As we’ve seen over and over again, rounding/truncation errors present problems in the real world. But what about the ways in which people (more specifically, the people in power) try to compensate for these errors?

Take this example, where drivers in Germany, the Netherlands, and the United Kingdom were overcharged for fuel. (Imagine how you would feel if you found out that rounding errors were making you pay even more on top of what we’re already paying for gas!) This unfortunate mistake was “caused by a rounding error in an automated pricing system”. To make up for the error, officials dropped oil prices by a penny per gallon. However, no one really knows how they came up with this suspicious price cut. Actually, due to the different prices in each country and without knowing the volume of gas sold in each country, it is impossible to know by how much the consumers were cheated.

Well, at least they are offering some sort of compensation, unlike in this story, where citizens’ electric bills produced errors caused by “the average daily energy use for the month in Kilowatt-hours [being] rounded to the nearest integer value”. In this case the company’s only response was to “assist [you] in understanding how the percentage … is calculated.”

A similar example of a rounding error correction is our internationally accepted timekeeping system. As I’ve learned in ASTRO 102, adding leap years is just another way to make up for rounding errors of the length of a year. From 325 A.D. to 1582, the Julian Calendar was used when a year was known to have 365.25 days. Thus, every fourth year had an extra day on February 29. Later, it was found that an average Julian year was actually about 11 minutes longer than a true year (365.25 vs. 365.2422). By 1582, the rounding error in “Julian’s method” had added up to 10 days! So, in 1582, under Pope Gregory XIII, October 5 became October 15! (The power of the Pope, I guess) And now, the only century years that are leap years are those that are divisible by 400 (ie. 2100 is not a leap year). Even now, our current timekeeping system still incurs an error for estimating Earth’s orbit around the Sun, and occasional leap seconds are even added as tweaks to our time.

Overall, “rounding errors” are just a part of life. As long as we try to apply the methods and systems of scientific computing to the real world, there will always be the element of chaos and randomness that cannot be predicted or recovered.