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I | {{Distinguish|Bhāskara I}} | ||
'''Bhāskara'''{{sfn|Pingree|1970|p=299}} (also known as '''Bhāskarāchārya''' ("Bhāskara the teacher") and as '''Bhāskara II''' to avoid confusion with [[Bhāskara I]]) (1114–1185), was an [[India]]n [[Indian mathematics|mathematician]] and [[astronomer]]. He was born near Vijjadavida (Bijapur in modern Karnataka). Bhāskara is said to have been the head of an [[astronomy|astronomical]] observatory at [[Ujjain]], the leading mathematical center of medieval India. He lived in the [[Western Ghats|Sahyadri]] region (Patnadevi, in Jalgaon district, Maharashtra).{{sfn|Pingree|1970|p=299}} | |||
Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.{{sfn|Chopra|1982|pp=52–54}} His main work ''[[Siddhānta Shiromani]],'' ([[Sanskrit]] for "Crown of treatises,"{{sfn|Plofker|2009|p=71}}) is divided into four parts called ''[[Lilavati|Lilāvati]]'', ''[[Bijaganita]]'', ''Grahaganita'' and ''Golādhyāya''.{{sfn|Poulose|1991|p=79}} These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named [[Karana Kautoohala]]. | |||
Bhāskara's work on [[calculus]] predates [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]] by over half a millennium.{{sfn|Seal|1915|p=80}}{{sfn|Sarkar|1918|p=23}} He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.{{sfn|Goonatilake|1999|p=134}} | |||
==Family== | |||
History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Mahesvara{{sfn|Pingree|1970|p=299}} was as an astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings. | |||
==Mathematics== | |||
Some of Bhaskara's contributions to mathematics include the following: | |||
* A proof of the [[Pythagorean theorem]] by calculating the same [[area]] in two different ways and then canceling out terms to get ''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>. | |||
* In ''Lilavati'', solutions of [[quadratic equation|quadratic]], [[cubic function|cubic]] and [[quartic equation|quartic]] [[indeterminate equation]]s are explained.<ref>Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy</ref> | |||
* Solutions of indeterminate quadratic equations (of the type ''ax''<sup>2</sup> + ''b'' = ''y''<sup>2</sup>). | |||
* Integer solutions of linear and quadratic indeterminate equations (''Kuttaka''). The rules he gives are (in effect) the same as those given by the [[Renaissance]] European mathematicians of the 17th century | |||
* A cyclic [[Chakravala method]] for solving indeterminate equations of the form ''ax''<sup>2</sup> + ''bx'' + ''c'' = ''y''. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the ''chakravala'' method. | |||
* The first general method for finding the solutions of the problem ''x''<sup>2</sup> − ''ny''<sup>2</sup> = 1 (so-called "[[Pell's equation]]") was given by Bhaskara II.{{sfn|Stillwell1999|p=74}} | |||
* Solutions of [[Diophantine equation]]s of the second order, such as 61''x''<sup>2</sup> + 1 = ''y''<sup>2</sup>. This very equation was posed as a problem in 1657 by the [[France|French]] mathematician [[Pierre de Fermat]], but its solution was unknown in Europe until the time of [[Leonhard Euler|Euler]] in the 18th century.<ref>Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy</ref> | |||
* Solved quadratic equations with more than one unknown, and found [[negative number|negative]] and [[irrational number|irrational]] solutions.{{citation needed|date=September 2012}} | |||
* Preliminary concept of [[mathematical analysis]]. | |||
* Preliminary concept of [[infinitesimal]] [[calculus]], along with notable contributions towards [[integral|integral calculus]].<ref>Students& Britannica India. 1. A to C by Indu Ramchandani</ref> | |||
* Conceived [[differential calculus]], after discovering the [[derivative]] and [[differential (calculus)|differential]] coefficient. | |||
* Stated [[Rolle's theorem]], a special case of one of the most important theorems in analysis, the [[mean value theorem]]. Traces of the general mean value theorem are also found in his works. | |||
* Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.) | |||
* In ''Siddhanta Shiromani'', Bhaskara developed [[spherical trigonometry]] along with a number of other [[trigonometry|trigonometric]] results. (See Trigonometry section below.) | |||
===Arithmetic=== | |||
Bhaskara's [[arithmetic]] text ''[[Lilavati|Leelavati]]'' covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, [[plane (geometry)|plane geometry]], [[solid geometry]], the shadow of the [[gnomon]], methods to solve [[Indeterminate (variable)|indeterminate]] equations, and [[combination]]s. | |||
''Lilavati'' is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include: | |||
* Definitions. | |||
* Properties of [[0 (number)|zero]] (including [[Division (mathematics)|division]], and rules of operations with zero). | |||
* Further extensive numerical work, including use of [[negative number]]s and [[Nth root|surds]]. | |||
* Estimation of [[Pi|π]]. | |||
* Arithmetical terms, methods of [[multiplication]], and [[Square (algebra)|squaring]]. | |||
* Inverse [[Cross-multiplication|rule of three]], and rules of 3, 5, 7, 9, and 11. | |||
* Problems involving [[interest]] and interest computation. | |||
* Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important{{citation needed|date=September 2012}}, since the rules he gives are (in effect) the same as those given by the [[renaissance]] European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of [[Aryabhata]] and subsequent mathematicians. | |||
His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore the ''Lilavati'' contained excellent recreative problems and it is thought that Bhaskara's intention may have be. | |||
===Algebra=== | |||
His ''Bijaganita'' ("''[[Algebra]]''") was a work in twelve chapters. It was the first text to recognize that a positive number has two [[square root]]s (a positive and negative square root).<ref>50 Timeless Scientists von K.Krishna Murty</ref> His work ''Bijaganita'' is effectively a treatise on algebra and contains the following topics: | |||
* Positive and [[negative number]]s. | |||
* [[0 (number)|Zero]]. | |||
* The 'unknown' (includes determining unknown quantities). | |||
* Determining unknown quantities. | |||
* [[Nth root#Working with surds|Surd]]s (includes evaluating surds). | |||
* ''Kuttaka'' (for solving [[indeterminate equation]]s and [[Diophantine equation]]s). | |||
* Simple equations (indeterminate of second, third and fourth degree). | |||
* Simple equations with more than one unknown. | |||
* Indeterminate [[quadratic equation]]s (of the type ax<sup>2</sup> + b = y<sup>2</sup>). | |||
* Solutions of indeterminate equations of the second, third and fourth degree. | |||
* Quadratic equations. | |||
* Quadratic equations with more than one unknown. | |||
* Operations with products of several unknowns. | |||
Bhaskara derived a cyclic, [[chakravala method|''chakravala'' method]] for solving indeterminate quadratic equations of the form ax<sup>2</sup> + bx + c = y.<ref>50 Timeless Scientists von K.Krishna Murty</ref> Bhaskara's method for finding the solutions of the problem Nx<sup>2</sup> + 1 = y<sup>2</sup> (the so-called "[[Pell's equation]]") is of considerable importance.{{sfn|Stillwell1999|p=74}} | |||
===Trigonometry=== | |||
The ''Siddhanta Shiromani'' (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered [[spherical trigonometry]], along with other interesting [[trigonometry|trigonometrical]] results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for <math> \sin\left(a + b\right) </math> and <math> \sin\left(a - b\right) </math>: | |||
===Calculus=== | |||
His work, the ''Siddhanta Shiromani'', is an astronomical treatise and contains many theories not found in earlier works{{citation needed|date=September 2012}}. Preliminary concepts of [[Non-standard calculus|infinitesimal calculus]] and [[mathematical analysis]], along with a number of results in [[trigonometry]], [[differential calculus]] and [[integral|integral calculus]] that are found in the work are of particular interest. | |||
Evidence suggests Bhaskara was acquainted with some ideas of differential calculus.<ref>50 Timeless Scientists von K.Krishna Murty</ref> It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement{{citation needed|date=September 2012}}. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of '[[infinitesimal]]s'.{{sfn|Shukla|1984|pp=95–104}} | |||
* There is evidence of an early form of [[Rolle's theorem]] in his work | |||
** If <math> f\left(a\right) = f\left(b\right) = 0 </math> then <math> f'\left(x\right) = 0 </math> for some <math>\ x </math> with <math>\ a < x < b </math> | |||
* He gave the result that if <math>x \approx y</math> then <math>\sin(y) - \sin(x) \approx (y - x)\cos(y)</math>, thereby finding the derivative of sine, although he never developed the notion of derivatives.{{sfn|Cooke|1997|pp=213–215}} | |||
** Bhaskara uses this result to work out the position angle of the [[ecliptic]], a quantity required for accurately predicting the time of an eclipse. | |||
* In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a ''truti'', or a {{Fraction|1|33750}} of a second, and his measure of velocity was expressed in this infinitesimal unit of time. | |||
* He was aware that when a variable attains the maximum value, its [[Differential (infinitesimal)|differential]] vanishes. | |||
* He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.{{citation needed|date=September 2012}} In this result, there are traces of the general [[mean value theorem]], one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by [[Parameshvara]] in the 15th century in the ''Lilavati Bhasya'', a commentary on Bhaskara's ''Lilavati''. | |||
[[Madhava of Sangamagrama|Madhava]] (1340–1425) and the [[Kerala school of astronomy and mathematics|Kerala School]] mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of [[calculus]] in India. | |||
==Astronomy== | |||
Using an astronomical model developed by [[Brahmagupta]] in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the [[sidereal year]], the time that is required for the Earth to orbit the Sun, as 365.2588 days which is the same as in Suryasiddhanta. {{citation needed|date=May 2013}} The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes. {{citation needed|date=May 2013}} | |||
His mathematical astronomy text ''Siddhanta Shiromani'' is written in two parts: the first part on mathematical astronomy and the second part on the [[sphere]]. | |||
The twelve chapters of the first part cover topics such as: | |||
* Mean [[longitude]]s of the [[planet]]s. | |||
* True longitudes of the planets. | |||
* The three problems of [[diurnal motion|diurnal rotation]]. | |||
* [[Syzygy (astronomy)|Syzygies]]. | |||
* [[Lunar eclipse]]s. | |||
* [[Solar eclipse]]s. | |||
* [[Latitude]]s of the planets. | |||
* [[Sunrise equation]] | |||
* The [[Moon]]'s [[crescent]]. | |||
* [[Conjunctions]] of the planets with each other. | |||
* Conjunctions of the planets with the fixed [[star]]s. | |||
* The paths of the Sun and Moon. | |||
The second part contains thirteen chapters on the sphere. It covers topics such as: | |||
*Praise of study of the sphere. | |||
*Nature of the sphere. | |||
*[[Cosmography]] and [[geography]]. | |||
*Planetary [[mean motion]]. | |||
*[[Eccentricity (mathematics)|Eccentric]] [[Deferent and epicycle|epicyclic]] model of the planets. | |||
*The [[armillary sphere]]. | |||
*[[Spherical trigonometry]]. | |||
*[[Ellipse]] calculations.{{Citation needed|date=September 2009}} | |||
*First visibilities of the planets. | |||
*Calculating the lunar crescent. | |||
*Astronomical instruments. | |||
*The [[season]]s. | |||
*Problems of astronomical calculations. | |||
==Engineering== | |||
The earliest reference to a [[perpetual motion]] machine date back to 1150, when Bhāskara II described a [[Bhāskara's wheel|wheel]] that he claimed would run forever.{{sfn|White|1978|pp=52–53}} | |||
Bhāskara II used a measuring device known as ''Yasti-yantra''. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.{{sfn|Selin|2008|pp=269–273}} | |||
==Legends== | |||
Bhaskara II conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity{{citation needed|date=September 2012}}. In his book ''[[Lilavati]]'', he reasons: "In this quantity also which has zero as its divisor there is no change even when many [quantities] have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]".{{sfn|Colebrooke|1817|p = }} | |||
==Notes== | |||
{{Reflist}} | |||
== References == | |||
* {{citation|title=Hindu achievements in exact science: a study in the history of scientific development|first=Benoy Kumar|last=Sarkār|publisher=Longmans, Green and co.|year=1918}} | |||
* {{citation|title=The positive sciences of the ancient Hindus|first=Sir Brajendranath|last=Seal|publisher=Longmans, Green and co.|year=1915}} | |||
* {{citation|title=Arithmetic and mensuration of Brahmegupta and Bhaskara|first=Henry T.|last= Colebrooke|year=1817}} | |||
* {{citation|first=Lynn Townsend| last=White|chapter=Tibet, India, and Malaya as Sources of Western Medieval Technology|title=Medieval religion and technology: collected essays|isbn=978-0-520-03566-9|publisher=University of California Press|year=1978}} | |||
* {{citation|chapter=Astronomical Instruments in India|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition)|editor-first=Helaine |editor-last=Selin |editor-link=Helaine Selin |publisher=Springer Verlag Ny|year=2008|isbn=978-1-4020-4559-2}} | |||
* {{citation | |||
| last =Shukla | |||
| first = Kripa Shankar | |||
| coauthors = | |||
| title = Use of Calculus in Hindu Mathematics | |||
| journal = Indian Journal of History of Science | |||
| volume = 19 | |||
| issue = | |||
| pages = 95–104 | |||
|year=1984 | |||
| url = | |||
| doi = | |||
| accessdate = }} | |||
* {{citation|first=Roger|last=Cooke|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Mathematics of the Hindus|pages=213–215|isbn=0-471-18082-3}} | |||
* {{citation|title=Scientific heritage of India, mathematics|first=K. G.|last=Poulose|volume=Volume 22 of Ravivarma Samskr̥ta granthāvali|publisher=Govt. Sanskrit College (Tripunithura, India)|editor=K. G. Poulose|year=1991}} | |||
* {{citation|last=Chopra|first=Pran Nath|year=1982|title=Religions and communities of India|publisher=Vision Books|isbn=978-0-85692-081-3}} | |||
* {{citation|title=Toward a global science: mining civilizational knowledge|first=Susantha|last=Goonatilake|publisher=Indiana University Press|year=1999|isbn=978-0-253-21182-8}} | |||
* {{citation|title=Mathematics across cultures: the history of non-western mathematics|volume=Volume 2 of Science across cultures|editor-first=Helaine|editor-last=Selin|editor-link=Helaine Selin|editor2-first=Ubiratan|editor2-last=D'Ambrosio|editor2-link=Ubiratan D'Ambrosio|publisher=Springer|year=2001|isbn=978-1-4020-0260-1}} | |||
* {{citation|title=Mathematics and its history, Undergraduate texts in mathematics|first=John|last=Stillwell|publisher=Springer|year=2002|isbn=978-0-387-95336-6}} | |||
==External links== | |||
{{Wikisource|Author:Bhāskara II|Bhāskara II}} | |||
* [http://www.4to40.com/legends/index.asp?p=Bhaskara Bhaskara Biography] | |||
* [http://www.canisius.edu/topos/rajeev.asp Calculus in Kerala] | |||
{{Indian mathematics}} | |||
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. --> | |||
| NAME = Bhaskara 2 | |||
| ALTERNATIVE NAMES = Bhaskaracharya | |||
| SHORT DESCRIPTION = Indian mathematician and astronomer | |||
| DATE OF BIRTH = 1114 | |||
| PLACE OF BIRTH = | |||
| DATE OF DEATH = 1185 | |||
| PLACE OF DEATH = | |||
}} | |||
{{DEFAULTSORT:Bhaskara 2}} | |||
[[Category:12th-century mathematicians]] | |||
[[Category:Medieval Indian mathematicians]] | |||
[[Category:Medieval Indian astronomers]] | |||
[[Category:People from Bijapur, Karnataka]] | |||
[[Category:1114 births]] | |||
[[Category:1185 deaths]] | |||
[[Category:Algebraists]] |
Revision as of 05:34, 7 March 2013
Template:Distinguish BhāskaraTemplate:Sfn (also known as Bhāskarāchārya ("Bhāskara the teacher") and as Bhāskara II to avoid confusion with Bhāskara I) (1114–1185), was an Indian mathematician and astronomer. He was born near Vijjadavida (Bijapur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of medieval India. He lived in the Sahyadri region (Patnadevi, in Jalgaon district, Maharashtra).Template:Sfn
Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.Template:Sfn His main work Siddhānta Shiromani, (Sanskrit for "Crown of treatises,"Template:Sfn) is divided into four parts called Lilāvati, Bijaganita, Grahaganita and Golādhyāya.Template:Sfn These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karana Kautoohala.
Bhāskara's work on calculus predates Newton and Leibniz by over half a millennium.Template:SfnTemplate:Sfn He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.Template:Sfn
Family
History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father MahesvaraTemplate:Sfn was as an astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings.
Mathematics
Some of Bhaskara's contributions to mathematics include the following:
- A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a2 + b2 = c2.
- In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.[1]
- Solutions of indeterminate quadratic equations (of the type ax2 + b = y2).
- Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century
- A cyclic Chakravala method for solving indeterminate equations of the form ax2 + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.
- The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhaskara II.Template:Sfn
- Solutions of Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.[2]
- Solved quadratic equations with more than one unknown, and found negative and irrational solutions.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
- Preliminary concept of mathematical analysis.
- Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.[3]
- Conceived differential calculus, after discovering the derivative and differential coefficient.
- Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.
- Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
- In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)
Arithmetic
Bhaskara's arithmetic text Leelavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.
Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:
- Definitions.
- Properties of zero (including division, and rules of operations with zero).
- Further extensive numerical work, including use of negative numbers and surds.
- Estimation of π.
- Arithmetical terms, methods of multiplication, and squaring.
- Inverse rule of three, and rules of 3, 5, 7, 9, and 11.
- Problems involving interest and interest computation.
- Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly importantPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park., since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.
His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore the Lilavati contained excellent recreative problems and it is thought that Bhaskara's intention may have be.
Algebra
His Bijaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root).[4] His work Bijaganita is effectively a treatise on algebra and contains the following topics:
- Positive and negative numbers.
- Zero.
- The 'unknown' (includes determining unknown quantities).
- Determining unknown quantities.
- Surds (includes evaluating surds).
- Kuttaka (for solving indeterminate equations and Diophantine equations).
- Simple equations (indeterminate of second, third and fourth degree).
- Simple equations with more than one unknown.
- Indeterminate quadratic equations (of the type ax2 + b = y2).
- Solutions of indeterminate equations of the second, third and fourth degree.
- Quadratic equations.
- Quadratic equations with more than one unknown.
- Operations with products of several unknowns.
Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y.[5] Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation") is of considerable importance.Template:Sfn
Trigonometry
The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for and :
Calculus
His work, the Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier worksPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.
Evidence suggests Bhaskara was acquainted with some ideas of differential calculus.[6] It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievementPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.Template:Sfn
- There is evidence of an early form of Rolle's theorem in his work
- He gave the result that if then , thereby finding the derivative of sine, although he never developed the notion of derivatives.Template:Sfn
- Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.
- In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a Template:Fraction of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
- He was aware that when a variable attains the maximum value, its differential vanishes.
- He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.
Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of calculus in India.
Astronomy
Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is the same as in Suryasiddhanta. Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes. Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.
The twelve chapters of the first part cover topics such as:
- Mean longitudes of the planets.
- True longitudes of the planets.
- The three problems of diurnal rotation.
- Syzygies.
- Lunar eclipses.
- Solar eclipses.
- Latitudes of the planets.
- Sunrise equation
- The Moon's crescent.
- Conjunctions of the planets with each other.
- Conjunctions of the planets with the fixed stars.
- The paths of the Sun and Moon.
The second part contains thirteen chapters on the sphere. It covers topics such as:
- Praise of study of the sphere.
- Nature of the sphere.
- Cosmography and geography.
- Planetary mean motion.
- Eccentric epicyclic model of the planets.
- The armillary sphere.
- Spherical trigonometry.
- Ellipse calculations.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
- First visibilities of the planets.
- Calculating the lunar crescent.
- Astronomical instruments.
- The seasons.
- Problems of astronomical calculations.
Engineering
The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever.Template:Sfn
Bhāskara II used a measuring device known as Yasti-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.Template:Sfn
Legends
Bhaskara II conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinityPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change even when many [quantities] have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]".Template:Sfn
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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42 year old Obstetrician and Gynaecologist Merle from Komoka, has interests including entertaining, property developers house in singapore singapore and train collecting. Just had a family trip to China Danxia.
- ↑ Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
- ↑ Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
- ↑ Students& Britannica India. 1. A to C by Indu Ramchandani
- ↑ 50 Timeless Scientists von K.Krishna Murty
- ↑ 50 Timeless Scientists von K.Krishna Murty
- ↑ 50 Timeless Scientists von K.Krishna Murty