Brahmagupta

Brahmagupta-theorem

Early Life

Ever wondered who would have thought that zero could be a real number? Well, the mathematician that came from India did.

Not much is known of the childhood and early years of Brahmagupta. However as obscure as his beginnings may be, his work in math and science stands out on its own merit.

In the year 598, Brahmagupta was born in Ujjain, which was an ancient city in India. An interesting bit of significance in his birth there is that many of the day considered Ujjain to have astronomical importance. Ujjain in its day was a hub for the study of mathematics and astronomy; basically, a mediation point of the importance of time and its understanding.

Apparently there is some discrepancy as to the possibility that Brahmagupta might have been born in Bhillamala; But it is a for a surety that he grew up in that city. He practiced the Hindu religion of Shaivite. This religion was more about how they projected themselves to their god, rather than following strict adherence to Hinduism tenets.

A final note about his birthplace is that there was even one consideration he came from the Abu region in India. Even though his birthplace may be in question; there is no question as to his place in mathematical history.

Studies and Discoveries

Brahmagupta loved the studies of the heavens and became an astronomer of one of four major Indian astronomy schools; the Brahmapaksha. His studies followed some of the notables in Indian astronomy such as Varahmihara, Aryabhata and Simha. These and other names in the Indian astronomy world were indicative of the focus that Brahmagupta found most interesting.

The young Indian’s passion for astronomy took a backseat to nothing of his day. He absorbed himself in learning all he could about the subject. There were five siddhanthas on the topic of Indian astronomy and Brahmagupta was well versed on all of them in a short period of time.

Brahmagupta was a well versed writer in areas of astronomy and mathematics. However, his work did not treat them as separate subjects; he played the two together considerably in much of his work. Through his research, he postulated a number of concepts that far surpassed many scholars in the arena of movement in the sky. He had established sound cycle motions of planetary motion across the sky and could accurately predict where they would rise and fall in the occurrence of each day.

Another area in astronomy that Brahmagupta excelled was in establishing clearer thought on the area of eclipses. For many cultures, the concept of the sun being blocked by the moon or alignment ideas was a spiritual matter. Where the Indian astronomer came in was to design computations that accurately dated eclipse happenings. His work in this arena caused many to look at the young mathematician with greater respect.

When Brahmagupta was 30 years of age, he wrote his first work which was a revised look at Brahma astronomy. Entitled Brāhmasphuṭasiddhānta, the 25-chapter manuscript looked at standard topics of period mathematical astronomy that was part of the Indian study. These chapters dealt with lunar and solar eclipses, longitudes of planets (both true and mean), shadowing of the moon and other areas of interest. These chapters seemed to be a repeat of what was considered to be known and simply restated.

The remaining chapters appeared to be a re-look at the viewed ideas and even believed to be refutations of some of the work. His purpose was not necessarily to debunk prior scholars, but to add original ideas on aspects of celestial concepts. His treatise considers more in the use of accurate examples of geometry, algebra, trigonometry to answer some of the questions that plagued astronomers for centuries. His take on things was a challenge to the siddhanta he has received in his earlier learning.

It should be noted that Brahmagupta’s postulations into the movement in the heavens considered the earth to be a stationary body. (There is a belief that Brahmagupta may have purposely held to the stationary earth idea to save his own life. The religion of the time held to the concept of the earth did not move as the earth was the center of all things.) Although his work articulated that motion was occurring with other planets and stars; everything hinged on this planet being a static position. In this book he sets forth the concept that one year equals 365 days, six hours, five minutes and 19 seconds. In the second book, to be discussed, he adds seven minutes to the idea; similar to that of Indian astronomer and mathematician Aryabhata.

A couple points of note about math in this first of his works. In one chapter, Brahmagupta creates a beginning for the mindset of arithmetic. He discusses the Indian math of “pati-ganita” or “mathematics of procedures”. Because the man had a deep passion of mathematics as a whole, this chapter seems to be an expose of himself in his love. He is frank about practical math and operations. It was sort of the math to the populace, or a look at “practical math”. He relates that this math had to be a part of a mathematician. Another chapter detailed the work involved in algebra. Although the name itself did not exist, Brahmagupta gives great detail to area itself for what became our view of algebraic equations.

Brahmagupta’s work with mathematics, including algorithms, in the area of astronomy opened up new frontiers for astronomers to follow. More than that, it seemed to open up the man himself to his peers. This was not to be his last publication.

One of Brahmagupta’s career positions was to become the lead (modern day position of director) over the astronomy observatory in his hometown of Ujjain. Prior directors had been notables of the day and had established a huge reputation for the observatory. It seemed to fit that Brahmagupta would continue the outstanding work that came from that institution.

Though there is no basis as to verify that his first book affected the Islamic look at math and astronomy, there is a good chance that it did. Some felt that his first treatise impressed the scholars in the learning center of Baghdad and may have obtained a copy of the mathematical astronomy publication. Then King Khalif Abbasid al-Mansoor is said to have requested the presence of a student of Brahmagupta’s teachings of astronomy to come give lectures. Translated into Arabic, the works of Brahmagupta would influence much of Arabic mathematical understanding.

This affect upon the math and sciences of the Middle East made its way to influence much of the same disciplines of Europe. The ancient world’s look at mathematics, via the realm of Brahmagupta’s involvement, impacted a considerable amount upon the western culture.

Another work that Brahmagupta published occurred when he was in his mid-60’s. Approximately in 665 he developed the treatise called Khandakhadyaka. This work was considered to be done in response to the work of one of Brahmagupta’s fellow scholars, Aryabhata who wrote Ardharatrikapaksa. Aryahata was considered to be one that stipulated that each day began with the midnight hour.

Khandakhadyakais made up of 8 chapters. These chapters look many of the same topics found in the first, such as eclipses, planet risings and settings, planet conjunctions and other ideas. Once again this seemed to be an attempt to enhance some of the ideas as put out by Aryabhata.This second work also paid an even closer look at mathematics in general. It puts a deeper emphasis on computing values that involved sines. Thirty years later Brahmagupta still researches and continues to improve his work with geometry and algebra.

He is one of the first in the realm of mathematics to use story problems to showcase his ideas. Without going into the actual story verbatim, Brahmagupta uses the idea loaning money to one entity and then some to another. The finish was to come up with the rate of interest on the monies distributed.

There is considerable in modern mathematics that can be traced back to the work of Brahmagupta. One such method was in solving quadratic equations. Brahmagupta came up with a theorem (named after him) for figuring out the area of cyclic quadrilateral. This included figuring diagonal lengths as well.

Ancient math, such as geometry and trigonometry, had remained a constant in many ways up to the time of Brahmagupta. He had a tenacity for delving into the research of commonly known equations and breaking them down to see if they were accurate. Over time this man would develop ideas that challenged and built upon the former ideas; similar to how he looked at mathematical astronomy.

The most notable innovations that Brahmagupta is remembered for is his look and pursuit of the number zero. Until this time, the zero had not been thought of as a number. The formation of the zero shape was a commonplace consideration; as was the concept of the value of places (numerical placement) to form importance to numbers. These had pre-dated Brahmagupta and were considerations from the ancient world. But it was Brahmagupta that established rules that put zero into the spotlight. He formulated equations that allowed zero to be used in positives and negatives. Although he did not use those specific words, Brahmagupta did follow with the importance in the need of the zero placement. Without the use of zero and its value defined, according to him, arithmetic really had nowhere to go.

When it came to the zero, the Indian mathematician even enhanced the knowledge of zero in relation to multiplication. When it came to division he seemed to have incorrect assumptions established. But, it cannot be denied that Brahmagupta brought to the world of arithmetic the value and need for a neutral digit that was zero.

Brahmagupta’s Look at Zero listed:

  • A debt minus zero is a debt.
  • A fortune minus zero is a fortune.
  • Zero minus zero is a zero.
  • A debt subtracted from zero is a fortune.
  • fortune subtracted from zero is a debt.
  • The product of zero multiplied by a debt or fortune is zero.
  • The product of zero multipliedby zero is zero.
  • The product or quotient of two fortunes is one fortune.
  • The product or quotient of two debts is one fortune.
  • The product or quotient of a debt and a fortune is a debt.
  • The product or quotient of a fortune and a debt is a debt.

The belief that the earth was flat as a pancake was a predominate belief in the ancient world; even into the middle ages. Although he was incorrect in his belief that the sun moved around the earth, since the earth had to be static in its existence; Brahmagupta was correct in giving a circular shape to the planet.

Other Insights of Brahmagupta

  1. The earth had a circumference of approximately 22,500 miles.
  2. Gave a close parameter to what pi is known to be as 3 or 3.16 if the measurement of a circle’s circumference to diameter ration needed to be more precise.
  3. Positive and negative numbers rule:
  • When a negative number is subtracted from a positive; it is the same as adding two positive numbers. (2- -3=5; 2+3=5)
  • Adding two negative numbers results in a negative number.
  • Multiplying two negatives is same as multiplying two positive numbers.
  • dividing a positive number by a negative, or a negative number by a positive result in a negative number.

Brahmagupta passed away in approximately 670. It appears part of his life’s work was a struggle between the Hindu religion and his own scientific mind’s research into astronomy from a mathematician’s perspective. Brahmagupta cannot be denied his place in establishing much of contemporary work. For example, his teachings of the property “zero” influenced work in modern thermodynamics. Brahmagupta in some ways was a man born out of time in the way he looked at early math and astronomy.

One of his quotes accurately portrays who he was. “As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them.”
F Cajori, “A History of Mathematics”

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