view of ancient Indian knowledge of multiplication.

Ancient Books Show the Story of Multiplication

By Raj Vedam

27 Oct 2019

Ancient Indians had powers of 10 enumerated in the Rg Veda, which names numbers as large as 10 to the power of 62 and more. Surya Siddhanta shows large and small numbers, measuring time from 10 to the power of 22 seconds up to 10 to the power of minus 7 seconds. The Yajur Veda discusses the nature of infinity in the famous “Purnamidam” mantra. Later Buddhist works name very large numbers, with 10 to the power of 421, and much larger. These instances show knowledge of doubling as a means for multiplication and diminishing as a means for division from the very early Vedic era thru the Buddhist era.

Baudhayana who is Western-dated questionably to 800 BCE shows knowledge of squares and square-roots in Sulba Sutras. Aryabhata (499 CE) in Aryabhatiya proposed an algorithm called “Kuttaka” to solve linear equations in integer variables which required multiplication and division. By the time of Brahmagupta (625 CE), ancient Indians had rules for multiplication with negative positive and zero numbers as seen in Brahmasputasiddhanta. From Bhaskara II (1125 CE) to the time of Madhava (1400 CE) saw the development of rudimentary calculus, as well as infinite series expansion of several functions, showing advanced knowledge.

It is instructive to see how multiplication was done in other societies in the ancient world.

The Egyptians used repeated doubling of numbers to multiply as seen in their earliest work, “Moscow papyrus” of 1800 BCE, a period after contact with migrants from India, around 1900 BCE.

The Sumerians who worked with the Harappans had a base-60 system which was inherited in neo-Babylonia, and between 300- 400 BCE, we see multiplication tables till 59 in Cuneiform tablets. Ancient Indian trade with the Sumerians, Kassites, Babylonians thru ports from Bharuch and Lothal to ancient Red Sea and Persian Gulf ports permitted transmission of Indian knowledge in various time periods, accounting for similarities seen in stories, medicine, astronomy and math.

The Greeks used repeated doubling similar to the Egyptians to do rudimentary multiplication, and are known to have origins in the Pythagorean School, influenced by ancient India.

The Chinese had multiplication tables at least by 300 BCE as seen in a recent find by Tsinghua Univ (see the picture). The book Nine Chapters on Mathematical Arts (between 100 CE – 250 CE) even discussed a method of solving simultaneous linear equations, indicating knowledge of multiplication and division. The Chinese benefited with Buddhist knowledge transfers of Indian math and medicine over an extended period of time.

The Arabs learnt arithmetic and astronomy including several other fields of study from Sanskrit works translated to Arabic and Persian in the 8th-12th centuries, notably using Al Khwarizmi’s 825 CE work, On the Calculation with Hindu Numerals, as well as translations of Brahmagupta’s works, which included decimal place value multiplication.

Europe learnt about Indian numerals via Arabic works translated to Latin such as Algoritmi de numero Indorum in the 11th-13thcenturies in Toledo, but were thwarted by the Church for use of “Satan’s numbers”, fearful of “zero” and “infinity”.

Fibonacci introduced Indian numerals to Western Europe in 1200s, but adoption was slow till invention of the printing press and works by Adam Ries, a German who wrote a popular text showing how to do arithmetic using Indian numerals in 1500s.

In Britain, the first evidence we have of place-value multiplication is in Robert Recorde’s works, for example in his 1556 book, The Castle of Knowledge, (see the picture below, where a number is cubed), many thousands of years after Indians were routinely multiplying large numbers.

One has to wonder how Britain matured in math from the elementary arithmetic in this 1556 book to Isaac Newton (1643-1727), allegedly inventing calculus in deep rivalry with Leibnitz, in about 100 years of gestation of mathematical ideas and principles.

And at the same time, India that was light-years ahead of Europe in math went behind by 1700s, and its educational system lay in ruins by 1850s.(See Sahana Singh's book, The Educational Heritage of Ancient India: How an Ecosystem of Learning Was Laid to Waste).

While there is no doubt about the development of modern mathematics post 1700s in Europe even as classical civilizations were being ravaged by the colonialists, their dearth of citation to earlier works that they built upon has led to a regrettable industry of Eurocentric works that posit the birth of all math in Babylon, Greece and Europe, with just grudging nods to Indian, Chinese and Arabic mathematicians of ancient times.