Despite evolving independently of Chinese and Babylonian mathematics, India made significant mathematical discoveries at a remarkably early stage in its history.
Mantras dating back to the early Vedic period, prior to 1000 BCE, reveal the invocation of powers of ten ranging from a hundred to a trillion. These mantras also provide evidence of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes, and roots. In the 4th Century CE, a Sanskrit text reported that Buddha enumerated numbers up to 1053 and described an additional six numbering systems, culminating in a number equivalent to 10421. To put this in perspective, it is remarkably close to the estimated 10^80 atoms in the entire universe, nearly approaching infinity by the standards of the ancient world. This text also detailed a series of iterations to illustrate the size of an atom, remarkably close to the actual size of a carbon atom (about 70 trillionths of a meter).
As early as the 8th Century BCE, well before Pythagoras, the “Sulba Sutras” (or “Sulva Sutras”) included a list of simple Pythagorean triples and presented a statement of the simplified Pythagorean theorem for squares and rectangles. In fact, it’s quite plausible that Pythagoras derived his foundational geometry knowledge from these sutras. Furthermore, the Sutras provided geometric solutions for linear and quadratic equations with a single unknown, as well as an impressively accurate approximation of the square root of 2. This approximation involved the sum of 1 + 1⁄3 + 1⁄(3 x 4) – 1⁄(3 x 4 x 34), yielding a value of 1.4142156, accurate to five decimal places.
By the 3rd or 2nd Century BCE, Jain mathematicians recognized five distinct types of infinities: one-directional, two-directional, infinite in area, infinitely pervasive, and perpetually infinite. Ancient Buddhist literature also demonstrated a forward-thinking understanding of indeterminate and infinite numbers, categorizing numbers into three types: countable, uncountable, and infinite.
Much like the Chinese, Indians discovered the advantages of a decimal place value number system, likely using it before the 3rd Century CE. They refined and perfected this system, particularly in the written representation of numerals, laying the foundation for the ancestors of the nine numerals that are now used worldwide. Thanks to its subsequent dissemination by medieval Arabic mathematicians, this contribution is considered one of the most significant intellectual innovations in history.
Earliest Recorded Usage of a Circle Character as Number Zero
The Indians also played a pivotal role in advancing mathematics through a groundbreaking development. The earliest known usage of a circular symbol to represent the number zero is typically attributed to an engraving in a 9th Century temple in Gwalior, central India. However, the ingenious conceptual leap of treating zero as a legitimate number, rather than merely a placeholder or empty space within a number, is commonly credited to Indian mathematicians of the 7th Century, such as Brahmagupta, and possibly another Indian mathematician, Bhaskara I. Although practical use of zero in mathematics may have existed for centuries prior, the introduction of zero as a number in its own right would prove to be a revolutionary advancement in the field.
Brahmagupta laid down fundamental mathematical rules for dealing with zero: 1 + 0 = 1; 1 – 0 = 1; and 1 x 0 = 0. The breakthrough that would provide a logical explanation for the seemingly nonsensical operation of 1 ÷ 0 would also come from an Indian mathematician, Bhaskara II, in the 12th Century. Brahmagupta further established rules for handling negative numbers and recognized that quadratic equations could theoretically have two solutions, one of which might be negative. He even attempted to document these rather abstract concepts, using the initials of color names to represent unknowns in his equations, foreshadowing what we now recognize as algebra.
The period often referred to as the “Golden Age of Indian mathematics” spanned from the 5th to the 12th Centuries. Many of the mathematical discoveries made during this era in India preceded similar findings in the Western world by several centuries. This has led to suggestions of potential influence and even claims of plagiarism by later European mathematicians, some of whom were likely aware of the earlier Indian achievements. It is evident that Indian contributions to mathematics have only received the acknowledgment they deserve relatively recently in modern history.
Indian astronomers used trigonometry tables
During the Golden Age of Indian mathematics, significant strides were made in the realm of trigonometry, a mathematical discipline that originally emerged from Greek origins. Indian mathematicians adeptly harnessed concepts like sine, cosine, and tangent functions, which establish connections between the angles of a triangle and the proportions of its sides. These mathematical tools found diverse applications, from land surveying and maritime navigation to celestial charting.
For example, Indian astronomers utilized trigonometry to compute the relative distances between the Earth, Moon, and Sun. They recognized that during a half-full Moon positioned directly opposite the Sun, the alignment of the Sun, Moon, and Earth forms a right-angled triangle. With precision, they measured this angle at 1⁄7°. Their sine tables provided a ratio for the sides of such a triangle as 400:1, signifying that the Sun resides 400 times farther from the Earth than the Moon.
While the Greeks had calculated the sine function for specific angles, Indian astronomers aspired to compute the sine function for any given angle. The “Surya Siddhanta,” a text of uncertain authorship dating back to around 400 CE, marks the origin of modern trigonometry. It introduces the practical use of sines, cosines, inverse sines, tangents, and secants.
As early as the 6th Century CE, the eminent Indian mathematician and astronomer Aryabhata provided precise definitions for sine, cosine, versine, and inverse sine. He also compiled comprehensive sine and versine tables, detailing values at 3.75° intervals from 0° to 90°, with an accuracy extending to four decimal places. Aryabhata additionally presented solutions for simultaneous quadratic equations and approximated the value of π as 3.1416, accurate to four decimal places. He used this approximation to estimate the Earth’s circumference, arriving at a figure of 24,835 miles, a mere 70 miles off its actual value. Remarkably, he seemed to grasp that π is an irrational number, signifying that any calculation can only ever be an approximation—a concept not proven in Europe until 1761.
Infinity as the Reciprocal of Zero
Bhaskara II, a prominent mathematician from the 12th Century, stands among India’s most accomplished mathematical minds. His notable achievement was demystifying the previously enigmatic concept of division by zero. He observed that dividing one unit into two equal parts results in a half, thus 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3 = 3. By extending this logic, dividing 1 by progressively smaller fractions yields a growing number of pieces. Ultimately, dividing one into infinitesimally small pieces would theoretically yield an infinite number of them, symbolized as 1 ÷ 0 = ∞ (the infinity symbol).
However, Bhaskara II’s contributions to mathematics spanned a wide array of areas. He ventured into solving quadratic, cubic, and quartic equations, even considering negative and irrational solutions. He tackled Diophantine equations of the second order and laid down the groundwork for early concepts of infinitesimal calculus and mathematical analysis. Bhaskara II made significant strides in spherical trigonometry and various aspects of trigonometry. Many of his discoveries predated similar findings in Europe by several centuries, and he excelled in systematizing existing knowledge while enhancing methods for established solutions.
In the late 14th Century, the Kerala School of Astronomy and Mathematics emerged, founded by Madhava of Sangamagrama, often regarded as the greatest mathematician-astronomer of medieval India. Madhava developed infinite series approximations for an array of trigonometric functions, including π and sine. His contributions extended to geometry and algebra, and he pioneered early forms of differentiation and integration for simpler functions. Some of Madhava’s work may have found its way to Europe through Jesuit missionaries, potentially influencing the later development of calculus in European mathematics.
