Much ado about nothing: ancient Indian text contains earliest zero symbol

Exclusive: one of the greatest conceptual breakthroughs in mathematics has been traced to the Bakhshali manuscript, dating from the 3rd or 4th century

Nowt, nada, zilch: there is nothing new about nothingness. But the moment that the absence of stuff became zero, a number in its own right, is regarded as one of the greatest breakthroughs in the history of mathematics.

Now scientists have traced the origins of this conceptual leap to an ancient Indian text, known as the Bakhshali manuscript a text which has been housed in the UK since 1902.

Radiocarbon dating reveals the fragmentary text, which is inscribed on 70 pieces of birch bark and contains hundreds of zeroes, dates to as early as the 3rd or 4th century about 500 years older than scholars previously believed. This makes it the worlds oldest recorded origin of the zero symbol that we use today.

The front page (recto) of folio 16 which dates to 224-383 AD. Photograph: Courtesy of Bodleian Libraries/ University of Oxford

Marcus du Sautoy, professor of mathematics at the University of Oxford, said: Today we take it for granted that the concept of zero is used across the globe and our whole digital world is based on nothing or something. But there was a moment when there wasnt this number.

The Bakhshali manuscript was found in 1881, buried in a field in a village called Bakhshali, near Peshawar, in what is now a region of Pakistan. It was discovered by a local farmer and later acquired by the Bodleian Library in Oxford.

Translations of the text, which is written in a form of Sanskrit, suggest it was a form of training manual for merchants trading across the Silk Road, and it includes practical arithmetic exercises and something approaching algebra. Theres a lot of If someone buys this and sells this how much have they got left? said Du Sautoy.

In the fragile document, zero does not yet feature as a number in its own right, but as a placeholder in a number system, just as the 0 in 101 indicates no tens. It features a problem to which the answer is zero, but here the answer is left blank.

Several ancient cultures independently came up with similar placeholder symbols. The Babylonians used a double wedge for nothing as part of cuneiform symbols dating back 5,000 years, while the Mayans used a shell to denote absence in their complex calendar system.

However the dot symbol in the Bakhshali script is the one that ultimately evolved into the hollow-centred version of the symbol that we use today. It also sowed the seed for zero as a number, which is first described in a text called Brahmasphutasiddhanta, written by the Indian astronomer and mathematician Brahmagupta in 628AD.

This becomes the birth of the concept of zero in its own right and this is a total revolution that happens out of India, said Du Sautoy.

The development of zero as a mathematical concept may have been inspired by the regions long philosophical tradition of contemplating the void and may explain why the concept took so long to catch on in Europe, which lacked the same cultural reference points.

This is coming out of a culture that is quite happy to conceive of the void, to conceive of the infinite, said Du Sautoy. That is exciting to recognise, that culture is important in making big mathematical breakthroughs.

Despite developing sophisticated maths and geometry, the ancient Greeks had no symbol for zero, for instance, showing that while the concept zero may now feel familiar, it is not an obvious one.

The Europeans, even when it was introduced to them, were like Why would we need a number for nothing? said Du Sautoy. Its a very abstract leap.

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Mathematical secrets of ancient tablet unlocked after nearly a century of study

Dating from 1,000 years before Pythagorass theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers

At least 1,000 years before the Greek mathematician Pythagoras looked at a right angled triangle and worked out that the square of the longest side is always equal to the sum of the squares of the other two, an unknown Babylonian genius took a clay tablet and a reed pen and marked out not just the same theorem, but a series of trigonometry tables which scientists claim are more accurate than any available today.

The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have cracked its secrets.

The team from the University of New South Wales in Sydney believe that the four columns and 15 rows of cuneiform wedge shaped indentations made in the wet clay represent the worlds oldest and most accurate working trigonometric table, a working tool which could have been used in surveying, and in calculating how to construct temples, palaces and pyramids.

The fabled sophistication of Babylonian architecture and engineering is borne out by excavation. The Hanging Gardens of Babylon, believed by some archaeologists to have been a planted step pyramid with a complex artificial watering system, was written of by Greek historians as one of the seven wonders of the ancient world.

Daniel Mansfield, of the universitys school of mathematics and statistics, described the tablet which may unlock some of their methods as a fascinating mathematical work that demonstrates undoubted genius with potential modern application because the base 60 used in calculations by the Babylonians permitted many more accurate fractions than the contemporary base 10.

The tablet could have been used in surveying, and in calculating how to construct temples, palaces and pyramids. Photograph: UNSW/Andrew Kelly

Mathematicians have been arguing for most of a century about the interpretation of the tablet known as Plimpton 322, ever since the New York publisher George Plimpton bequeathed it to Columbia University in the 1930s as part of a major collection. He bought it from Edgar Banks, a diplomat, antiquities dealer and flamboyant amateur archaeologist said to have inspired the character of Indiana Jones his feats included climbing Mount Ararat in an unsuccessful attempt to find Noahs Ark who had excavated it in southern Iraq in the early 20th century.

Mansfield, who has published his research with his colleague Norman Wildberger in the journal Historia Mathematica, says that while mathematicians understood for decades that the tablet demonstrates that the theorem long predated Pythagoras, there had been no agreement about the intended use of the tablet.

The huge mystery, until now, was its purpose why the ancient scribes carried out the complex task of generating and sorting the numbers on the tablet. Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.

The tablet not only contains the worlds oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry. This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3,000 years, but it has possible practical applications in surveying, computer graphics and education. This is a rare example of the ancient world teaching us something new.

The tablet also long predates the Greek astronomer Hipparchus, traditionally regarded as the father of trigonometry.

Wildberger said: Plimpton 322 predates Hipparchus by more than 1,000 years. It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own.

He and Mansfield believe there is more to learn of Babylonian maths, still buried in untranslated or unstudied tablets.

A treasure trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.

They suggest that the mathematics of Plimpton 322 indicate that it originally had six columns and 38 rows. They believe it was a working tool, not as some have suggested simply a teaching aid for checking calculations. Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids, Mansfield said.

As far back as 1945 the Austrian mathematician Otto Neugebauer and his associate Abraham Sachs were the first to note that Plimpton 322 has 15 pairs of numbers forming parts of Pythagorean triples: three whole numbers a, b and c such that a squared plus b squared equal c squared. The integers 3, 4 and 5 are a well-known example of a Pythagorean triple, but the values on Plimpton 322 are often considerably larger with, for example, the first row referencing the triple 119, 120 and 169.

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What kind of eclipse are you likely to see? Let our visualizations show you

The path of the eclipse is 70 miles wide meaning most of the US will see a partial eclipse. Mona Chalabis visualization will give you a sense of what you should prepare yourself for

A total solar eclipse will take place on Monday, the first in 38 years. As the moon passes in front of the sun, a shadow will be cast across all of North America. But the way that will look depends on where you are in the country.

The path of the total eclipse (when none of the sun is visible) is only 70 miles wide, so unless youre an eclipse chaser and willing to travel, youre more likely to see a partial eclipse.

(Note: these visualizations are inspired by the word mooning, which has been in use since about 1600 and means exposing to moonlight.)

The totality of the eclipse in various parts of the US Photograph: Mona Chalabi for the Guardian

If the image above looks ever so slightly imprecise, it should. Even Nasa concedes that perfectly calculating the mathematics of the eclipse is difficult because the universe is more subtle than that.

Nasa explains:The earth is neither smooth nor perfectly spherical, nor does it rotate at a perfectly constant, predictable speed. The moon isnt smooth, either, which means that the shadow it casts isnt a simple circle. And our knowledge of the size of the sun is uncertain by a factor of about 0.2%, enough to affect the duration of totality by several seconds.

In other words, theres still a lot we dont understand about our universe. Contemplate that as you enjoy Mondays moon miracle.

Contribute with

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Guardian of the galaxy: Nasa seeks new ‘planetary protection officer’

Role involves safeguarding Earth from extra-terrestrial infection, and stopping other planets being contaminated by robotic or human explorers

Nasa is looking for a planetary protection officer who will help safeguard Earth from alien bacteria.

No, it isnt the script of an elaborate science fiction film, but an actual job advertisement on the US governments website.

According to the site, the unusual role involves creating policies to ensure the avoidance of organic-constituent and biological contamination in human and robotic space exploration.

The job description says the three-year position involves frequent travel and comes with an annual handsome salary of up to $187,000 (141,000). The roles security clearance level is secret.

NASA People (@NASApeople)

Interested in @NASA‘s opportunity to become a Planetary Protection Officer?! Vacancy is open! Learn more on @USAJOBS

August 2, 2017

The successful candidate, who must be a US citizen or national and hold a degree in physical science, engineering, or mathematics, will make sure that no microbial life travels from Earth to infect other planets, and vice versa.

The planetary protection officer will oversee all space flight missions that may intentionally or unintentionally carry Earth organisms and organic constituents to the planets or other solar system bodies, and any mission employing spacecraft, which are intended to return to Earth and its biosphere with samples from extraterrestrial targets of exploration, the ad says.

The ad was created on 13 July but this week started to gain more attention after it was posted on Twitter, prompting a slew of mildly amusing jokes and faux job applications.

Townie Bagels (@TownieBagels)

.@NASA was this the idea for planetary protection officer you had in mind?

August 2, 2017

Steve Rogers (@Supes252)

Dear .@nasa I hear you’re looking for ‘Planetary Protection Officer’ we’ll now submit our interest in the position

August 2, 2017

jillthrash (@jillthrash)

There’s one other name you may know me by . . . Planetary Protection Officer. #nasa #PlanetaryProtectionOfficer

August 3, 2017

The job was created in 1967 in order to make the US compliant with the International Outer Space Treaty.

In 2014, Catherine Conley, the current previous planetary protection officer, told Scientific American that one of her concerns was that humans travelling to Mars could contaminate the planet if they died there.

She said it was important not to pollute other planets and repeat the mistakes humans have made on Earth.

If you wanted to drill into an aquifer on Mars, it would be in the interest of future colonists that you keep the drilling clean because organisms can grow in the aquifer and change the conditions so that it is no longer available. Weve seen that happen on Earth. That would be really unfortunate.

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Can you solve it? Are you smarter than a forester?

A puzzle about planting trees

Hello guzzlers,

Your mission today is to design an arrangement of trees on a desert island, like the one below.

An aerial view of five trees on an island.

When there is a single tree, no matter where you stand on the island you will always be able to see exactly one tree.

An island with a single tree. From each of the two black dots you can see a single tree.

With two trees, however, there are some places where you can see two trees, and there are some places where you can see only a single tree, since the other one is blocked from view.

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Can you solve it? Are you smarter than an architect?

A puzzle that tests 3D thinking

Hi guzzlers,

Todays puzzle was sent in by a reader who remembers it from his days as an architecture student.

Draw a 3-dimensional picture of a shape that goes through each of these holes, exactly touching all sides as it passes through.

A triangle with sides 1 unit. A square with sides 1 unit. A circle with diameter 1 unit.

Architects will surely find the answer obvious. The heads of the rest of us will look rather like the house in the picture above, since it requires you to visualise an object in three dimensions, which is a challenge if your brain isnt trained to do it.

If you want to email me your answer, or post it on Twitter with the hashtag #MondayPuzzle, Ill send the author of my favourite image a copy of my puzzle book Can You Solve My Problems?

Ill be back at 5pm UK time with the solution.


I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. Im always on the look-out for great puzzles. If you would like to suggest one, email me.

My puzzle book Can You Solve My Problems? is just out in paperback.

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Maryam Mirzakhani, first woman to win mathematics’ Fields medal, dies at 40

Stanford professor, who was awarded the prestigious prize in 2014, had suffered breast cancer

Maryam Mirzakhani, a Stanford University professor who was the first and only woman to win the prestigious Fields medal in mathematics, has died. She was 40.

Mirzakhani, who had breast cancer, died on Saturday, the university said. It did not indicate where she died.

In 2014, Mirzakhani was one of four winners of the Fields medal, which is presented every four years and is considered the mathematics equivalent of the Nobel prize. She was named for her work on complex geometry and dynamic systems.

Mirzakhani specialized in theoretical mathematics that read like a foreign language by those outside of mathematics: moduli spaces, Teichmller theory, hyperbolic geometry, Ergodic theory and symplectic geometry, the Stanford press announcement said.

Mastering these approaches allowed Mirzakhani to pursue her fascination for describing the geometric and dynamic complexities of curved surfaces spheres, doughnut shapes and even amoebas in as great detail as possible.

Her work had implications in fields ranging from cryptography to the theoretical physics of how the universe came to exist, the university said.

Mirzakhani was born in Tehran and studied there and at Harvard. She joined Stanford as a mathematics professor in 2008. Irans president, Hassan Rouhani, issued a statement praising Mirzakhani.

The grievous passing of Maryam Mirzakhani, the eminent Iranian and world-renowned mathematician, is very much heart-rending, Rouhani said in a message that was reported by the Tehran Times.

Irans foreign minister, Mohammad Javad Zarif, said her death pained all Iranians, the newspaper reported.

The news of young Iranian genius and math professor Maryam Mirzakhanis passing has brought a deep pang of sorrow to me and all Iranians who are proud of their eminent and distinguished scientists, Zarif posted in Farsi on his Instagram account.

I do offer my heartfelt condolences upon the passing of this lady scientist to all Iranians worldwide, her grieving family and the scientific community.

Mirzakhani originally dreamed of becoming a writer but then shifted to mathematics. When she was working, she would doodle on sheets of paper and scribble formulas on the edges of her drawings, leading her daughter to describe the work as painting, the Stanford statement said.

Mirzakhani once described her work as like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.

Stanford president Marc Tessier-Lavigne said Mirzakhani was a brilliant theorist who made enduring contributions and inspired thousands of women to pursue math and science.

Mirzakhani is survived by her husband, Jan Vondrk, and daughter, Anahita.

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Can you solve it? Are you smarter than a cat?

Feline clever? This moggy mystery will mess with your mind

Hi guzzlers,

Todays puzzle requires you to demonstrate superior intelligence to a contrary cat.

A straight corridor has 7 doors along one side. Behind one of the doors sits a cat. Your mission is to find the cat by opening the correct door. Each day you can open only one door. If the cat is there, you win. You are officially smarter than a cat. If the cat is not there, the door closes, and you must wait until the next day before you can open a door again.

If the cat was always to sit behind the same door, you would be able to find it in at most seven days, by opening each door in turn. But this mischievous moggy is restless. Every night it moves one door either to the left or to the right.

How many days do you now need to make sure you can catch the cat?

A cat sits behind one of these doors. Whats the best strategy to find it?

(First some clarifications. The 7 doors are in a line, so if the cat is behind the first or the last door, it has only one option for where it can move during the night. Otherwise, each night it decides randomly whether to move to the left or to the right.)

I purr with delight at this puzzle. At first it appears almost impossible that you will be able to get your hands on the furtive feline. But if you begin by trying the puzzle with a smaller number of doors, you will hopefully be able to work out the correct strategy.

Ill get you started. If there are only THREE doors, then it is possible to catch the cat in two days:

  • Day 1: open the middle door.
  • Day 2: open the middle door.

This strategy guarantees you will get the cat, since if it is not behind the middle door on Day 1, then it must be behind either of the end doors. And if it is behind either of the end doors on Day 1, then in both cases it will move to behind the middle door on Day 2. Caught!

If there are FOUR doors, it is possible to catch the cat in four days. But now its up to you to work out how.

The cat puzzle originally appeared in the New York Times now defunct Numberplay column as The Princess Problem, where a prince was knocking on doors and a flighty princess moving from room to room. This version has become a staple problem for maths teachers in Singapore. Toh Pee Choon, of Singapores National Institute of Education, told me that the princess context had great effect in stirring up interests in young girls.

I rephrased the puzzle with a cat to make it non gender specific, and also because people on the internet like looking at pictures of cats.


Ill be back at 5pm with the solution.

UPDATE: Read the solution here.

I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. Im always on the look-out for great puzzles. If you would like to suggest one, email me.


My puzzle book Can You Solve My Problems is out in paperback this week. You can get it from the Guardian bookstore or other online retailers.

Thanks to Charlie Gilderdale from maths resource project NRICH for first alerting me to this puzzle.

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Googles Tensor2Tensor makes it easier to conduct deep learning experiments

Googles brain team is open sourcing Tensor2Tensor, a new deep learning library designed to help researchers replicate results from recent papers in the field and push the boundaries of whats possible by trying new combinations of models, datasets and other parameters. The sheer number of variables in AI research combined with the fast pace of new developments makes it difficult for experiments run in two distinct settings to match. This is a pain for researchers and a drag on research progress.

The Tensor2Tensor library makes it easier to maintain best practices while conducting AI research. It comes equipped with key ingredients including hyperparameters, data-sets, model architectures and learning rate decay schemes.

The best part is that any of these components can be swapped in and out in a modular fashion without completely destroying everything. From a training perspective, this means that with Tensor2Tensor you can bring in new models and data sets at any time a much simpler process than would ordinarily be possible.

Google isnt alone in its pursuits to help make research more reproducible outside the lab. Facebook recently open sourced ParlAI, its tool to facilitate dialog research that comes prepackaged with commonly used datasets.

Similarly, Googles Tensor2Tensor comes with models from recent Google research projects like Attention Is All You Need and One Model to Learn Them All. Everything is available now on Github so you can start training your own deep learning-powered tools.

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Can you solve it? Pythagoras’s best puzzles

Three teasers from the vaults

Hi guzzlers,

The most famous theorem in maths is named after the Greek thinker Pythagoras. So is the most famous recreational mathematics publication in the Netherlands.

Pythagoras Magazine was founded in 1961, and to celebrate its half century it recently published a selection of its best brainteasers in English. Ive selected three of them here, in increasing order of difficulty.

1) Dollar bills. In a bag are 26 bills. If you take out 20 bills from the bag at random, you have at least one 1-dollar bill, two 2-dollar bills, and five 5-dollar bills. How much money was in the bag?

2) Yin and Yang. The Yin-Yang symbol is based on the figure below, bordered by three semi-circles. How can you divide this shape into two identical shapes?

Big yin

3) Huge pie. A huge pie is divided among 100 guests. The first guest gets 1% of the pie. The second guest gets 2% of the remaining part. The third guest gets 3% of the rest, etc. The last guest gets 100% of the last part. Who gets the biggest piece?

Ill be back later today with the solutions.


I set a puzzle here every two weeks on a Monday. Send me your email if you want me to alert you each time I post a new one. Im always on the look-out for great puzzles. If you would like to suggest one, email me.

Thanks to the editors of Pythagoras Magazine for todays puzzles. You can check out more of them in Half a Century of Pythagoras Magazine.


Football School, which I which I co-wrote with Ben Lyttleton, is a book for 7 to 13-year olds children that uses football to explain subjects like English, maths, physics, geography, philosophy and zoology. You (by which I mean any 7-13-year-olds you may know) can check out the Football School YouTube channel, in which Ben and I answer all questions about football and life. Submit your questions and subscribe!

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