Spain starting in the 10th century. <br />Vance Painterhttps://www.blogger.com/profile/13211389044326396600noreply@blogger.comtag:blogger.com,1999:blog-3948816397620270756.post-62824924781541981832009-11-14T00:49:06.399-08:002009-11-14T00:49:06.399-08:00Al-Khwarizmi and quadratic equations.
The treatis...Al-Khwarizmi and quadratic equations.<br /><br />The treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. In this book, al-Khwarizmi gives a complete solution to all possible types of quadratic equation. <br /><br />Here is an example to al-Khwarizmi's solution of the equation: <br /><br />x2 + 21 = 10x. <br /><br />In this case the quadratic has two solutions.<br /><br />He commented " Halve the number of the roots. It is 5. Multiply this by itself and the product is 25. Subtract from this the 21 added to the square term and the remainder is 4. Extract its square root, 2, and subtract this from half the number of roots, 5. There remains 3. This is the root you wanted, whose square is 9. <br /><br />Alternatively, you may add the square root to half the number of roots and the sum is 7. This is then the root you wanted and the square is 49. When you meet an instance which refers you to this case, try its solution by addition, and if that does not work subtraction will. In this case, both addition and subtraction can be used, which will not serve in any other of the three cases where the number of roots is to be halved. <br /><br />Know also that when, in a problem leading to this case, you have multiplied half the number of roots by itself, if the product is less than the number of dirhams added to the square term, then the case is impossible. On the other hand, if the product is equal to the dirhams themselves, then the root is half the number of roots".Vance Painterhttps://www.blogger.com/profile/13211389044326396600noreply@blogger.com