Every year the Indian seaside town of Puri fills with a million pilgrims. They come for one of the most spectacular festivals in the Hindu calendar – the Rath Yatra, in which three chariots the size of carnival floats are pulled through the town. When I visited, the streets were crowded with cymbal-crashing, mantra-chanting devotees, barefoot holy men with long beards and middle-class Indian tourists with fashionable T-shirts and neon saris. It was midsummer, the beginning of the monsoon season, and in between downpours festival-workers sprayed the faces of passers-by with water to cool them down. Smaller Rath Yatra processions take place simultaneously all over India, although Puri’s is the focal event and its chariots the biggest.

The festival gets under way only when the local holy man, the Shankaracharya of Puri, stands in front of the crowds and blesses them. The Shankaracharya is one of Hinduism’s most important sages, the head of a monastic order that dates back more than a thousand years. He was also the reason I had travelled to Puri. As well as being a spiritual leader, the Shankaracharya is a published mathematician. I was also a pilgrim in search of enlightenment.

Right away in India, I noticed something unfamiliar about their use of numbers. In my hotel reception I picked up a copy of The Times of India. With the paper’s corners flapping from gusts of competing metal fans, the front-page headline caught my eye:

Crore is the Indian-English word for ten million, so the article was saying that India had just discovered 50 million citizens it never knew it had – a number roughly comparable to the population of England. It was startling that a country could overlook such a large number, even if it represented less than 5 percent of the overall population. Yet I was more puzzled by the word crore. Indian English has different words for high numbers than British or American English. For example, the word ‘million’ is not used. A million is instead expressed as ten lakh, where lakh is 100,000. Since ‘million’ is unheard of in India, the Oscar-winning film Slumdog Millionaire was released there as Slumdog Crorepati. A very rich person is someone who has a crore of dollars or rupees, not a million of them. The table of Indian equivalents for British/American number words is as follows:

Note that above a thousand, Indians introduce a comma after every two digits, while in the rest of the world the convention is every three.

The use of lakh and crore is a legacy of the mathematics of ancient India. The words come from the Hindi lakh and karod, which in turn came from the Sanskrit words for those numbers, laksh and koti. In ancient India coining words for large numbers was a scientific and religious preoccupation. For example, in the Lalitavistara Sutra, a Sanskrit text that dates from the beginning of the fourth century at the latest, the Buddha is challenged to express numbers higher than a hundred koti. He replies:

Just as in contemporary India, the Buddha went up the list in multiples of a hundred. Since a koti is ten million, the value of a tallakshana is ten million multiplied by a hundred 23 times, which works out as 10 followed by 52 zeros, or 10⁵³. This is a phenomenally large number, so large, in fact, that if you measure the entire universe from end to end in metres, and then square that number, you are roughly around 10⁵³.

But Buddha didn’t stop there. He was just warming up. He explained that he had described only the tallakshana counting system, and above it there was another one, the dhvajâgravati system, made up of the same number of terms. And above that, another one, the dhavjâgranishâmani, again with 24 number words. In fact, there were another six systems to go – which the Buddha, of course, listed perfectly. The last number in the final system is equivalent to 10⁴²¹: one followed by 421 zeros.

It is worth taking a breath to consider the view. There are an estimated 10⁸⁰ atoms in the universe. If we take the smallest measurable unit of time – known as Planck time, which is a second divided into 10⁴³ parts – then there have been about 10⁶⁰ units of Planck time since the Big Bang. If we multiply the number of atoms in the universe by the number of Planck times since the Big Bang – which gives us the number of unique positions of every particle since time began – we are still only on 10¹⁴⁰, which is way, way smaller than 10⁴²¹. The Buddha’s big number has no practical application – at least not for counting things that exist.

Not only was the Buddha able to fathom the impossibly large, he was also proficient in the realm of the impossibly tiny, explaining how many atoms there were in the yojana, an ancient unit of length around 10km. A yojana, he said, was equivalent to:

This was, in fact, a pretty good estimate. Just say that a finger is 4cm long. The Buddha’s ‘first atoms’ are, therefore, 4cm divided by seven ten times, which is 0.04m×7^–10 or 0.0000000001416m, which is more or less the size of a carbon atom.

The Buddha was by no means the only ancient Indian interested in the incredibly large and the unfeasibly small. Sanskrit literature is full of astronomically high numbers. Followers of Jainism, a sister religion to Hinduism, defined a raju as the distance covered by a god in six months if he covers 100,000 yojana in each blink of his eye. A palya was the amount of time it takes to empty a giant yojana-sized cube filled with the wool of newborn lambs if one strand is removed every century. The obsession with high (and low) numbers was metaphysical i nature, a way of groping towards the infinite and of grappling with life’s big existential questions.

Before Arabic numerals became an international lingua franca, humans had many other ways of writing down numbers. The first number symbols that emerged in the West were notches, cuneiform bird tracks and hieroglyphics. When languages developed their own alphabets, cultures began to use letters to represent numbers. The Jews used the Hebrew aleph () to mean one, bet () to be two and so on. The tenth letter, yod (), was ten, after which each letter went up in tens, and on reaching 100 went up in hundreds. The twenty-second and final letter of the Hebrew alphabet, tav (), was 400. Using letters for numbers was confusing and also encouraged a numerological approach to counting. Gematria, for example, was the practice of adding up the numbers of the letters in Hebrew words to find a value and using this number for speculations and divinations.

The Greeks used a similar system, with alpha () being one, beta (ß) being two, and so on to the twenty-seventh letter of their alphabet, sampi (), which was 900. Greek mathematical culture, the most advanced in the classical world, did not share the Indian hunger for jumbo numbers. The highest-value number word they had was myriad, meaning 10,000, which they wrote as a capital M.

Roman numerals were also alphabetic, although their system had more ancient roots than those of either the Greeks or the Jews. The symbol for one was I, probably a relic of a notch on a tally stick. Five was V, maybe because it looked like a hand. The other numbers were X, L, C, D and M for 10, 50, 100, 500 and 1000. All the other numbers were generated using these seven capital letters. The Roman system’s provenance from the tally stick made it a very intuitive way of writing out numbers. It was also efficient – using just seven symbols compared to 22 in Hebrew and 27 in Greek – and Roman numerals were the predominant number system in Europe for well over a thousand years.

Roman numerals, however, were very poorly suited to arithmetic. Let’s try to calculate 57×43. The best way to do this is with a method known as Egyptian or peasant multiplication, because it dates back at least to ancient Egypt. It is an ingenious method, though slow.

You first decompose one of the numbers being multiplied into powers of two (which are 1, 2, 4, 8, 16, 32 and so on, doubling each time) and make a table of the doubles of the other number. So, for the example 57×43, let us decompose 57 and draw up a table of doubles of 43. I’m using Arabic numerals to show how it’s done, and will translate into Roman numerals afterwards.

Decomposition: 57 = 32 + 16 + 8 + 1

Table of doubles:

The multiplication of 57×43 is equivalent to the addition of the numbers in the table of doubles that correspond to amounts in the decomposition. This sounds like a mouthful but is fairly straightforward. The decomposition contains a 32, a 16, an 8 and a 1. In the table, 32 corresponds to 1376, 16 corresponds to 688, 8 corresponds to 344 and 1 corresponds to 43. So, we can rewrite the initial multiplication as 1376 + 688 + 344 + 43, which equals 2451.

Now for the Roman numerals: 57 is LVII and 43 is XLIII. The decomposition and the table becomes:

and

so,

Oof! By breaking down the calculation into digestible morsels involving only doubling and adding, Roman numerals are just about up to the task. Still, we did much more work than we needed to. I mentioned earlier that the Roman system was intuitive and efficient. I’m taking that back. The Roman system quickly becomes counter-intuitive since the length of the number is not dependent on value. MMCDLI is larger than DCLXXXVIII, but uses fewer numerals, which goes against common sense. And any efficiency gained by using only seven symbols is forfeited by the inefficiency of how they are used. Often long strings are required to signify small numbers: LXXXVI uses six symbols, compared to the Arabic equivalent, 86, which uses two.

Compare the calculation above with the method of ‘long’ multiplication we all learned at school:

There is a very simple reason why our method is easier and quicker. Neither the Romans nor the Greeks or Jews had a symbol for zero. When it comes to sums, nothing makes all the difference.

The Vedas are Hinduism’s sacred texts. They have been passed down orally for generations until being transcribed into Sanskrit about 2000 years ago. In one of the Vedas a passage about the construction of altars lists the following number words:

With names for every multiple of ten, large numbers can be described very efficiently, which provided astronomers and astrologers (and, presumably, altar builders) with a suitable vocabulary for referring to the enormous quantities required in their calculations. This is one reason why Indian astronomy was ahead of its time. Consider the number 422,396. The Indians started at the smallest digit, at the right, and enumerated the others successively from right to left: Six and nine dasa and three sahasra and two ayuta and two niyuta and four prayuta. It is not too much of a step to realize that you can leave out the names for the powers of ten, since the position of the number in the list defines its value. In other words, the number above could be written: six, nine, three, two, two, four.

This type of enumeration is known as a ‘place-value’ system, which we discussed earlier. An abacus bead has a value dependent on which column it is in. Likewise, each number in the above list has a value dependent on its position in the list. Place-value systems, however, require the concept of a ‘place-holder’. For example, if a number has two dasa, no sata and three sahasra it cannot be written two, three since that refers to two dasa and three sata. A place-holder is needed to maintain the correct positions, to make it clear that there are no sata, and the Indians u a e word shunya – meaning ‘void’ – to refer to this place-holder. The number that is just two dasa and three sahasra would be written as two, shunya, three.

The Indians were not the first to introduce a place-holder. That honour probably went to the Babylonians, who wrote their number symbols in columns with a base 60 system. One column was for units, the next column was for 60s, the next for 3600s and so on. If a number had no value for that column, it was initially left blank. This, however, led to confusion, so they eventually introduced a symbol that denoted the absence of a value. This symbol, however, was used only as a marker.

After adopting shunya as a place-holder, the Indians took the idea and ran with it, upgrading shunya into a fully fledged number of its own: zero. Nowadays, we have no difficulty in understanding that zero is a number. But the idea was far from obvious. The Western civilizations, for example, failed to come up with it even after thousands of years of mathematical enquiry. Indeed, the scale of the conceptual leap achieved by India is illustrated by the fact that the classical world was staring zero in the face and still saw right through it. The abacus contained the concept of zero because it relied on place value. For example, when a Roman wanted to express 101, he would push a bead in the first column to signify 100, move no beads in the second column, indicating no tens, and push a bead in the third column to signify a single unit. The second, untouched column was expressing nothing. In calculations, the abacist knew he had to respect untouched columns just as he had to respect ones in which the beads were moved. But he never gave the value expressed by the untouched column a numerical name or symbol.

Zero took its first tentative steps as a bonafide number under the tutelage of Indian mathematicians such as Brahmagupta, who in the seventh century showed how shunya behaved towards its number siblings:

If ‘fortune’ is understood as a positive number, a, and ‘debt’ as a negative number, –a, Brahmagupta has written out the statements:

Numbers had emerged as tools for counting, as abstractions that described amounts. But zero was not a counting number in the same way; understanding its value required a further level of abstraction. Yet the less that maths was tied to actual things, the more powerful it became.

Treating zero as a number meant that the place-value system that had made the abacus the best way to calculate could be properly exploited using written symbols. Zero would enhance mathematics in other ways too, by leading to the ‘invention’ of negative numbers and decimal fractions – concepts we learn effortlessly at school and are intrinsic to our needs in daily life, but which were in no way self-evident. The Greeks made fantastic mathematical discoveries without a zero, negative numbers or decimal fractions. This was because they had an essentially spatial understanding of mathematics. To them it was nonsensical that nothing could be ‘something’. Pythagoras was no more able to imagine a negative number than a negative triangle.

Evolution of modern numerals.

Of all the innovative ways that numbers were treated in ancient India, perhaps none was more curious than the vocabulary used to describe the numbers from zero to nine. Rather than each digit having a unique name, there was a colourful lexicon of synonyms. Zero, for example, was shunya, but it was also ‘ether’, ‘dot’, ‘hole’ or the ‘serpent of eternity’. One could be ‘earth’, ‘moon’, ‘the pole star’ or ‘curdled milk’. Two was interchangeable with ‘arms’, three with ‘fire’ and four with ‘vulva’. The names that were chosen depended on context and conformed to Sanskrit’s strict rules of versification and prosody. For example, the following verse is a piece of number-crunching from an ancient astrological text:

While at first it might seem confusing to have flowery alternatives for each digit, it actually makes perfect sense. During a period in history when manuscripts were flimsy and easily spoiled, astronomers and astrologers needed a backup method to remember significant numbers accurately. Strings of digits were more easily memorized when described in verse with varied names, rather than when using the same number names repeatedly.

The nine Indian figures are:

9     8     7     6     5     4     3     2     1

With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written, as will be demonstrated.

77

97

Using the third sutra, Vertically and Cross-wise, we can find the answer quite handily as follows.

Vertically and Cross-wise, or ‘cross-multiplication’, is faster, uses less space and is less laborious than long multiplication. Kenneth Williams told me that whenever he explains the Vedic method to school pupils they find it easy to understand. ‘They can’t believe they weren’t taught it before,’ he said. Schools favour long multiplication because it spells out every stage of the calculation. Vertically and Cross-wise keeps some of the machinery hidden. Williams thinks this is no bad thing, and may even help less bright pupils. ‘We have to steer a path and not insist that kids have to know everything all of the time. Some kids need to know how [multiplication] works. Some don’t want to know how it works. They just want to be able to do it.’ If a child ends up not being able to multiply because the teacher insists on teaching a general rule that he or she cannot grasp, he said, then the child is not being educated. For the smarter kids, added Williams, Vedic Maths brings arithmetic alive. ‘Mathematics is a creative subject. Once you have a variety of methods, children realize you can invent your own and they become inventive too. Maths is a really fun, playful subject and [Vedic Maths] brings out a way to teach it that way.’

My first audience with the Shankaracharya had not covered all intended topics of discussion, so I was granted a second one. At the beginning of the session, the senior disciple had an announcement to make: ‘We would like to say something about zero,’ he said. The Shankaracharya then spoke for about ten minutes in Hindi in an animated manner, with the disciple then translating: ‘The present mathematical system considers zero as a non-existent entity,’ he declared. ‘We want to rectify this anomaly. Zero cannot be considered a non-existent entity. The same entity cannot be existing in one place and non-existing somewhere else.’ The thrust of the Shankaracharya’s argument was, I think, the following: people consider the 0 in 10 to exist, but 0 on its own not to exist. This is a contradiction – either something exists or it does not. So zero exists. ‘In Vedic literature zero is considered as the everlasting number,’ he said. ‘Zero cannot be annihilated or destroyed. It is the indestructible base. It is the basis of everything.’

By now I was used to the Shankaracharya’s distinctive mix of mathematics and metaphysics. I had given up asking him to clarify certain points since by the time my comments had been translated into Hindi, discussed and then translated back, the answers inevitably added to my confusion. I decided to stop concentrating on the details of his speech, and let the translated words just float over me. I looked at the Shankaracharya closely. He was wearing an orange robe today, tied with a big knot behind his neck, and his forehead had been daubed with beige paint. I wondered what it would be like to live the way he did. I had been told that he sleeps in an unfurnished room, eats the same bland curry every day, and that he has no need or desire for possessions. Indeed, at the beginning of the session a pilgrim had approached him to give him a bowl of fruit, and as soon as he received it, he had given the fruit away to the rest of us. I got a mango, which was by my feet.

Trying to experience the Shankaracharya’s wisdom in a different way, I thought of the phrase ‘zero is an existent entity’ and repeated it like a mantra in my head. I let go. Suddenly I am lost in my thoughts. And it all makes sense. ‘Zero is an existent entity’ is not just the Shankaracharya’s mathematical point of view, but a pithy phrase of self-description. Sitting in front of me is Mr Zero himself, the embodiment of shunya in flesh and bone.

It was a moment of clarity, maybe even of enlightenment. Nothing was not nothing in Hindu thought. Nothing was everything. And the monastic, self-abnegating Shankaracharya was a perfect ambassador for this nothingness. I thought about the deep connection between Eastern spirituality and mathematics. Indian philosophy had embraced the concept of nothingness just as Indian maths had embraced the concept of zero. The conceptual leap that led to the invention of zero happened in a culture that accepted the void as the essence of the universe.

The symbol that emerged in ancient India for zero perfectly encapsulated the Shankaracharya’s overriding message that mathematics cannot be separated from spirituality. The circle, 0, was chosen because it portrays the cyclical movements of the face of heaven. Zero means nothing, and it means eternity.

Pride in the invention of zero has helped make mathematical excellence an aspect of Indian national identity. Schoolchildren must learn their times tables up to 20, which is twice as high as I was taught growing up in the UK. In previous decades Indians were required to learn their tables up to 30. One of India’s top non-Vedic mathematicians, S.G. Dani, attested to this: ‘As a child I did have this impression of mathematics being extremely important,’ he told me. It was always common for elder people to set children mathematical challenges, and it was greatly appreciated if they got the answers right. ‘Irrespective of whether it is useful or not, maths is something that is valued in India by one’s peer group and friends.’

Dani is senior professor of mathematics at the Tata Institute of Fundamental Research in Bombay. He has an academic’s comb-over frizz, rimmed tortoiseshell glasses and a moustache that frames the length of his upper lip. And he is no fan of Vedic Mics; he neither believes that Tirthaji’s arithmetical methods can be found in the Vedas nor does he believe it is particularly helpful to say that they do. ‘There are many better ways to bringing interest into mathematics than resorting to inputting them into ancient texts,’ he said. ‘I don’t believe that they are making mathematics interesting. The selling point is that these algorithms make you fast, not that it makes it interesting, not that it makes you internalize what is going on. The interest is in the end, not the process.’ He is doubtful they do make calculation quicker, since real life does not throw up such perfectly formed problems as finding the decimal breakdown of 1/19. At the end of the day, he added, the conventional method is more convenient.

So, I was surprised that Dani spoke empathetically of Tirthaji’s mission with Vedic Maths. Dani related to Tirthaji on an emotional level. ‘The dominant feeling that I had for him is that he had this inferiority complex that he was trying to conquer. As a child I also had this kind of attitude. In India in those days [shortly after Independence] there was a strong feeling that we needed to get back [from the British] what we lost by hook or by crook. It was mostly in terms of artefacts, stuff that the British might have taken away. Because we lost such a lot, I thought we should have the equivalent back of what we lost.

‘Vedic Mathematics is a misguided attempt to claim arithmetic back for India.’

Some of the tricks of Vedic Mathematics are so simple that I wondered if I might come across them anywhere else in arithmetical literature. I thought that Fibonacci’s Liber Abaci would be a good place to start. When I got back to London I found a copy at the library, opened the chapter on multiplication and Fibonacci’s first suggested method is none other than…Vertically and Cross-wise. I did some more research and discovered that multiplication using All from 9 and the last from 10 was a favoured technique in several books from sixteenth-century Europe. (In fact, it has been suggested that both methods might have influenced the adoption of×as the multiplication sign. When×made its first appearance as a notation for multiplication in 1631, books had already been published illustrating the two multiplication methods with large ×s drawn as cross-lines.)

Tirthaji’s Vedic Mathematics is, it would seem, at least in part, a rediscovery of some very common Renaissance arithmetical tricks. They may or may not have come from India originally, but whatever their provenance, the charm of Vedic Mathematics for me is the way it encourages a childlike joy in numbers and the patterns and symmetries they hold. Arithmetic is essential in daily life and important to do properly, which is why we are taught it so methodically at school. Yet in our focus on practicalities we have lost sight of quite how amazing the Indian number system is. It was a dramatic advance on all previous counting methods and has not been improved upon in a thousand years. We take the decimal place-value system for granted, without realizing how versatile, elegant and efficient it is.