Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

Pepsi took active steps to protect itself from future problems and re-released the ad with the Harrier increased in value to 700 million Pepsi Points. I find it amazing that they did not choose this big number in the first place. It’s not like 7 million was funnier; the company just didn’t bother to do the math when choosing an arbitrary large number.

I had to go on the BBC News in 2012 to explain how big a trillion is. The UK debt had just gone over £1 trillion, and they wheeled me out to explain that that is a big number.

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.

Those surprising numbers actually make perfect sense after a moment’s thought. Million, billion, and trillion are each a thousand times bigger than each other. A million seconds is roughly a third of a month, so a billion seconds is on the order of 330 (a third of a thousand) months. And if a billion is around 31 years, then of course a trillion is around 31,000 years.

Humans instinctively perceive numbers logarithmically, not linearly. A young child or someone who has not been indoctrinated by education will place three halfway between one and nine.

Why?

So the “multiplication middle” is three, and that is what humans do by default, until we are taught otherwise.

...what? I'd like a citation

Even after a lifetime of education dealing with small numbers, there is a vestigial instinct that larger numbers are logarithmic; that the gap between a trillion and a billion feels about the same as the jump between a million and a billion—because both are a thousand times bigger.

the skills that allow us to survive and form communities do not necessarily match formal mathematics. A logarithmic scale is a valid way to arrange and compare numbers, but mathematics also requires the linear number line.

All humans are stupid when it comes to learning formal mathematics. This is the process of taking what evolution has given us and extending our skills beyond what is reasonable.

We now have school systems that force students to study math and, through enough exposure, our brains can learn to think mathematically. But if those skills cease to be used, the human brain will quickly return to factory settings.

As a species, we have learned to explore and exploit mathematics to do things beyond what our brains can process naturally. They allow us to achieve things well beyond what our internal hardware was designed for. When we are operating beyond intuition, we can do the most interesting things, but this is also where we are at our most vulnerable.

Mistakes like the ones in the following pages aren’t just amusing; they’re revealing. They briefly pull back the curtain to reveal the mathematics that is normally working unnoticed behind the scenes.

The radios were down for about three hours, during which time the controllers used their personal cell phones to contact other traffic control centers to get the aircraft to retune their communications.

Four hundred flights on the ground were delayed and a further six hundred canceled. All because of a math error.

It seems the air-traffic control system kept track of time by starting at 4,294,967,295 and counting down once a millisecond. Which meant that it would take 49 days, 17 hours, 2 minutes, and 47.295 seconds to reach 0.

What tf

So, in 2004, it accidentally ran for fifty days straight, hit zero, and shut down. Eight hundred aircraft traveling through one of the world’s biggest cities were put at risk because, essentially, someone didn’t choose a big enough number.

Whenever you started the program, Windows would count up once every millisecond to give the “system time” that would drive all the other programs. But once the Windows system time hit 4,294,967,295, it would loop back to zero.

It is unclear if Windows itself was directly to blame or if it was a new piece of computer code within the control center system itself. But, either way, we do know that the number 4,294,967,295 is to blame.

Oh, and it was not big enough in 2015 for the Boeing 787 Dreamliner aircraft.

Dear God

The problem with the Boeing 787 lay in the system that controlled the electrical power generators. It seems they kept track of time using a counter that would count up once every 10 milliseconds (so, a hundred times a second) and it topped out at 2,147,483,647 (suspiciously close to half of 4,294,967,295).

This means that the Boeing 787 could lose electrical power if turned on continuously for 248 days, 13 hours, 13 minutes and 56.47 seconds. This was long enough that most planes would be restarted before there was a pr...

Their official line on airworthiness was the requirement of “repetitive maintenance tasks for electrical power deactivation.” That is to say, anyone with a Boeing 787 had to remember to turn it off and on again.

Why would Microsoft, Los Angeles Air Route Traffic Control Center, and Boeing all limit themselves to this seemingly arbitrary number of around 4.3 billion (or half of it) when keeping track of time? It certainly seems to be a widespread problem. There is a massive clue if you look at the number 4,294,967,295 in binary. Written in the 1s and 0s of computer code, it becomes 11111111111111111111111111111111; a string of thirty-two consecutive ones.

If you had space for only five digits on a piece of paper, the largest number you could write down would be 99,999. You’ve filled every spot with the largest digit available. What the Microsoft, air-traffic control, and Boeing systems all had in common is that they were 32-bit binary-number systems, which means the default is that the largest number they can write down is thirty-two 1s in binary, or 4,294,967,295 in base-10.

Counting only centiseconds rather than milliseconds bought them some time—but not enough.

Modern computer systems are generally 64-bit, which allows for much bigger numbers by default. The maximum possible value is of course still finite, so any computer system is assuming that it will eventually be turned off and on again. But if a 64-bit system counts milliseconds, it will not hit that limit until 584.9 million years have passed.

so thought the Russian shooting team as they arrived at the 1908 Olympic Games in London a few days before the international shooting was scheduled to start on July 10. But if you look at the results of the 1908 Olympics, you’ll see that all the other countries did well but there are no Russian results for any shooting event. And that is because what was July 10 for the Russians was July 23 in the UK (and indeed most of the rest of the world). The Russians were using a different calendar.

The universe has given us only two units of time: the year and the day.

The orbit we ended up in gave us the length of the year, and the rate of the Earth’s spin gave us the length of the day. Except they don’t match. There is no reason they should!

This goes from being a minor inconvenience to becoming a major problem because the Earth’s orbital year controls the seasons.

To fix this, we had to tweak the calendar to have the same number of days as the orbit. Somehow, we needed to break away from having the same number of days every year, but without having a fraction of a day; people get upset if you restart the day at a time other than midnight. We needed to link a year to the Earth’s orbit without breaking the tie between a day and the Earth’s rotation. The solution that most civilizations came up with was to vary the number of days in any given year so there is a fractional number of days per year on average.

If you ever have access to a friend’s phone, go into the settings and change their calendar to the Buddhist one. Suddenly, they’re living in the 2560s. Maybe try to convince them they have just woken up from a coma.

Our main modern calendar is a descendant of the Roman Republican calendar. They had only 355 days, which was substantially fewer than required, so every few years an entire extra month was inserted between February and March, adding an extra twenty-two or twenty-three days to the year. In theory, this adjustment could be used to keep the calendar aligned with the solar year. In practice, it was up to the reigning politicians to decide when the extra month should be inserted. As this decision could either lengthen their year of ruling or shorten that of an opponent, the motivation was not always to keep the calendar aligned.

The years leading up to 46 BCE were known as the “years of confusion,” as extra months came and went, with little relation to when they were needed.

In 46 BCE 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!

There was an initial clerical error, by which the last year in a four-year period was double-counted as the first year of the next period, so leap years were actually put in every three years. But this was spotted, fixed, and by 3 CE, everything was on track.

But Julius Caesar was betrayed—albeit long after his death—by the difference of 11 minutes and 15 seconds between the 365.25 days per year his calendar gave and the actual length of 365.242188792 days.

There is an oft-repeated statement that the Julian calendar years of 365.25 days were too long compared to the Earth’s orbit. But that is incorrect! The Earth’s orbit is 365 days, 6 hours, 9 minutes and 10 seconds: slightly more than 365.25 days. The Julian calendar is too short compared to the orbit.

it is too long compared to the seasons. Bizarrely, the seasons don’t even exactly match the orbital year.

As the Earth orbits, the direction it is leaning also changes, going from pointing directly at the sun to pointing away every 13,000 years. A calendar perfectly matching the Earth’s orbit will still swap the seasons every 13,000 years.

The movement of the Earth’s tilt buys us an extra 20 minutes and 24.43 seconds per orbit. So the true sidereal (literally, “of the stars”) year based on the orbit is longer than the Julian calendar, but the tropical year based on the seasons (which we actually care about) is shorter.

the seasons depend on the tilt of the Earth relative to the sun, not on the actual position of the Earth.

This slight mismatch between the Julian and tropical years was unnoticeable enough that, by 1500 CE, pretty much all of Europe and parts of Africa were using the Julian calendar. But the Catholic Church was sick of Jesus’s death (celebrated according to the seasons) drifting away from his birth (celebrated on a set date).

What we now know as the Gregorian calendar was not actually designed by Pope Greg—he was too busy doing pope things and convincing people to change their behavior—but by the Italian doctor and astronomer Aloysius “Luigi” Lilius.

Luigi’s breakthrough was to keep the standard every-fourth-year leap year of the Julian calendar but to take out three leap days every four hundred years.

Leap years were all the years divisible by four, and all Luigi suggested was to remove the leap days from years which were also a multiple of 100 (except those that were also a multiple of 400).

Despite it being a mathematically better calendar, because this new system was born out of Catholic holidays and promulgated by the pope, anti-Catholic countries were duly anti–Gregorian calendar.

This problem was exacerbated because the Gregorian calendar was backdated, recalibrating the year as if it, rather than the Julian option, had always been used. Through the use of pope power, it was decreed that ten dates would be taken from October 1582 and so, in Catholic countries, October 4, 1582, was directly followed by October 15.

When the English forces landed on Île de Ré on July 12, 1627, as part of the Anglo-French War, the French forces were ready to fight back on July 22. That is, on exactly the same day. At least, for both armies, it was a Thursday.

England (which still included parts of North America) swapped over in 1752, realigning its dates by removing eleven days from September. Thus, September 2, 1752, was followed by September 14, 1752.