Brahmagupta - 598 AD
Born in the year 598 AD, Brahmagupta is argued to be one of the most influential mathematicians and astronomer of ancient India. He was the head of the astronomical observatory in Ujjain as well. In his book Brahmasphutasiddhānta, he explains the base of a lot of modern-day algebra, defines negative numbers, explores rules of 0, method of calculating square roots, outlines the basis of the Pythagorean theorem, further advances Aryabhata's theory and explains (his own) Brahmagupta's theorem.
Fun Fact: Brahmagupta never considered himself a mathematician. Apparently, this guy did maths for his own pleasure... wouldn't he be fun at parties.
To understand Brahmagupta's work, one needs to understand basic arithmetic operations. You may be patting yourself on the back and thinking "I'm smart enough to understand the works of a genius", if you are, you deserve an award for being overconfident. At the beginning of his 12th chapter called Calculation, he describes operations on fractions, how to find the cube and cube root of an integer and then gives rules to help find squares and square roots.
He also gives the sum of the squares of the first n natural numbers is n(n + 1)(2n + 1)/6 and the sum of the cubes of the first n natural numbers as (n(n + 1)/2)^2, finding the answer to your pattern investigation which you cried over.
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Brahmagupta also provides rules for using 0 in chapter 18, which at that time was a strange and unusual concept. What may seem really straight forward to you, was revolutionary at that time. His verses on describing addition and subtraction:
18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero [...]
18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added
This is also where he describes negative numbers, a never before heard concept, which is the basis for our society today. Without his work on negative numbers, there would be no concept of debt or monetary loss, and you'd be sitting at home with $0 instead of -$20 in your bank account.
Describing multiplication:
18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero
However, his description of division differs from ours, as numbers divided by 0 are undefined but he did not commit to that and left it as a/0, with no further explanation. Despite that, his work on 0 helped make sense of the mystical number which would be the basis of so many mathematical theories today.
Bhramagupta was also the first to write an arithmetic solution in words instead of symbols (including zero and negative solutions) to quadratic equation. He called this avyakt ganit.
Brahmagupta also provides a formula for generating the Pythagorean triplets in a convoluted riddle "12.39. 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". Meaning, if a right-angled triangle has a base of length of a = mx and the altitude of length b = m+d, then the length, c, of its hypothesis is c = m(1+x) - d. And with algebraic manipulation, you get the formula a^2 + b^2 = c^2.
One of his most impressive accomplishments was to generate solutions to certain parts of Diophantine equations of the second degree such as Nx2 + 1 = y^2. Now known as Pell's equation (which was discovered by John Pell in the 17th century). This method was known as Pulverizer since it broke down numbers into smaller pieces. For a better explanation, here is a snippet from the book (pg.29) Pride of India: A Glimpse Into India's Scientific Heritage
Some scholars argue (I would too if I understood), that Brahmagupta's most famous results were a formula for the area of a cyclic quadrilateral (a four-sided polygon whose vertices all reside on some circle) and the length of its diagonals in terms of its sides. In geometry, Brahmagupta's theorem states if a cyclic quadrilateral is an orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side, as seen in the picture.
He also creates a sine table in his second chapter called "Planetary Longitudes" and devised and used a special case of the Newton–Stirling interpolation formula of the second-order.
References:
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"Brahmagupta". En.Wikipedia.Org, 2020, https://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007. Accessed 2 Dec 2020.
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"Brahmagupta Theorem". En.Wikipedia.Org, 2020, https://en.wikipedia.org/wiki/Brahmagupta_theorem. Accessed 2 Dec 2020.
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"Brahmagupta - Biography". Maths History, 2020, https://mathshistory.st-andrews.ac.uk/Biographies/Brahmagupta/. Accessed 2 Dec 2020.
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Hayashi, Takao, "Brahmagupta | Indian Astronomer". Encyclopedia Britannica, 2020, https://www.britannica.com/biography/Brahmagupta. Accessed 9 Dec 2020.
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"Brahmagupta - Indian Mathematics - The Story Of Mathematics". Lukemastin.Com, 2020, http://www.lukemastin.com/storyofmathematics/indian_brahmagupta.html. Accessed 9 Dec 2020.
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"Algebra". En.Wikipedia.Org, https://en.wikipedia.org/wiki/Algebra#Early_history_of_algebra. Accessed 9 Dec 2020.
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“Mathematics.” Pride of India: A Glimpse into India's Scientific Heritage, by Pradeep Kolhe et al., Samskrita Bharati, 2006, pp. 27–55.
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“Brahmagupta - Biography, Facts and Pictures.” Famousscientists.org, 2014, www.famousscientists.org/brahmagupta/. Accessed 31 Jan. 2021.