Humble Pi: A Comedy of Maths Errors

by Matt Parker

A million seconds from now is just shy of eleven days and fourteen hours. Not so bad. I could wait that long. It’s within two weeks. A billion seconds is over thirty-one years. A trillion seconds from now is after the year 33700CE.

In 46BCE Julius Caesar decided to fix this with a new, predictable calendar. Every year would have 365 days – the closest whole number to the true value – and the bonus quarter days would be saved up until every fourth year, which would have a single bonus day. The leap year with an extra leap day was born! To get everything back into alignment in the first place, the year 46BCE had a possible-world-record 445 days. In addition to the bonus month between February and March, two more months were inserted between November and December. Then, from 45BCE, leap years were inserted every four years to keep the calendar in synch.

If you have a Mac within reach, open up the app Terminal, which is the gateway to how your computer actually works. Type in date +%s and hit Enter. Staring you in the face will be the number of seconds that have passed since 1 January 1970.

So, when playing the piano, going up a ‘third’ means going up two notes and going up a ‘fifth’ is going up only four notes. Put together, the whole transition is a ‘seventh’, giving us 3 + 5 = 7. Counting the dividers and not the intervals means that the note between the transitions is double-counted. It is also why an ‘octave’ of seven notes (and seven intervals) is named ‘oct’ for eight. The upside is that I can blame my terrible lack of musical ability on the numbers not behaving normally.

Peter Cameron went to a pancake restaurant in Canada he noticed that they advertised a choice of ‘1,001 toppings’. Being a combinatorist, he recognized that 1,001 is the total number of ways to pick four things out of a total of fourteen options, so he figured they had fourteen toppings and customers could choose four. Actually, the restaurant had twenty-six toppings (he asked) and had picked ‘1,001’ just because it sounded big. Had they done the maths correctly, those twenty-six toppings in fact allowed for 67,108,864 options. A rare case of marketing understatement.

Gandhi was a bit of a jerk. Once he developed atomic technology, he would start dropping nuclear bombs on other nations. This was because of a mistake in the computer code. The game designers had deliberately given Gandhi the lowest non-zero aggression rating possible: a score of 1. Classic Gandhi. But later in the game, when all the civilizations were becoming more, well, civilized, every leader had their aggression rating dropped by two. For Gandhi, starting from 1, this calculation played out as 1 − 2 = 255, suddenly setting him to maximum aggression. Even though this error has since been fixed, later versions of the game have kept Gandhi as the most nuke-happy leader as a tradition.

Trains in Switzerland are not allowed to have 256 axles.

This roughly translates as ‘In order to avoid the danger of an unintentional all-clear signal for a railway section because of the reset to zero of the axles counter, a train must not have an effective total number of axles equal to 256.’

Steward Nigel Ogden rushed back on to the flight deck, to find the co-pilot trying to regain control of the aircraft because the pilot had been sucked out of the window, colliding with the control column on the way out and disengaging the autopilot. Well, he’s almost out of the window. He’s caught on the windscreen frame, so his legs are still inside the aircraft. Ogden managed to grab the pilot’s legs to stop him from flying out of the window completely. The co-pilot, Alistair Atcheson, was able to regain control of the aircraft and land it, with Captain Tim Lancaster dangling half out of the window. The crew had taken it in turns to hold on to his legs. Everyone survived, including Captain Lancaster, who spent twenty-two minutes outside the aircraft, made a full recovery and went back to being a pilot.

German physicist Daniel Fahrenheit proposed the scale which bears his name in 1724, and the zero-point was based on a frigorific mixture. If ‘frigorific’ has not instantly become your new favourite word, you’re cold and dead inside.

A frigorific mixture is a pile of chemical substances which will always stabilize to the same temperature, so they make for a good reference point. In this case, if you give ammonium chloride, water and ice a good stir, they will end up at 0°F. If you mix just water and ice, it will be 32°F, and the far less frigorific mixture of human blood (while still inside a healthy human) is 96°F. While these were Fahrenheit’s original reference points, the modern Fahrenheit scale has since been adjusted and is now pinned to water freezing at 32°F and boiling at 212°F. Frigorific!

The Celsius scale began around the same time, with Swedish astronomer Anders Celsius, except he counted the wrong way. Celsius started with zero as the boiling point of water at normal atmospheric pressure then counted up as the temperature went down, with water eventually freezing at 100°C. Meanwhile, other people went with the more popular convention of starting with zero at water’s freezing point and counting up to a hundred as its boiling point, and then they all argued over who had the idea first. There was no clear winner as to whose idea it was, but the unit itself caught on and was give the neutral name Centigrade. Celsius had the last laugh, however, when the name Centigrade clashed with a unit for measuring angles (a centigrade, or gradian, is one four-hundredth of a circle) so, in 1948, it was named after him after all. Celsius is now used almost universally to measure temperature, except s...

In the Apothecaries system of weight units, a pound can be split into 12 ounces, which each consist of 8 drams. A dram is then 3 scruples, each made from 20 grains. I hope that made sense. A grain is one 5,760th of a pound. But not a normal pound: this is a Troy pound. Which is different to a normal pound.

A patient taking Phenobarbital (an anti-epileptic drug) was prescribed 0.5gr per day (32.4 milligrams), and this was mistaken for 0.5 grams per day (500 milligrams). After three days on over fifteen times their normal dose, the patient started to have respiratory problems. Thankfully, when they were taken off the dose, they made a full recovery. This was a case of no grain, no pain.

After the 2011 census the Australian Bureau of Statistics published who the average Australian was: a thirty-seven-year-old woman who, among other things, ‘lives with her husband and two children (a boy and a girl aged nine and six) in a house with three bedrooms and two cars in a suburb of one of Australia’s capital cities’. And then they discovered that she does not exist. They scoured all the records and no one person matched all the criteria to be truly average. As they rightly pointed out: While the description of the average Australian may sound quite typical, the fact that no one meets all these criteria shows that the notion of the ‘average’ masks considerable (and growing) diversity in Australia. – Australian Bureau of Statistics

The US constitution requires a nationwide census every ten years. But by 1880, due to the increase in population and in the census questions, it was taking eight years to process all the data. To fix the problem, electromechanical tabulating machines were invented that could automatically total up data which had been stored on punch cards. Tabulating machines were used in the 1890 census and were able to complete the data analysis in only two years.

Between 1993 and 2008 the police in Germany were searching for the mysterious ‘phantom of Heilbronn’, a woman who had been linked to forty crimes, including six murders; her DNA had been found at all the crime scenes. Tens of thousands of police hours were spent looking for Germany’s ‘most dangerous woman’ and there was a €300,000 bounty on her head. It turns out she was a woman who worked in the factory that made the cotton swabs used to collect DNA evidence.

Otherwise, when you hear a statistic such as the fact that cancer rates have been steadily increasing, you could assume that people are living less healthy lives. The opposite is true: longevity is increasing, which means more people are living long enough to get cancer. For most cancers, age is the biggest risk factor and, in the UK, 60 per cent of all cancer diagnoses are for people aged sixty-five or older. As much as it pains me to say it, when it comes to statistics, the numbers are not everything.

Benford’s Law,

YouTube video for the Numberphile

The CHI+MED team who are researching the computer–human interactions with medical devices actually came up with a new version of the Swiss Cheese model which I’m quite partial to: the Hot Cheese model of accident causation. This turns the Swiss cheese on its side: imagine the slices of cheese are horizontal and mistakes are raining down from the top. Only mistakes which fall down through holes in every layer make it out the bottom to become accidents.

In 1907 a combination road-and-railway steel bridge was being built across a section of the St Lawrence River in Canada, which was over half a kilometre wide. Construction had been going on for some time but on 29 August one of the workers noticed that a rivet he had put in place about an hour earlier had mysteriously snapped in half. Then, suddenly, the whole south section of the bridge collapsed, with a noise that was heard up to 10 kilometres away. Of the eighty-six people working on the bridge at the time, seventy-five died. There had been a miscalculation as to how heavy the bridge would be, partly because, when the bridge design was increased from 1,600 feet to 1,800 feet, the forces had not been recalculated, so the lower support beams buckled and eventually failed completely.

Because of the Quebec Bridge disaster, starting in 1925 any student graduating from an engineering degree in Canada can attend a voluntary Ceremony of the Calling of an Engineer, where they are given a steel ring to remind them of the humility and fallibility of engineers.