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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Ivy League Academic Removed From Plane And Questioned After Passenger Spotted His Equations

Finally, the world is safe from Italian economists doing mathematics on a plane.

Alarm bells were rung last Thursday on a flight from Philadelphia to Ontario, after a passenger saw aman suspiciously writing down a complicated looking formula on a piece of paper and notified cabin crew. The passenger told flight attendants she was feeling ill, causing the flight to turnaround on the runway.

After some confusion, the mysterious mathematicsenthusiastwas taken off the flight and questioned by security agents.

Fortunately for international security, the man was actually Guido Menzio, an Italian-born associate professor in Economics at the University of Pennsylvania, who also happened to be a young, dark-haired, bearded, and slightly tanned male with a foreign accent on a plane.

Menzio told theAssociated Press: “I thought they were trying to get clues about her illness. Instead, they tell me that the woman was concerned that I was a terrorist because I was writing strange things on a pad of paper.”

His scrawlings were actually some last minute work on a differential equation that he was preparing for a lecture on Search Theory in Canada.

Aftertwo hours of questioning,Menzio,who said he was treated with respect,was able to explain himself to the security officials andwas allowed back onto the flight. The passenger who complained, however, did not return to the flight.

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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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10 Truth Bombs to Drop at your Next Dinner Party

1. There are more trees on Earth
than stars in the Milky Way Galaxy


Photograph by Layne Lawson

In a September 2015 paper published in the scientific journal Nature entitled “Mapping tree density at a global scale†an estimate for the number of trees on Earth was approximately 3.04 trillion.
Meanwhile, according to NASA the generally accepted answer for number of stars in the Milky way is between 100 and 400 billion. [source]

2. Speaking of stars, guess how many miles (or km)
the nearest star (after our Sun) to Earth is?


Photograph by ESO/Y. Beletsky

Like objects in your side-view mirror, stars in the night sky seem a lot closer than they are. Alpha Centauri is the closest ‘star system’ to us at an approximate distance of 4.37 light-years which works out to roughly 25 trillion miles or 40 trillion km away. 😳

3. Alaska is simultaneously the most northern,
the most western, and the most eastern state in the US


Wait, what? Look on a map and it’s easy to see that Alaska is the United States’ most northern and western state. But eastern? That’s because the Aleutian Islands are part of Alaska and stretch beyond the 180° line of longitude (which is measured from Greenwich) thus placing some of the islands technically in the Eastern hemisphere, since the dividing line for the eastern/western hemisphere is at 180° (source)

4. ‘Oxymoron’ is an oxymoron


The term was first recorded as latinized Greek oxymÅrum and is derived from the Greek where ‘oxys’ means “sharp, keen, pointed” and ‘moros’ means “dull, stupid, foolish”. Oxymoron is also an autological word, which means it expresses a property that is also possesses (e.g. the word “noun” is a noun, “English” is English, “pentasyllabic” has five syllables, and “word” is a word) [source]

5. If you start counting at one and spell out the numbers
as you go, you won’t use the letter “A” until you reach 1,000


6. Oxford University is (way) Older than the Aztec Empire


Photograph by Chensiyuan

Older, like it’s not even close. As the oldest university in the English-speaking world, Oxford is a unique and historic institution. While there is no clear date of foundation, teaching existed at Oxford in some form in 1096 and developed rapidly from 1167, when Henry II banned English students from attending the University of Paris. [source]
Conversely, the Aztec Empire, or the Triple Alliance, began as an alliance of three Nahua “altepetl” city-states: Mexico-Tenochtitlan, Texcoco, and Tlacopan. These three city-states ruled the area in and around the Valley of Mexico from 1428 until they were defeated by the combined forces of the Spanish conquistadores and their native allies under Hernán Cortés in 1521. [source]

7. The official animal of Scotland is… the Unicorn


Royal Coat of Arms of the Kingdom of Scotland used from the 12th century to 1603


According to The Scotsman: in Celtic mythology, the Unicorn of Scotland symbolized innocence and purity, healing powers, joy and even life itself, and was also seen as a symbol of masculinity and power. It has been a Scottish heraldic symbol since the 12th century and today, the Royal Coat of Arms of the United Kingdom of Great Britain and Northern Ireland still has the English lion on the left and the Scottish unicorn on the right. [source]

8. There was a third Apple co-founder, Ronald Wayne.
He sold his 10% stake for $800 in 1976.
Today it would be worth roughly $75.5 billion


Ronald Wayne worked with Steve Jobs at Atari before he, Jobs, and Wozniak founded Apple Computer on April 1, 1976. Serving as the venture’s “adult supervision”, Wayne drew the first Apple logo, wrote the three men’s original partnership agreement, and wrote the Apple I manual.
Wayne received a 10% stake in Apple. Less than two weeks later, on April 12, 1976 he relinquished his equity for US$800. Legally, all members of a partnership are personally responsible for any debts incurred by any partner; unlike Jobs and Wozniak, then 21 and 25, Wayne had personal assets that potential creditors could seize. The failure of a slot machine company, which he had started five years earlier also contributed to his decision to exit the partnership.
Later in 1976, venture capitalist Arthur Rock and Mike Markkula helped develop an Apple business plan and converted the partnership to a corporation. A year after leaving Apple, Wayne received $1,500 for his agreement to forfeit any claims against the new company. [source]

9. With just 70 people, there is a 99.9% chance
that two people share the same birthday


23 people is all it takes for there to be a 50/50 chance that two of the people share a birthday. The ‘birthday paradox‘ provides a valuable lesson in probability and reveals our tendency to think linearly instead of exponentially.
You can find a thorough mathematical explanation of the birthday paradox here, but at it’s core, we tend to think of our birthday compared to the 22 other people so there are 22 chances. But when all 23 birthdays are compared against each other, it makes for much more than 22 comparisons.
So the first person has 22 comparisons to make, but the second person was already compared to the first person, so there are only 21 comparisons to make. The third person then has 20 comparisons, the fourth person has 19 and so on. If you add up all possible comparisons (22 + 21 + 20 + 19 + … +1) the sum is 253 comparisons, or combinations. Check out the table below to see how the probability increases as the number of people do. [source]

The following table shows the probability for some other values of n (this table ignores the existence of leap years, as described above, as well as assuming that each birthday is equally likely)

10. There’s enough water in Lake Superior
to cover North and South America in a foot of water


Photograph by Lorie Shaull

To talk of Lake Superior is to talk in superlatives. Its 3 quadrillion gallons are enough to cover both North and South America under a foot of water; it holds 10% of the world’s surface fresh water supply; at 31,700 square miles (82,100 sq km) it’s roughly the size of Maine.
If all 7 billion people on Earth drank a gallon of water per day it would collectively take us 1,174 years to drain it. [source]

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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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In Math, Profs Use This Puzzle To Teach a Valuable Lesson About Problem Solving

Graphic by Krauss

The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry.
It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it. [source]

Graphic by Trekky0623

The key to the puzzle is the fact that neither of the 13×5 “triangles” is truly a triangle, because what appears to be the hypotenuse is bent. In other words, the “hypotenuse” does not maintain a consistent slope, even though it may appear that way to the human eye. [source]

Graphic by Krauss

According to Martin Gardner, this particular puzzle was invented by a New York City amateur magician, Paul Curry, in 1953. However, the principle of a dissection paradox has been known since the start of the 16th century. [source]

Graphic by Krauss

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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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Crunch Report | Caldbeck and Mazzeo leave Binary Capital

Todays Stories

  1. Binary Capital co-founder Justin Caldbeck quits as Matt Mazzeo steps away from thefirm
  2. Pandora co-founder and CEO Tim Westergren will step down according toreports

  3. Houzz raises a huge $400M round at a $4Bvaluation

  4. Avis signs on to manage Waymos self-driving vehicle fleet in Phoenix

  5. MITs new drones switch between flying and driving for optimal urbantransport


Written and Hosted by:John Mannes
Filmed by: Gregory Manalo
Edited by:Chris Gates


Tito Hamze is gone seeing how many times we can pass the show back and forth between NYC and SF before he comes back

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