Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

An error of 0.000095 percent may not feel like much; it is only off by one part in a million. And when the time value is small, the error is also small. But the problem with a percentage error is that, as the value gets bigger, the error grows with it. The longer the Patriot system was running, the larger the time value became and the bigger the error accumulated was. When the Scud missile was launched that day, the Patriot system near the target had been on for about a hundred continuous hours, roughly 360,000 seconds. That’s about a third of a million seconds. So the error was about a third of a second. A third of second does not feel very long until you’re tracking a missile going 3,800 mph. In a third of a second a Scud missile can move almost 2,000 feet. It is very hard to track and intercept something that is a third of a mile away from where you expect it to be. The Patriot system was unable to stop the Scud missile, and it hit the US base, killing twenty-eight soldiers and injuring about a hundred other people.

Ironically, if the system had consistently been off from the correct time, it could still have worked OK. Tracking a missile requires accurate tracking of time differences, so a consistent error would cancel out. But now different parts of the system were using different levels of precision in their conversion, and a discrepancy slipped in. The incomplete upgrade is why the system could not track the incoming missile. Even more depressing is the fact that the US Army knew about this problem and, on February 16, 1991, it had released a new version of the software to fix it. As this would take a while to distribute to all the Patriot systems, a message was also sent out to warn Patriot users not to let the system run continuously for long periods of time. But what constituted a “long period of time” was not specified. As well as the mathematical problems, those twenty-eight deaths were also the result of poorly fixed code and the lack of a message simply saying to restart once a day. The software fix arrived at the base in Dhahran on February 26. The day after the missile attack.

In mathematics, it is impossible to divide numbers by zero. Many an internet argument has raged over this, with well-meaning people maintaining that the answer to dividing by zero is infinity. It is not. The argument is that if you take 1/x and let x get closer and closer to zero, the value shoots off to be infinitely large. Which is half true. It only works if you come at it from the positive direction; if x starts negative and then approaches zero from below, the value of 1/x races toward negative infinity, completely the opposite direction of before. If the limiting value is different depending on the direction you approach it from, then, in mathematics, we say that limit is “undefined.” You cannot divide by zero. The limit does not exist.

But, as always, some calculators slip through the gaps. And not just calculators: US Navy warships can get division by zero wrong too. In September 1997, the cruiser USS Yorktown lost all power because its computer control system tried to divide by zero. It was being used by the navy to test their Smart Ship project: putting computers running Windows on warships to automate part of the ship’s running and reduce the crew by around 10 percent. Given that it left the ship floating dead in the water for over two hours, it certainly succeeded in giving the crew some time off. The prevalence of military examples of math going wrong is not because the armed forces are particularly bad at mathematics. It’s partly because the military is big on research and development, so they are at the bleeding edge of what can be done, which tends to invite mistakes. Moreover, they have some level of public obligation to report on things that go wrong. Obviously, a lot of undoubtedly fascinating math mistakes never get declassified, but within private companies, even more mistakes are completely hushed up. I’m largely limited to talking about mistakes that have been openly reported on. In the case of USS Yorktown, the details are still a bit hazy. It’s not clear whether the ship had to be towed back to port or if it eventually regained power in the water. But we do know that the problem was a divide-by-zero error. The mistake seems to have started when someone entered a zero in a database somew...

Unlikely events can happen. On June 7, 2016, Colombia was playing Paraguay in the 2016 Copa América. The referee flipped a coin to see which soccer team got to choose which end of the field their goal would be. Except the coin landed perfectly on its edge. After only a moment’s hesitation, and a few laughs from the nearby players who saw it happen, the referee picked the coin up and managed to flip it successfully. I will accept that falling into grass makes an edge landing more probable. A coin landing on its edge on a hard surface is almost impossible. I believe the coin with the highest chance of landing on its edge is the UK old-style £1 coin (in circulation from 1983 to 2017), which is the thickest coin I’ve seen in daily use. To check how likely it is to land on its edge, I sat down and spent three days flipping one. After ten thousand flips, it had landed on its edge fourteen times. Not bad. I suspect the new £1 coin will have similar odds, but I’ll leave those ten thousand flips to someone else. For something like the much thinner US nickel, I suspect it would take tens of thousands of flips for even a single edge case. But it is still possible. If you want something unlikely to occur, you simply need the patience to create enough opportunities to allow it to happen. Or, in my case, patience, a coin, a lot of free time, and the kind of obsessive personality that keeps you sitting in a room flipping a coin by yourself, despite the desperate pleas of your friends and...

In 1999 a British woman was sentenced to life in prison for the murders of two of her children. However, the two deaths could have been entirely accidental; every year, just under three hundred babies in the UK die unexpectedly from sudden infant death syndrome (SIDS). During the trial, the jury had to decide if she was guilty of murder beyond reasonable doubt. Was she the perpetrator or the victim in this emotionally charged case? The jury was presented with stats that seemed to imply that two siblings both dying of SIDS was extremely rare. They returned a verdict of guilty (with a majority of ten to two), but the defendant’s conviction was later overturned. At the trial, erroneous statistics were presented, which gave the false impression that there was only a 0.0000014 percent chance (around one in 73 million) of two babies in such a family dying from SIDS. The Royal Statistical Society claimed there was “no statistical basis” for this figure and was concerned by the “misuse of statistics in the courts.” Upon a second appeal in 2003, the conviction was quashed, the woman having already spent three years in prison. How could math have gone so far wrong as to convict an innocent woman? The prosecution had taken the probability of a crib death in a family such as this woman’s at 1 in 8,543 and multiplied 1⁄8,543 by 1⁄8,543 to estimate the probability of two deaths.

Take a look at my predictions: the final two choices of mine are the same as the first two of my opponent. My goal is to cut them off at the pass. Sure, one player could win on the very first three flips of the coin (a 12.5 percent chance for each person), but after that, the winning run of heads and tails will be preceded by a run of three that overlaps it. I want to choose that preceding group. For something like TTT, it either has to be the first three flips, or it will definitely be beaten by HTT coming directly before it. A midsequence run of three tails will always have a head directly before it, giving HTT before TTT. The game is rigged. This is a game known as Penney ante (named for its inventor, Walter Penney), and it has been used to separate humans from their cash for years. In the game of Penney ante, people get thrown because every option of three heads and tails has a different combination that is more likely to win. It is unsettling that there is no best option to pick that is more likely to win than all the others. But this exact oddity is the basis of the game rock-paper-scissors. Any option picked can be beaten by one of the other options.

This is the difference between transitive and nontransitive relations. A transitive relation is one that can be passed along a chain. The size of real numbers is transitive: if nine is bigger than eight and eight is bigger than seven, then we can assume that nine is bigger than seven. Winning in rock-paper-scissors is nontransitive. Scissors beats paper and paper beats rock but that does not imply that scissors can beat rock.

The second probability game you can use to trick humans was invented by mathematician James Grime. He developed a set of nontransitive dice, which now bear his name: Grime dice (which is now the second-best boy-band name in this book). They come in five colors (red, blue, green, yellow, and magenta), don’t have the normal numbers 1 to 6 on them, and you can use them to play a game of highest number wins. You and your opponent choose a dice* each and roll them at the same time to see who gets the highest number. But for every dice there is a different-colored dice that will beat it more often than not. On average: red beats blue; blue beats green; green beats yellow; yellow beats magenta; and then magenta beats red. My contribution to the dice was to wait for James to work the numbers out and then suggest a range of colors to help remember the red-blue-green-yellow-magenta order; each color is one letter longer than the previous one. Now, you let your opponent choose their color first, and you pick the dice with one fewer letters. Red (3) rolls over to magenta (7). If you use these dice to win drinks and money off your friends and family, you may eventually have to come clean about their nontransitiveness. Maybe even teach them the order of the dice. But then suggest doubling the dice up and rolling two of each color together. Because rolling the dice in pairs perfectly reverses the order of which beats which. Instead of red beating blue, blue now beats red more often than ...

Nontransitive dice are a relative newcomer to the world of mathematics. They appeared on the math scene only in the 1970s, but they quickly made a big impact. Multibillionaire investor Warren Buffett is a big fan of nontransitive dice and brought them out when he met a fellow multibillionaire, computer guy Bill Gates. The story goes that Gates’s suspicion was aroused when Buffett insisted that Gates pick his dice first, and upon a closer inspection of the numbers, he in turn insisted that Buffett choose first. The link between people who like nontransitive dice and billionaires may be only correlation and not causation.

There are people online trying to sell their secrets to winning the lottery. Much of the pseudoscience around lottery draws tries to cloak itself as being mathematical and is normally a variation on the gambler’s fallacy. This logical fallacy is that, if some random event has not happened for a while, then it is “due.” But if events are truly random and independent, then an outcome cannot be more or less likely based on what has come before it. Yet people track which numbers have not come up in the lottery recently to see which ones are due an appearance. This reached fever pitch in Italy in 2005 when the number 53 had not been seen for a very long time. The Italian lottery in 2005 was a bit different from those in other countries: it had ten different draws (named after different cities), in each of which participants chose five numbers from a possible ninety. Unusually, players didn’t have to choose a complete set of numbers. They could opt to bet on a single number coming out of a certain draw. And the number 53 had not come out of the Venice draw for nearly two years. Loads of people felt that the Venice 53 ball was due. At least €3.5 billion was spent buying tickets with the number 53; that’s €227 per family in Italy. People were borrowing money to place bets as 53 continued not to get drawn and so was, apparently, more and more overdue. Those with a system kept increasing their stake each week so when 53 finally arrived they would recoup all previous losses. Players ...

Another thing that happens is that people mistakenly think that recent results are unlikely to happen again. I’ve seen advice online like “Don’t choose numbers that have won the big jackpot before” and “Using a combination that has gone through the system already will stack the odds even higher.” It’s all rubbish. In 2009 the Bulgarian lottery drew the same numbers—4, 15, 23, 24, 35, and 42—two draws in a row, on September 6 and 10. They were drawn in a different order but, in a lottery, the order does not matter. Amazingly, no one won the jackpot the first time they were drawn but, the following week, eighteen people had chosen them, in the hope they would come up again. The Bulgarian authorities launched an investigation to check that nothing untoward was going on, but the lottery organizers said that it was just random probability. And they were right.

But we owe it to ourselves to try to work out these probabilities as best we can. This is what Richard Feynman was faced with during the investigation into the shuttle disaster. The managers and high-up people in NASA were saying that each shuttle launch had only a 1-in-100,000 chance of disaster. But, to Feynman’s ears, that did not sound right. He realized it would mean there could be a shuttle launch every day for three hundred years with only one disaster. Almost nothing is that safe. In 1986, the same year as the disaster, there were 46,087 deaths on roads in the United States—but Americans drove a total of 1,838,240,000,000 miles in that year. Which means a journey of around 400 miles had a 1-in-100,000 chance of ending in a fatal disaster (for comparison, in 2015 it was 882 miles). The shuttle was cutting-edge space travel, which is always going to be more dangerous than driving 400 miles in a car. The odds of 1 in 100,000 was not a sensible estimate of the probability. When Feynman asked the actual engineers and people working on the space shuttle what they thought the chance of disaster was on any given flight, they gave answers of around 1 in 50 to 1 in 300. This is very different from what the manufacturers (1 in 10,000) and NASA management (1 in 100,000) believed. In hindsight, we now know that, of the 135 flights (before the shuttle program was ended in 2011), 2 of them ended in disaster—a rate of 1 in 67.5. Feynman came to realize that the 1-in-100,000 probab...

Far from getting a string of faultless test flights, NASA was seeing signs of possible failure during tests. There were also some noncritical failures during actual launches, which did not cause any problems with the flight itself but showed that the chance of things going wrong was higher than NASA wanted to admit. They had calculated their probability based on what they wanted and not on what was actually happening. But the engineers had used the evidence from testing to try to calculate the actual risk, and they were about right. When humankind puts its mind to it and doesn’t let its judgment be clouded by what people want to believe, humans can be pretty good at probability. If we want to.

What counts as a mistake in finance? Of course, there are the obvious ones, where people simply get the numbers wrong. On December 8, 2005, the Japanese investment firm Mizuho Securities sent an order to the Tokyo Stock Exchange to sell a single share in the company J-COM Co., Ltd. for ¥610,000 (around $5,100 at the time). Well, they thought they were selling 1 share for ¥610,000 but the person typing in the order accidentally swapped the numbers and put in an order to sell 610,000 shares for ¥1 each. They frantically tried to cancel it, but the Tokyo Stock Exchange was proving resistant. Other firms were snapping up the discount shares and, by the time trading was suspended the following day, Mizuho Securities was looking at a minimum of ¥27 billion in losses (well over $200 million at the time). It was described as a “fat-fingers error.” I would have gone with something more like “distracted fingers” or “‘should learn to double-check all important data entry but is probably now fired anyway’ fingers.” The wake of the error was wide-reaching: confidence dropped in the Tokyo Stock Exchange as a whole, and the Nikkei Index fell 1.95 percent in one day. Some, but not all, of the firms that bought the discount shares offered to give them back. A later ruling by the Tokyo District Court put some of the blame on the Tokyo Stock Exchange because their system did not allow Mizuho to cancel the erroneous order. This only serves to confirm my theory that everything is better with a...

These clay tablets are from the Sumerian city of Uruk (in modern-day southern Iraq) and were made between 3400 and 3000 BCE, so over five thousand years ago. It seems the Sumerians developed writing not to communicate prose but rather to track stock levels. This is a very early example of math allowing the human brain to do more than it was built for. In a small group of humans you can keep track of who owns what in your head and have basic trade. But when you have a city, with all the taxation and shared property that it requires, you need a way of keeping external records. And written records allow for trust between two people who may not personally know each other. (Ironically, online writing is now removing trust between humans, but let’s not get ahead of ourselves.) Some of the ancient Sumerian records were written by a person seemingly named Kushim and signed off by a supervisor, Nisa. Some historians have argued that Kushim is the earliest human whose name we know. It seems the first human whose name has been passed down through millennia of history was not a ruler, a warrior, or a priest . . . but an accountant. The eighteen existing clay tablets that are signed by Kushim indicate that their job was to control the stock levels in a warehouse that held the raw materials for brewing beer.

Kushim and Nisa are particularly special to me not because they are the first humans whose names have survived but because they made the first ever mathematical mistake, or at least the earliest that has survived (at least, it’s the earliest one I’ve managed to find; let me know if you locate an earlier error). Like a modern trader in Tokyo incorrectly entering numbers into a computer, Kushim entered some cuneiform numbers into a clay tablet incorrectly. From the tablets we can find out a bit about the math that was being used so long ago. For a start, some of the barley records cover an administration period of thirty-seven months, which is three twelve-month years plus one bonus month. This is evidence that the Sumerians could have already been using a twelve-month lunar calendar with a leap month once every three years. In addition, they did not have a fixed number-base system for numbers but rather a counting system using symbols that were five, six, ten, or three times bigger than the next lower in the scale. Once you get through the alien number system, the mistakes are so familiar, they could have been made today. On one tablet, Kushim simply forgets to include three symbols when adding up a total amount of barley. On another one, the symbol for one is used instead of the symbol for ten. I think I’ve made both those mistakes when doing my own bookkeeping. As a species, we are pretty good at math, but we haven’t gotten any better over the last few millennia. I’m sure...

In August 2012 the New York Stock Exchange started a new Retail Liquidity Program, which meant that, in some situations, traders could offer stocks at slightly better prices to retail buyers. This Retail Liquidity Program received regulatory approval only a month before it went live, on August 1. Knight Capital rushed to update its existing high-frequency trading algorithms to operate in this slightly different financial environment. But during the update Knight Capital somehow broke its code. As soon as it went live, the Knight Capital software started buying stocks of 154 different companies on the New York Stock Exchange for more than it could sell them for. It was shut down within an hour, but once the dust had settled, Knight Capital had made a one-day loss of $461.1 million—roughly as much as the profit they had made over the previous two years. Details of what exactly went wrong have never been made public. One theory is that the main trading program accidentally activated some old testing code, which was never intended to make any live trades—and this matches the rumor that went around at the time that the whole mistake was because of “one line of code.” Whatever the case, an error in the algorithms had some very real real-world consequences. Knight Capital had to offload the stocks it had accidentally bought to Goldman Sachs at discount prices and was then bailed out by a group including investment bank Jefferies in exchange for 73 percent ownership of the firm. T...

In high-speed trading, data is king. If a trader has exclusive information about what the price of a commodity is likely to do next, they can place orders before the market has a chance to adjust. Or rather, that data can go straight into an algorithm that can place the order, making decisions at incredible speeds. These times are measured in milliseconds. In 2015 Hibernia Networks spent $300 million laying a new fiber-optic cable between New York and London to try to reduce communication times by 6 milliseconds. A lot can happen in a thousandth of a second. Let alone six.

For financial data, time is literally money. The University of Michigan publishes an Index of Consumer Sentiment, which is a measure of how Americans are feeling about the economy (produced after phoning roughly five hundred people and asking them questions), and this information can directly affect financial markets. So it was important how this data was released. Once the new figures were ready, Thomson Reuters would put them on its public website at precisely 10 a.m. so everyone could access them at once. In exchange for this exclusive deal to release the data for free, Thomson Reuters paid the University of Michigan over $1 million dollars per year. Why were they paying to give the data away for free? In the contract, Thomson Reuters was allowed to give the numbers out five minutes early to its subscribers. So anyone who paid to subscribe to Thomson Reuters could get the data five minutes before the rest of the market and start trading accordingly. And subscribers to their “ultra-low-latency distribution platform” received the data two seconds earlier than that, at 9:54:58 (plus or minus half a second), ready to be fed straight into trading algorithms. In the first half second after this data is released, more than $40 million worth of trades can already have occurred in a single fund. The chumps who waited to get the data for free at 10 a.m. would find that the market had already adjusted. The ethics (and probably legality) are a bit blurry. Private institutions are a...

The two competing technologies are fiber-optic cables and microwave relays. Light going down a fiber-optic cable travels at about 69 percent of the maximum speed of light in a vacuum, which is still blisteringly fast, covering about 128,000 miles every second. Microwaves move through the air at almost the full 186,282 miles per second maximum speed of light, but they have to be bounced from base station to base station to follow the curvature of the Earth. There are also problems about where microwave base stations can be built and where fiber-optic cable can be laid. So the path from DC to Chicago taken by the data will not be the shortest possible route. But to get a lower limit, we can assume that the data follows the shortest line between the Fed Building in DC to the Chicago Mercantile Exchange building (593.8 miles) at the full speed of light (new hollow-core fiber-optic cables can reach 99.7 percent of the speed of light) and calculate a time of 3.19 milliseconds. A similar calculation for the shorter DC–to–New York City journey gives 1.09 milliseconds. Those times assume that the data is racing down a fiber-optic cable following the curvature of the Earth. Straight-line travel would be slightly faster. There are already line-of-sight laser communication systems for financial data where the beginning and end points have nothing but air between them—for example, those that relay information between buildings in New York and buildings in New Jersey. To travel between ...

It would be remiss of me not to say something about the global financial crisis of 2007–8. It was kicked off by the subprime-mortgage crisis in the United States, then rapidly spread to countries all around the world. And there are some interesting bits of mathematics that fed into it. My personal favorites are collateralized-debt-obligation (CDO) financial products. A CDO groups a bunch of risky investments together on the assumption that they couldn’t possibly all go wrong. Spoiler: they all went wrong. Once CDOs could themselves contain other CDOs, a mathematical web was built that few people understood. I love my math, but, looking back at the global financial crisis as a whole, I don’t claim to understand what went wrong. If you want to look into it in greater depth, there are countless books out there dedicated to just this one topic. Or (if you’re old enough) watch the film The Big Short. I’m not going to say anything about it. Instead I’ll talk about a more interesting and concrete example of people not understanding math: company boards giving chief executive officers (CEOs), that is, the people in charge of companies, pay awards.