Sunday, October 28, 2012

173: Digital Roots

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Back in episode 128, you may have heard me rightly make fun of New Age numerologists for their propensity to draw great significance into adding up the digits of a number, ignoring the various deep and interesting aspects of numerical structure. I still reserve the right to make fun of them for this, as it's generally pretty silly to prognosticate the future that way. But, being a thoughtful Number 7 personality, it occurred to me that it might be worth talking about the legitimate uses for summing the digits of a number. While it doesn't have quite the universal significance taught by the numerologists, adding the digits of a number is a legitimate mathematical operation, known as the “Digital Root”, and does have real applications.

To start with, let's take a closer look at what you get when you add a number's digits, repeating the process if the result is multi-digit, until you have digested the original number down to a 1-digit value. At first you might think the result is totally arbitrary, but as with many things in math, if we look a little closer we can find a surprising pattern. Try adding the digits of a few examples, and you may start to notice that the digital root is congruent modulo 9 to the original number. In other words, if you take the number you started with, divide it by 9, and look at the remainder, you'll get the same value as you did by this digit-summing process. (Note that for this purpose, as is standard in modular arithmetic and bargain sales prices, we are treating 9 as equivalent to 0.) For example, look at 56. Since 5+6=11. and 1+1=2, its digital root is 2. And it's also 2 more than the closest multiple of 9, 54. How did that happen?
 
It's actually pretty easy to prove that this is always the case. Suppose we have a 4 digit number abcd. Remember that in our place-based number system, the value of this number is equivalent to 1000a + 100b + 10c + d. Let's rearrange this a bit, and write it as 999a + 99b + 9c + a +b+c+d. When you write it this way, you can see that the original number is equal to some multiple of 9 plus the sum of the digits. If the sum of the digits is itself multi-digit, you can repeat the process for another iteration or two. In any case, you can see that our result is proven: this sum really is congruent mod 9 to the starting number.
 
If you're lucky enough to have gone to elementary school before the age of calculator addiction, you may vaguely recall one powerful use for this property of digital roots: the old trick of “casting out 9s”, to check if the result of an arithmetic operation you did by hand is accurate. This takes advantage of the fact that the mod 9 value is preserved by addition, subtraction, multiplication, and division: so if P is congruent to 3, and Q is congruent to 2, then P+Q is congruent to 5. So if you find the digital root of P and add it to the digital root of Q, the sum should match the digital root of your result for P+Q-- if it didn't, you made a mistake somewhere. Be careful though-- about 1/9 of your mistakes will probably result in an error that is congruent mod 9, and will be missed by this method. The term “casting out 9s” comes from the fact that you can take a shortcut and cross out any sets of digits that sum to 9 before adding up the digital roots, without changing the results.
 
There are also many cases where certain types of numbers leave “tracks” in the digital roots that can be used to quickly eliminate candidates that might or might not be numbers of a certain type. As the simplest example, numbers divisible by 9 are identifiable by their digital root of 9. Related to this is that numbers divisible by 3 always have digital roots of 3, 6, or 9. All squares have a digital root of 1, 4, 7, or 9, and cubes have a digital root of 1, 8, or 9. On the wikipedia page you can see the patterns for other cases such as perfect numbers, prime numbers, twin primes, factorials, and more.
 
A cooler application of digital roots is the figure known as the Vedic Square, an ancient Indian variant of the traditional 9x9 multiplication table, with the digital roots in each square instead of the product. So for example, in row 7 and column 8, instead of displaying the product 56, we would write 2, since 5+6 = 11 and 1+1 = 2. I know, you're probably thinking to yourself, “Since when are multiplication tables cool?” But try drawing one, and then coloring in all the squares with a particular set of numbers-- color in all the 1s, or all the 2s, or all the 4s and 5s, etc. You will find that you generate intricate symmetrical patterns, which you can see at some links in the show notes if you're too lazy to start sketching out squares yourself. Such patterns have been observed since ancient times and have been used as the foundations of various forms of abstract Islamic art. I also found one web page claiming the Sistine Chapel, the I Ching, and the game of chess were all influenced by Vedic Square patterns, but these sound like a bit of a stretch.
 
We should also keep in mind that the concept of a digital root is closely tied with the fact that our number system is Base 10. If, for example, we decided to use a Base 12 number system, then the digital roots of any number would be congruent modulo 11 rather than modulo 9 to it. The process of casting out 9s would be replaced by casting out 11s. And we would be able to generate an 11x11 Vedic square that would be totally different from our base-10 9x9 one. Next time you're in the mood to doodle some art & feeling uninspired, just choose an abitrary base and draw a Vedic square of your own, and you'll generate a new fresh set of visually interesting patterns.
 
And this has been your math mutation for today.
References: 
 
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Sunday, August 5, 2012

169: Isaac Newton, Supercop

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Before we start, I'd like to thank listener DeeTeeEnn, who posted another nice review on iTunes. Remember, you can have your name immortalized in podcast form as well, if you follow Dee's lead & post a review of your own. Anyway, on to our main topic.

You may have been amused or disgusted by the recent release of a cinematic bomb known as "Abraham Lincoln, Vampire Hunter". Kind of ridiculous to take an accomplished historical figure and use him as a prop in an adventure totally unrelated to his accomplishments, isn't it? But perhaps the silliest thing about this idea is that there are plenty of real-life unexpected juxtapositions that could have been mined to create much better, and less repulsive, movies. For example, suppose you saw a marquee with the title "Isaac Newton, Supercop." What would you think? Believe it or not, such a movie might very well be a historically accurate documentary. Isaac Newton, the founder of modern physics and co-inventor of calculus, was also a pioneer in many aspects of modern law enforcement, and an amazingly successful detective.

Newton's unexpected detective career began when he was appointed Warden of the Royal Mint in 1696. If you're like me, you may have read about this in the last line of some biographical article on Newton, and assumed this was a symbolic office or some kind of semi-retirement. In the past many in this post had been influential nobles who got there through connections, and didn't care about the job-- but Newton was meticulous in everything he did, and the nation was in a monetary crisis. One estimate was that 20% of the coinage was counterfeit, and low confidence in the value of money caused a resurgence of medieval-style barter. In order to restore trust in the monetary system, a great recoining was planned, where new coins would be issued that had guaranteed value and authenticity. The first thing Newton did was what you might have guessed: he carefully managed the manufacturing processes involved in recoinage, performing some of the first time-motion studies and making the mint more efficient than it had ever been, increasing its output by a factor of 10.

But taking a wider view of the problem, Netwon realized that the new coins would not solve the nation's monetary problems on its own: after they were issued, the state needed to defend their integrity. Reading the details of his job description, he discovered that he was the primary official entrusted with catching counterfeiters. At first he tried to get out of this duty, but once he realized he was stuck with it, he dove into it with the same skill he devoted to his other pursuits. Looking around and studying the situation, he figured out a few things. Creating counterfeit money was a crime that always involved multiple people: it required a location to produce the coins, accomplices to acquire the raw materials and put the coins into circulation afterwards, and complicit neighbors who would not question the noise and smoke. But anyone who you might catch red-handed with a counterfeit coin in the street would be at best a small player: getting to the source of the problem required human intelligence, direct testimony from people involved in coining conspiracies or close to them. So he recruited a network of undercover agents and informers who would hang out among the seedy locales where these criminals might be found, bringing him information about counterfeiting plots as they were being hatched. In some cases Newton himself even frequented such locations to get a feel for the atmosphere in which such conspiracies occurred. Learning from his informers, he carefully waited until he had enough independent evidence, and only then took the criminals into custody. He carefully interrogated them with methods that would generally not be too far out of place in an episode of Law and Order: after questioning them in detail about their crimes and their accomplices, he would record the results in writing and ask the subject to sign and verify that the recorded information was accurate. He conducted over 200 cross-examinations, and managed to convict 28 counterfeiters.

His most famous case was the one of William Chaloner, a sometimes successful counterfeiter who was a little too audacious for his own good. Chaloner had made a profitable career both of creating fake coinage, and of tricking others into counterfeiting plots so he could turn them in to the government and earn a reward. This techique of playing both sides of the fence worked for him for a number of years, his so-called "service to the Crown" saving him on the rare occasions when he was caught commiting crimes. When the recoinage was announced in 1696, he began scheming to use it to his advantage. He made some personal connections and managed to make a presentation to Parliament on the various ways of manufacturing false coins, which he knew well from experience, attempting to make the case that his extensive knowledge made him the perfect candidate to supervise coin production, superior to the current Warden of the Mint. If it had worked, Chaloner would have been able to counterfeit coins right at the source, from within the mint-- no doubt leading to a fortune in illicit profits. This was a massive strategic blunder on his part, though, since it brought him to Newton's attention, and Newton then took a closer look at Chaloner's colorful career. Using his skills in human intelligence, he soon connected the various dots of Chaloner's life, confirming his supicion that the supposed public servant was actually a professional criminal. Soon he was able to neutralize Chaloner's deceptive claims and convict him of his crimes.

While we might argue that these law enforcement activites are pretty trivial in comparison to his mathematical and physical contributions, it would be foolish to underestimate the importance of England's economy and monetary system at this critical time in history. You can learn a lot more about Newton's career at the Mint and his battle with Chaloner in the book "Newton and the Counterfeiter" by Thomas Levenson, which is where I first learned of this story. So next time you take some time from your day to ponder Isaac Newton's accomplishments, which you should really be doing pretty often given all that he did accomplish, don't forget that on top of the physics and the math, you should also think about the law enforcement. And next time you are having lunch with your favorite Hollywod producer, be sure to pitch Isaac Newton as an ideal subject for the next historical action thriller.

And this has been your math mutation for today.


References:
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Wednesday, March 14, 2012

162: The Mathematician Who Wasn't There

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Nicholas Bourbaki was an influential 20th-century mathematician who published a series of texts begnning in the 1930s, trying to rigourously describe the major core areas of mathematics based on the foundations of set theory. Many of his books became standard references in their fields. Among the Bourbaki contributions that have become familiar to modern math students are the use of the slashed zero to represent an empty set, and the terms injective, surjective, and bijective. The ill-fated New Math movement in education, which I discussed back in podcast 145 (http://mathmutation.blogspot.com/2011/12/145-why-johnny-couldnt-add.html), was also largely inspired by Bourbaki ideas. But if you try to find out about the life of Bourbaki himself, you will be in for a bit of a surprise: Nicholas Bourbaki did not actually exist.

Actually, to be more precise, Bourbaki was not an individual, but a secretive society of mathematicians established by Andre Weil, Henri Cartan, and other young mathematicians in Paris in the 1930s. They originally got together because they believed there was a serious gap in the available mathematics texts of the time, a problem magnified by the fact that a generation of potential leaders in the field had been wiped out by World War I. They hoped to re-establish the rigorous foundations of mathematics, while at the same time to provide a standard series of reference works. While secret societies tend to sound sinister in concept, their secrecy actually stemmed from rather noble motives: they wanted to ensure that any works they produced would be judged on the basis of their content, not on factors related to personal egos. So they created the ficticious persona of Nicholas Bourbaki, member of the Royal Academy of Poldavia, and agreed that he would be credited as the author of all books they produced.

　
The members of the Bourbaki group enjoyed having fun with the semi-secret nature of their small club. To anyone who asked, they would claim that Bourbaki was a real person of their acquaintance, and they even printed up a set of mathematical-pun-laden wedding invitations from Bourbaki's daughter, to show to anyone who doubted his authenticity. According to the invitation, "The trivial isomorphism (aka the sacrament of matrimony) will be given to them by P. Adic, of the Diophantine Order, at the Principal Cohomology of the Universal Variety, the 3 Cartember, year VI, at the usual hour.". You wouldn't think that would fool too many people-- but this prank backfired horribly during World War II, when Andre Weil (who had fled to Finland) was arrested on suspicion of spying. This apparently encoded letter from a strange foreign contact was considered a key piece of evidence. Weil was sentenced to death, though at the last minute a friend with government contacts managed to intervene and get him pardoned.

You would think they would be done with practical jokes after that, but the Bourbaki group tried to continue the whimsical spirit, perhaps a welcome break from the incredbly serious work they were attempting. In the late 1940s, American mathematician Ralph Boas was contacted by the Encyclopedia Britannica to assist with an article on modern mathematics, and since he knew Weil, he mentioned in his article that Bourbaki was actually a pseudonym for a collaboration rather than an actual person. Boas and the Encylopedia then received a letter, claiming to have been written by Bourbaki from an ashram in the Himalayas, asking "You miserable worm, how dare you say that I do not exist?" A series of letters went back and forth on the topic, though the encyclopedia editors were eventually convinced of the truth. But then Bourbaki members began to spread a new rumor, that Ralph Boas did not actually exist, and was actually a collective pseudonym for a group of American mathematicians!

The Bourbaki group lasted for several generations, producing a series of comprehensive texts on the foundations of modern mathematics. While they made many lasting contributions, they were also criticized for focusing too narrowly on foundations, omitting areas relevant to widely applicable topics such as logic and mathematical physics. They also had a strange aversion to pictures and illustrations, which meant that although some of their books were very useful as references, they ironically were not very good as textbooks. The group gradually declined after the 1970s, with their last major text, "Spectral Theory", being published in 1983. The decline may have been hastened by a long legal battle with their publishing company over royalties and translation rights: Pierre Cartier, one of the members during the final productive period, described the result as "both parties lost and the lawyer got rich."

In the show notes, you can find a link to a 1997 interview with Cartier, who shares some fascinating thoughts on the rise and fall of the Bourbaki group. Aside from the legal issues, one of the other major reasons he gave for Bourbaki's decline was that they had successfully achieved their objectives. As Cartier describes it, "In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. Now we are again at the beginning of a new revolution."

And this has been your math mutation for today.

　

References:


　

　
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Sunday, February 19, 2012

161: The Numbers Of Love

Math Mutation 161:  The Numbers Of Love

Before we get started, I'd like to thank listeners Foxy McLovin and The Devonian Kid, who recently posted nice reviews on iTunes.  Remember, you too can experience the thrill of having your name, or bizarre iTunes nickname, mentioned on Math Mutation by posting a review.  Or you can send a donation to your favorite charity in honor of Math Mutation, and email me to tell me about it.  Anyway, on to today's topic.

During this Valentine's week, it occurred to me that there was not yet a Math Mutation episode focusing on amicable numbers, said to be numbers that represent friendship and love.  Yes, there are such things.   And yes, math geeks do occasionally experience such emotions.  So what exactly are amicable numbers?

To start with, let's review the concept of a perfect number.  A perfect number, like 6 or 28, is precisely equal to the sum of its factors.  So 6, for example, is perfect because its factors are 1, 2, and 3, and 1 + 2 + 3 = 6.   The concept of amicable numbers is related to perfect numbers, except that they are pairs of numbers, such that each is the sum of the other's factors.  The smallest pair of amicable numbers is 220 and 284.  The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which add up to 284.  And the factors of 284 are 1, 2, 4, 71 and 142, which add up to 220.   If you stretch your sense of analogy a but, you can kind of see how the way in which two numbers are unexpectedly intertwined might suggest love and friendship.

The pair 220 and 284 was known to the ancient Greeks.  But as you can probably guess, amicable numbers are relatively rare, and can be tricky to find.  In 850 A.D, Arabic mathematician Thabit ibn Kurrah discovered if n > 1 and each of p = 3*2^n-1-1, q = 3*2^n-1, and r = 9*2^2n-1-1 are prime, then (2^n)pq and (2^n)r are amicable numbers.  This was a nice advance, but finding numbers that meet the preconditions of this formula wasn't very easy-- several centuries later, it led to discovery of the pairs 17,296 and 18,416, resulting from n=4, and 9,363,584 and 9,437,056, which you get from n=7.  Euler later published a list of 64 new amicable pairs, but he actually made some mistakes, and the list had 2 bogus entries, which was not discovered until the 20th century.   But most surprisingly, the second-lowest pair of amicable numbers, 1184 and 1210, was not discovered until 1866, by a 16-year old Italian boy named Nicolo Paganini.   With the help of modern computers, millions of amicable pairs have been discovered, though the conjecture that there are an inifnite number of them has not yet been proven.

Amicable numbers have been thought to symbolize love and friendship since ancient times.   It is said that when Pythagoras was asked to define a friend, his definition was "one who is the other I, such as are 220 and 284."   In Genesis verse 32:14, Jacob gave his brother 220 sheep when he feared he was planning to kill him, and some Torah scholars interpret this as due to its status as an amicable number, to reinforce their brotherly love.  I think this last interpretation is a bit of a stretch though-- he also gave him 30 camels, 40 cows + 10 bulls, and 30 donkeys, which don't seem to have anything to do with amicable numbers.  And to get the number 284 to appear somehow, scholars have interpreted one partial word in the verse as a Hebrew encoding of a number, which really seems to be reaching.

According to one online site called "Renassance Astrology", you can make a magical love talisman by the following method:  "Make two images, and put one of the Fortunes at the ascendant and the Moon in Taurus conjunct Venus. Write on one image a number ... for 220 in the proper place, and on the other image write the same kind of figure for 284 in the proper place. Join the two figures together in an embrace, and then there will be perfect and lasting love between the two."  They go on to supply further mystical powers of these numbers: "If the aforesaid numbers were carved in wood, and bread or anything else edible was sealed with them and you gave it to someone to eat, he will delight in you with a great love. If those numbers are written on your clothing, your garments cannot be taken away from you; and if you write them on banners that are put in the street to draw business, they will draw business to you."   I'm thinking of writing some of these numbers on my socks, to stop them from disappearing in the dryer.

And at least some modern geeks still believe that amicable numbers represent love:  with a quick online search, I was able to find a pair of half-heart pendants, one labeled 220 and the other labeled 284, for sale at a site called mathsgear.com .    But math geekiness can only get you so far in a marriage-- when it comes to Valentine's day, I made the safe choice and got my wife some non-math-related flowers.  Perhaps you will be more courageous on your next romantic occasion.
And this has been your math mutation for today.


References:

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Thursday, December 29, 2011

156: The Ultimate Answer

I celebrate my 42nd birthday this week.  This is an important milestone, since in his classic Hitchhikers series of science fiction novels, the late Douglas Adams described 42 as the ultimate answer to life, the universe, and everything.  This was figured out by a massive computer, Deep Thought, after 7.5 million years of calculation.  Once they heard the answer, people realized that maybe they should have figured out what the question was first.  But we can ask a more pertinent question in real life:  why did Adams settle on 42 as the answer in his books?  It somehow seems to have a nice ring to it, but what makes 42 a good "ultimate answer"?

Intrinsically, 42 does have a few things going for it.  One fun fact is that you can build a 3x3x3 magic cube with each number from 1 to 27 appearing exactly once, and every row, column, and diagonal summing up to 42.  42 is also the 5th Catalan Number, which means that 42 is the number of triangulations of a heptagon, or the number of ways you can cut a heptagon into triangles using straight lines.  (You may recall that a heptagon is a 7-sided regular polygon.)  It's a Harshad number, or an integer divisible by the sum of its digits.   There are also many more obscure properties of this number, too numerous to list here. 

Science provides some more cool applications of this number.  In 1268, English philosopher Roger Bacon calculated the geometric properties of rainbows, and discovered that the summit of a rainbow cannot appear more than 42 degrees above the horizon.  More recently, in 1966, mathematician Paul Cooper calculated that if you bore a frictionless hole all the way through the earth and try to travel to China by just jumping in and using gravity, the trip would take you exactly 42 minutes.  Surprisingly, this calculation works even if your hole doesn't pass through the center of the earth:  the reduction in gravitational force and distance traveled exactly balance each other out.

The number 42 also appears in various places in religion and culture.  It's unlucky in Japan, since the name of it sounds like the words for "unto death".  In ancient Egypt, there were 42 principles of Ma'at, or religious laws.  In the Christian book of Revelations, the Beast will hold dominion over the Earth for 42 months.  The Jewish Talmud references the "42-lettered name of god", and perhaps most relevant, Kabbalistic tradition makes the related claim that this number was somehow used by God to create the universe. 

The Wikipedia article also points out that Lewis Carroll, popular 19th-century author of lighthearted math-influenced works, used the number 42 numerous times:  42 illustrations in "Alice in Wonderland", the rule 42 in the same book (requiring all persons more than a mile high to leave the court), 42 boxes in The Hunting of the Snark, and a few other places. 

One more intriguing possiblity is that a semi-serious guide book from the 1970s and source of Adams's title, "The Hitchhiker's Guide to Europe", mentioned that travelers to the UK looking for family roots were likely to find the answers disappointing.  And that comment was on page 42.

So, were one of these the reasoning as to why Adams chose 42 as the ultimate answer?  These possibilities, mostly included on the wikipedia page linked in the show notes, all have some level of plausibility.   That page also includes various other references to 42, too numerous to include here.  But these just scratch the surface-- while doing online research for this podcast, I found that author Peter Gill wrote a whole book on the topic, which I haven't yet read. 

On the other hand, most of these references to the number are what I would describe as, well, very miscellaneous at best.   There's no overarching theme that really convinces me that 42 is a good candidate for the answer to the universe.  I would bet that with a little research, I could find an equal number of properties and references to any other two-digit number you might select.  Think about it:  in any instance where a number less than 100 appears in history, religion, or even some mathematical grouping, there's roughly a 1% chance that it's the number you want!  

So my inclination is to guess that Adams was playing a massive joke on the public, and really did choose the number randomly, just to see what interpretations his readers would come up with.  Since the author is no longer with us, we never will know for sure.

And this has been your math mutation for today.

  • Review of Gill book
  • 42 at Wikipedia
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    153: Cracking Pythagoras


    A few days ago I was browsing the infotainment site cracked.com, and saw an amusing article.  It was titled "6 Famous Firsts That You Learned In History Class (And Are Total BS)."  One of these 'famous firsts' was Pythogras's discovery of the Pythagorean Theorem.  As you probably recall, this is the theorem that if the legs of a right triangle are A and B, and the hypotenuse is C, A squared plus B squared equals C squared.  Pythagoras is known for having first proven this theorem around 550 BC.  Is it really the case that this is total BS?

    Actually, I wasn't too surprised to see this theorem on such a list.  This theorem is critical for measuring distances and amounts of materials in construction, real estate boundaries, and many other real cases where life doesn't always provide you with straight lines.  Because of this, it was discovered much earlier than Pythagoras-- ancient Babylonians had been using it for at least 1000 years by Pythagoras's time, and there are also references to it in ancient India, China, and Egypt.  Cracked.com even included a photo of an ancient Babylonian stone tablet inscribed with Pythagorean triples, sets of integers such that A squared plus B squared equals C squared.

    But there's a huge difference between having observed the theorem experimentally and actually knowing for sure that it will always be true.  The history of math and science is full of examples of supposedly true facts that were later overturned; just ask your doctor if he has any leeches in stock.  If you read your high school math textbook a little more carefully, you'll see that Pythagoras is credited not with discovering the theorem, but with *proving* it.  So if you really want to claim it is illegitimate to cite Pythagoras, you need to supply an earlier proof of the theorem.  The Cracked guys are on their toes though-- they have done this as well, pointing to an earlier Chinese text known as the Chou Pei Suan Ting.

    This book does indeed contain a diagram which seems to illustrate a common geometric proof of the theorem.  You can see the picture if you follow the links in the show notes.  The way it works is that you put four copies of the triangle together to form a large square, in such a way that each side contains one of each of the A and B legs of the original triangle.  This forms a large square whose sides are each of length (A+B), and in its center is a smaller square whose side is of length C.  The total area of the large square is (A+B), the quantity, squared, or A squared + 2AB + B squared.  The four triangles are each of area AB/2, using the standard fomula for area of a right triangle, so together their area adds up to 2AB.  But the remaining square in the middle has sides each of length C, with total area C squared.   So we have A squared + 2AB + B squared equals C squared + 2AB--- or A squared + B squared equals C squared!   Does this illustration thus establish that the Chinese proved the theorem before Pythagoras?  Cracked seems to think so.

    But not so fast-- there are a few problems with the Chinese illustration.  Most glaringly, it has grid markings that show it is addressing a particular case, of 3-4-5 right triangles, with no clear evidence that it was thought of as a general proof.  There is even some dispute as to whether the diagram was actually part of the original book, or transcribed by later commentators.  While the book was begun as early as 1046 BC, annotations and additions continued until 220 A.D.  And the book is not part of any kind of general document created to supply axiomatic proofs of mathematical discoveries; it's a collection of 246 specific problems encountered by the Duke of Zhou and his astrologer.

    But most importantly, we need to keep in mind that the Pythagorean Theorem is not taught as a standalone discovery, a single isolated contribution of the Greeks.  It was one of the beginnings of a systematic approach to mathematics, of not just being satisfied with observations, but of proving hypotheses based on fundamental assumptions, that culminated in Euclid's Elements.  Whether earlier societies knew some of the facts discussed by Greek mathematicians, or even came up with a few isolated proofs, is beside the point.  And if you want to unseat the Greeks as founders of modern mathematics, you need to point out systematic efforts to prove theorems on the basis of fundamental axioms, not grab a few random factoids out of other societies' texts.   I love cracked.com, but I think they truly are cracked on this one.

    And this has been your math mutation for today.

  • Cracked article
  • Chou Pei Suan Ching at Wikipedia
  • Pythagoream Theorem at Wikipedia
  • Another Pythagorean Theorem article
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    150: A Podcast About Nothing

    Before we start, I thought might be nice to highlight a few ways you can show your support for Math Mutation, since I received a recent query about donations.   I don't actually accept money, as I like the idea of total independence.  But if you find the power of the podcast to be so overwhelming as to loosen your wallet, please donate some money to your favorite charity in honor of Math Mutation, and send me an email about it.  I also love hearing directly from fans, either in the form of an email to erik(e r i k)@mathmutation.com, or by posting a review to iTunes.  And don't forget to sign up as a fan on Facebook.

    Anyway, you may recall that in the last episode, I asked for topic suggestions for Episode 150.  While I received a few unrelated emails from listeners, nobody was brave enough to actually sugget a topic.  So, being in a glass-half-full kind of mood, I decided to treat the lack of suggestions as a suggestion in itself, and do a topic that has been on my back burner for a while, the history of the number zero.

    To start with, we should clarify a few things about zero.  In one sense, it is a simple representation of nothing.  But perhaps even more importantly, it plays the critical role of a placeholder in our positional number system.  For example, how do you know you are listening to episode 150 and not episode 15?  It's because of that 0 in the ones place, which pushes the 5 into the tens place and the 1 into the hundreds place.  This seems like an obvious idea now, but that wasn't always the case.  You may recall that the Roman number system was basically non-positional:  with a few minor exceptions, you essentially wrote a bunch of symbols down in any order, added their values, and got a total.  Systems like the Roman numerals quickly grow cumbersome in the face of large numbers, and led to the confusing situation of many possible representations for the same number.

    One of the earliest positional number systems was used by the Babylonians, well-established by the second millenium B.C.  Their system was base-60 instead of base-10, and we still hear echoes of it today when we measure time or angles.  Initially, they did not have a symbol for zero, which meant that written numbers were inherently ambiguous:  you could not tell whether you had written 61 or 3601, 60 squared plus 1, because there was no symbol marking any non-used powers of 60 in the number.  By the first millenium B.C., the system had been enhanced by several authors to use placeholder symbols such as a pair of wedges, but these were only used internally between two digits: trailing zeros in the lower places could only be identified by context.   There must have been a lot of arguments with the waiter over the check in ancient Babylonian resturants.

    Strangely, even when used by a few mathematicians, this placeholder concept did not take hold very quickly.   Despite all their advances in mathematics, the Greeks didn't develop a true positional number system.  This may have been partially due to the Euclidean emphasis on geometry, where numbers were respected mainly for their usefulness when talking about drawn figures.  A few Greek and Roman astronomers, such as Ptolemy in 130 A.D., used the Babylonian system enhanced by a placeholder zero when recording their observations, but this was still considered an esoteric usage. 

    We should also mention that the native Olmecs and Mayans of the Americas independently developed a mixed base-20 and base-18 positional system as early as the 1st century B.C., which also included a true zero placeholder symbol.  Their Long Count calendar basically counted the days from 3114 B.C, when Raised-up-Sky-Lord caused three stones to be set by associated gods at Lying-Down-Sky, First-Three-Stone-Place.  While their numbers were mostly base-20, the second digit from the end rolled over whenever it hit 18 rather than 20.  The mixed base is kind of strange, until you think about the fact that 20x18 is 360, very close to an actual year.

    Most sources seem to agree that the widespread use of a true zero originated in India between 500-700 A.D.  In Brahmagupta's treatise "The Opening Of The Universe", he laid out a number of rules for mathematical operations on numbers including zero.  Some of his rules are very familiar to us today, such as adding zero to a negative number gives you a negative, and adding zero to a positive number gives you a positive.  But oddly, he tried to define division by zero, claiming that zero over zero equals zero.  Today we see that is clearly wrong:  if a calculation results in 0/0, you need more context to figure out a reasonable interpretation.  For example, look at the function y = x/x.  You can see that this is 1 for all nonzero values-- so shouldn't it also be 1 for x=0, thus showing that 0/0 = 1?  But on the other hand, look at y=2x/x.  With the same reasoning, we find that 0/0 equals two! 

    The Hindu-Arabic number system, a positional base-10 system with zero, was originally brought into Spain in the 11th century by the Moors.  Apparently it had spread into common usage among merchants in that civilization.   It was popularized in Christian Europe by Leonardo of Pisa, or Fibonacci, in 1202.  Even then it did not exactly take the continent by storm:  while mathematicians embraced it, merchants continued using the non-positional Roman numeral system for several more centuries before it slowly died out in the face of a superior competitor.  Except, of course, for the critical tasks of expressing motion picture copyright dates, or naming Popes.

    As usual, I've barely scratched the surface here.  Whole books have been written about the number zero, such as Robert Kaplan's highly regarded "The Nothing That Is".  And you can also find some excellent online articles linked in the show notes. But hopefully this podcast has given you a non-zero number of things to think about.

    And this has been your math mutation for today.

  • Robert Kaplan's book
  • Zero at Wikipedia
  • Another online history of zero
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