• Mohor Sengupta

Ever wondered where numbers came from?

Updated: Oct 16, 2019

This article is really about the history, and not the math. But let’s begin with a bit of math refresher.

Modern arithmetic uses the numbers 0 to 9 for all calculations. This system of numerals has a base 10 and is a place value (positional) system, where the position of a number tells its value. For example, in 2019, 2 is in thousand’s place and has the highest value (2000); 9 is in one’s place, so its value is 9. Brings back memories of the first-grade class-teacher?

Since 2019 has no number in the hundred’s place and has 1 in ten’s place, their positions are denoted by a 0 and 1 respectively. Zero is a place holder, and one of the most ingenious mathematical inventions. This system of numerals began to be widely used in the Western world after the fifteenth century CE. However, its origins lie far away from Europe and long back in time. Also known as the Hindu-Arabic numeral system, 0-9 originated and evolved in India, was adopted by travelers from the Arab world, and disseminated by them into Europe, where it was accepted much later.

We will travel back and forth in time, but let’s start our story with the Arab traveler al-Biruni. In the 1020s when he was traveling across India, then a new conquest of his ruler, the Mahmud of Ghazni, al-Biruni came across inscriptions in caves and on coins in central and eastern parts of the country. Historians have dated these inscriptions and coins to sometime in the 3rd and 4th centuries BCE. This time roughly corresponds to the Mauryan Empire that ruled a large part of the Indian subcontinent and had an iconic ruler who left his mark in numerous edicts found across modern day Afghanistan to Bihar, and Tibet to Madhya Pradesh. But more on that later. What al-Biruni saw was not an ancient language, but numbers. This was clear because 1, 2 and 3 were represented by vertical lines (I, II, III), like Roman numerals.

The first Brahmi numerals used by Ashoka in his edicts circa 250 BCE. Source: Wikimedia Commons.

The script used in northern India at that time was Brahmi. Therefore, these numerals are called Brahmi numerals. This wasn’t a place value system. That’s why there are distinct symbols for every number from 1 to 9, and onwards, in multiples of 10, such as 50 and 200, as shown above. It was a complicated system with many symbols, each denoting a number! In one of his books written in 1030s, al-Biruni wrote: “Whilst we use letters for calculation according to their numerical value, the Indians do not use letters at all for arithmetic. And just as the shape of the letters that they use for writing is different in different regions of their country, so the numerical symbols vary.”

al-Biruni gazing at Ashoka's inscriptions on a pillar. Illustration by Arghya Manna.

The Brahmi numerals had the base 10. We know this because there are separate signs for 1 to 9, 10, 20, 30…to 90, and 100, 200, 300…to 1000. A multiple of 100 or 1000 was denoted by modifying the individual signs to accommodate a multiplier sign. A very complicated system indeed! Also, zero was unknown at that time. In fact, the first written evidence of the use of zero, as we will see, is from at least from 600 years later.

The Brahmi script was deciphered early to mid-19th century when some enthusiastic employees of the East India Company discovered edicts of the Mauryan emperor Ashoka strewn across the subcontinent. One man devoted his life’s passion to cracking the Brahmi script. His name was James Princep. Brahmi numerals on Ashoka’s edicts and coins from that period proved to historians beyond doubt that these bore the same number system as al-Biruni’s descriptions. Over time the Brahmi numerals evolved into different shapes. Most notably, 1, 2 and 3 became a set of horizontal lines instead of vertical.

Brahmi numerals from 2nd century CE. Source: Wikimedia Commons.

In the summer of 1881, in a village called Bakhshali, now near Peshawar in Pakistan, a local person discovered several leaves of birch-bark, some in precarious condition. The leaves found their way to the Lahore Museum and eventually on advice of General Alexander Cunningham, who recognized great value in this ancient manuscript, they landed up in the Calcutta Madrasa to be studied. A year after its discovery, the manuscript was presented to the Asiatic Society of Bengal, the only institution in the world collecting and studying Indian antiquities at the time. The Bakhshali manuscript now rests in the Bodleian Library in Oxford.

What did everyone see in this set of dried up birch-bark? The manuscript was a mathematical document with illustrative examples of mathematical rules and their solutions. This manuscript happens to be the first identified one that uses place value system with a dot (ancestor of zero) denoting an empty place. 2019 would be thus be written as 2.19, for example.

After much debate, the Bakhshali manuscript was dated to around 400 CE. This period closely predates the classical age of Indian astronomy when Aryabhata-I devised a place value number system without a zero. In 628 CE, the most prominent Indian mathematician of his time, Brahmagupta, attempted to form rules of mathematics using zero and negative numbers. In his work Brahma Sphuta Siddhanta (The Opening of the Universe), he defined zero as the number you’d get by subtracting a number from itself.

The first written record of the use of zero unequivocally dates to 876 CE in an inscription on a stone tablet found in the town of Gwalior in central India.

Bakhshali numerals. Source: Wikimedia Commons.

Brahma Sphuta Siddhanta is said to have traveled from India into the Middle East at the time when Islamic caliphates began taking roots in Persia. A lost work of the 13th century Egyptian historian al-Qifti, tells the story of an Indian presenting a mathematical work in the court of Caliph al-Mansur, legendary ruler and founder of Baghdad, in around 776 CE. Contemporary French historian of mathematics, Georges Ifrah, believes that al-Qifti was referring to Brahma Sphuta Siddhanta, but no one knows for sure.

Although it is controversial, the first point of introduction of Hindu numerals into Persia is believed to be via the book On the Calculation with Hindu Numerals written in around 820 CE by al-Khwarizmi, a Persian scholar from whose name the Latin world ‘algorithm’ was derived. The original Arabic text is lost, but the work is survived by a twelfth century Latin translation called Algoritmi de numero Indorum. The Latin text describes the Indian place value system and mentions zero as a place holder. Scholars believe that the Latin text had additions to the original Arabic, so it is uncertain if the original did tackle the Indian numeral system as it is presented in Algoritmi de numero Indorum.

The earliest surviving text dealing with Indian numeral system is Kitab al-fusul fi al-hisab al-Hindi written by an Arab mathematician al-Uqlidisi in around 952 CE. He is accredited with the invention of decimal fractions, which he introduced in this work.

By the end of tenth century CE, Indian numerals were widespread in the Arab world. Just like in India, the numerals underwent changes in shape and pattern in Persia over time. The tenth century Persian astronomer and mathematician, al-Sijzi copied the Indian numerals 0-9 in the way they were written in Persia at the time. This work was completed in Shiraz in modern day Iran, and dated 969 CE. Al-Sijzi was a contemporary of al-Biruni. The two corresponded regularly and one of their famous correspondences discusses about the Earth being spherical and rotating round its axis. Out of context in this article, but that brings our story to a full circle.

The Hindu numerals gained widespread acceptance with Arab merchants who found it easier to use in their calculations. The numerals traveled into Europe from the Arab world via divergent routes. They first appeared in Europe in the 976 CE work, Codex Vigilanus, a compilation of historical narratives, law codes, decrees, and illustrations by three monks of the Riojan Monastery in Spain.

Arabic numerals in Codex Vigilanus. Source: Wikimedia Commons.

The thirteenth century Italian mathematician Fibonacci, who studied in Algeria, introduced Arabic numerals in Europe with his book Liber Abaci, published in 1202 CE in Pisa, Italy. A passage from this famous book reads: “When my father, who had been appointed by his country as public notary in the customs at Bugia [in Algeria] acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms”.

A page from Liber Abaci. The Fibonacci series is shown at the top right. Note that the numbers 2, 8 and 9 closely resemble the Indian numerals and modern versions of these numbers. Source: Wikimedia Commons.

Liber Abaci was the first European work to describe the Hindu-Arabic numeral system, and the first to use the traditional Arabic numeral symbols. The book addressed both commercial tradesmen and mathematicians. It played a part in convincing Europeans of the superiority of Hindu-Arabic numerals over the use of abacus in calculations.

A medieval illustration of abacus versus the Hindu-Arabic numeral system. Source: Wikimedia Commons.

Hindu-Arabic numerals started gaining acceptance in Europe only after the invention of printing. It appears in the 1482 CE World map from Ptolemy, Cosmographia, printed in Germany. The modern numbers, as we know today, appeared first on the title page of the Libro Intitulado Arithmetica Practica by Juan de Yciar, a calligrapher and mathematician in Zaragoza in 1549 CE.

The Hindu-Arabic numeral system replaced all other systems of counting in the Western world. We cannot imagine any other way of counting today. Yet, its origins in India and journey to Europe through Persia is unknown to many. An Opinion article in the New York Times from June this year talked about a survey in which 3200 Americans were asked whether Arabic numerals should be taught in schools. More than half of them said ‘no’ and fifteen percent of the respondents had no answer. In the current geopolitical climate, the word ‘Arab’ might have been a decisive factor for the respondents, but the fact remains that most people don’t know that their very own 0-9 are in fact Arabic numerals.