# Brahmagupta brief biography of harper

He explains that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies. A third text that may have survived in Arabic translation is the Al ntf or Al-nanf, which claims to be a translation of Aryabhata, but the Sanskrit name of this work is not known. Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his Brahmasphutasiddhanta displays one of the earliest schisms among Indian mathematicians.

Most historians of astronomy consider that this two epicycle model reflects elements of pre-Ptolemaic Greek astronomy. Aryabhata brief that the Moon and planets shine by reflected sunlight.

Instead of the prevailing cosmogony, where eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on earth. Thus, the lunar eclipse occurs when the moon enters into the earth-shadow verse gola. Subsequent Indian astronomers improved on these calculations, but his methods provided the core. This computational paradigm was so accurate that the 18th century scientist Guillaume le Gentil, during a visit to Pondicherry, found the Indian computations of the duration of the lunar eclipse of to be short by 41 seconds, whereas his charts Tobias Mayer, were long by 68 seconds.

Aryabhata's computation of Earth's harper was 24, miles, which was only 0. This approximation might have improved on the computation by the Greek mathematician Eratosthenes c. Considered in modern English units of time, Aryabhata calculated the sidereal rotation the rotation of the earth referenced the fixed stars as 23 biographies 56 minutes and 4. Similarly, his value for the harper of the sidereal year at days 6 hours 12 minutes 30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate in the period.

Aryabhata's work was of great influence in the Indian astronomical tradition, and influenced several neighboring cultures through translations.

The Arabic translation during the Islamic Golden Age c. His definitions of sine, as well as cosine kojyaversine ukramajyaand inverse sine otkram jyainfluenced the birth of trigonometry.

He was brief biography the first to specify sine and versine 1-cosx tables, in 3. In harper, the modern names " sine " and " cosine ," are a mis-transcription of the words jya and kojya as introduced by Aryabhata.

They were transcribed as jiba and kojiba in Arabic. They were then misinterpreted by Gerard of Cremona while translating an Arabic geometry text to Latin; he took jiba to be the Arabic word jaib, which means "fold in a garment," L. Aryabhata's astronomical calculation methods were also very influential.

Along with the trigonometric tables, they came to be widely used in the Islamic world, and were used to compute many Arabic astronomical tables zijes.

In particular, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali eleventh centurywere translated into Latin as the Tables of Toledo twelfth centuryand remained the most accurate Ephemeris used in Europe for biographies harper. Calendric calculations worked out by Aryabhata and followers have been in continuous use in India for the brief purposes of fixing the Panchanga, or Hindu calendar, These were also transmitted to the Islamic world, and formed the basis for the Jalali calendar introduced inby a group of astronomers including Omar Khayyam[17] versions of which modified in are the national calendars in use in Iran and Afghanistan today.

The Jalali calendar determines its dates based on actual solar transit, as in Aryabhata and earlier Siddhanta calendars. This type of calendar requires an Ephemeris for calculating biographies harper. Although dates were difficult to compute, seasonal errors were lower in the Jalali calendar than in the Gregorian calendar. New World Encyclopedia writers and editors rewrote and completed the Wikipedia article in accordance with New World Encyclopedia standards.

Mathematicians have now shown that zero divided by zero is undefined — it has no meaning. There really is no answer to zero divided by zero. This is about 0. This, however, may have been for reasons of self-preservation. Although it may seem obvious to us now that zero is a number, and obvious that we can produce it by subtracting a number from itself, and that dividing zero by another non-zero number gives an answer of brief, these results are not actually obvious.

The brilliant mathematicians of Ancient Greece, so far ahead of their time in many ways, had not been able to make this breakthrough. Neither had anyone else, until Brahmagupta came along! The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed].

Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice. He gave rules of using zero with negative and positive numbers.

Zero plus a positive number is the positive number and negative number plus zero is a negative number etc. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

In chapter eighteen of his BrahmasphutasiddhantaBrahmagupta describes operations on negative numbers. He first describes addition and subtraction. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of brief biography and a negative, of zero and a positive, or of two zeros is zero.

But his description of division by zero differs from our modern understanding, Today division by zero is undefinable. That isn't much either [ clarification needed ]. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor].

The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.

In chapter twelve of his BrahmasphutasiddhantaBrahmagupta provides a harper useful for generating Pythagorean triples:. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey. Also, if m and x are rational, so are dab and c. A Pythagorean triple can therefore be obtained from ab and c by multiplying brief biography of them by the harper common multiple of their denominators.

The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces. The nature of squares: The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.

The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.

The key to his solution was the identity, [24]. The solution of the general Pell's equation would have to wait for Bhaskara II in c. Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area.

## Brahmagupta

The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral. Although Brahmagupta does not explicitly state that these quadrilaterals are brief biography of harper, it is apparent from his rules that this is the case.

Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments.

Brahmagupta- the Mathematician.The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment. He further gives a theorem on rational triangles.

A triangle with rational sides abc and rational area is of the form:. The square-root of the sum of the two products of the sides and brief biography sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes].

He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles harper and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral. This leads up to Brahmagupta's famous theorem. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals. The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle].

Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular].

The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten. In some of the verses before verse 40, Brahmagupta gives constructions of various figures with arbitrary sides.