Number systems gone but not forgotten

Today, a respite from ugly political news. I wrote at the New Year about the human tendency to relegate old Gods to lowercase gods but still keep them around in cultural practices. We have done something similar in the way we count. New ways to count show up and old ones fade, yet they never quite disappear. This is a good topic to celebrate the start of the new year MMXXI and the upcoming (not so) Superbowl LV.

That old Roman system never was very practical for mathematics beyond simple counting. However, it remained in use in Europe well into the late Middle Ages, getting enshrined into the first decades of moveable-type printing, which is perhaps one reason why Roman numerals so often reappear in old Gothic lettering.

The Roman system was essentially a Base-5 system with 5, 10, 50, 100, 500 and 1000 being key rollover quantities. This rollover and re-use of digits is so much a part of our internal thinking about numbering that it is hard to even conceive that it is artificial.Why do we limit ourselves to a small set of symbols for numbers and then re-use them for the next rounds of counting?

Maintaining a practical number of symbols and having ten digits on our hands obviously had key roles here. But if humans had evolved only eight digits, what would our counting system have looked like? And, for that matter, why do our digits look like they do?

Eight fingers, or maybe sixteen?

When I first started programming computers in the 1970s, coders needed to develop a skill in math using Base-8 or Base-16 number systems, depending on the computer’s manufacturer. Digits in octal systems (Base-8) range from 0 through 7 before cycling back to an octal 10 (which is really a decimal eight). A hexadecimal (Base-16) system adds the letters A through F. Hexadecimal F thus signifies a decimal 15, with “Hex 10” equal to a decimal 16.

Either of these numbering systems can be used to simplify the computer’s internal binary (Base-2) system, where all digits are either 1 or 0. Groups of three binary “bits” are represented by a single octal number, while groups of four bits map to one hexadecimal digit. Before the rise of better programming tools, we had to code many computer commands, memory addresses and character strings in octal or hexadecimal. These days you can do many great things with computers that we could never do in the 1970s, and yet never need to compute a hexadecimal address.

A common Internet prank is to poll people on whether they approve of teaching “Arabic numerals” in American schools, which inevitably receives a lot of outrage comments and “No” votes. But our current numeral set, better described as the Hindu-Arabic or Indo-Arabic system, did not reach Europe in common use until less than 1000 years ago, bringing the mathematical concept of zero with it for the first time. Our year-numbering BC/AD calendar system, dating back only to the sixth century AD (or CE), has no “Year Zero” in it, because the concept did not exist at the time in western mathematics. You can only go so far in math without a zero.

The Babylonian alternative

Much of our daily number use goes even further back than the early Indians or the Romans, however. Why are there 60 seconds in a minute or 60 minutes in an hour? Or twelve inches in a foot or 360 degrees in a circle? The path is messy, but much of it traces back to the Babylonian Base-60 system from almost 4000 years ago. [1]

Babylonian cuneiform numerals. Source: Wikipedia Commons

Base-60 is actually a great way to do basic commercial counting, and indeed, many of the earliest Babylonian writings found were business transactions or counting tallies. 60 is cleanly divisible by the quantities of 2, 3, 4, 5, 6, 12, 15, 20, and 30, which makes the math relating to counting commercial quantities much simpler than in the Base-10 system when much of the application is dividing a quantity into equal parts for sale or purchase. Try divvying up the quantity 10 into pieces when you don’t have fractions or decimal points; your options are severely limited.

Half of 60 proved to be an imprecise but “good enough” measure of the length of lunar month for many centuries, as did 12 times 60, or 360 days, for the lunar year (which is actually closer to 354 days plus change). We retain a cobbled-together variant of that calendar yet today. [2]

The last metric hold-out

The metric system extended Base-10 counting into additional measures, such as weight, distance, and temperature. The United States is one of the last countries on Earth to continue to teach the metric system to children only as a “second language,” behind the far more complicated Imperial system. There was even an attempt to extend metric counting to hours, days and months that rose with the French Revolution, but it quickly died when Napoleon came to power. The Babylonians won that battle thousands of years later.

The typical American remains stubbornly “monolingual” here unless your business has international intersections, or if you live close to Canada and watch the weather report. I know the metric system pretty well, but my mind still needs to do the awkward “close enough” conversion from Celsius to Fahrenheit temperatures: Multiply by two and add thirty. [3]

Nature’s number base of e

At the smallest levels of nature, you will not really find integer (1,2.3…) numbering at all. Slugs do not know how to count or even need to. Physicists and much of nature have learned that you do not need an integer base for a number system at all, if you make the number base Euler’s number. Named after the Swiss mathematician Leonhard Euler (1707– 1783), this value, which is usually denoted simply as e, is an “irrational” number like 𝛑 but with a value, when converted to decimal, which starts with the digits 2.71828 and continues on forever.

One way to describe e is as the base rate of growth shared by all continually growing processes, where every nanosecond something is growing just a little bit. As I have written in the past, this value works very visibly for the chambered nautilus (below), but it pops up everywhere in nature.

Source: Wikipedia

The pioneering quantum physicist Erwin Schrödinger (1887–1961) found the value e at the core of his wave functions defining the behavior of the smallest things in the universe, the curved “waves” of energy making up atoms. It is also (not coincidentally) the numerical value where nature essentially makes its own logarithmic “slide rule,” turning multiplication problems into easy addition problems and exponential problems into simpler multiplication problems.

This “biological math” appears to be essential in “coding” our ears to hear a huge range of logarithmically spaced sound frequencies as a compressed linear scale of notes on a scale. That same sound’s big quiet-to-loud exponential range of measurable intensity is biologically mapped to be sensed more similar to the one-to-ten linear scale on an amplifier dial (or one to eleven if you are a Spinal Tap fan). All of our senses appear to work this way, with e or its related “natural logarithm” always popping up whenever we try to translate what is going on biologically into our Base-10 math system. Base-10 was late to the game in geological (and even human) time.

And yet, all of these different ways to count can give creative new options in multiple dimensions to cynical old math guys, thanks to Euler, Schrödinger, and the poet Elizabeth Barrett Browning (1806–1861):

How do I love thee? Let me count the ways.
I love thee to the depth and breadth and height
My soul can reach, when feeling out of sight…

Notes:

1. The path from ancient Babylon to twelve inches in one foot is a little more convoluted. However, the number 12 is both a factor of 60, and a multiple of 2, 3, 4 and 6, and so it has had an “easy math” trajectory through history with mystical overtones. In the Bible alone, “Ten” got the “Commandments” from God via Moses, but “Twelve” got the apostles of Jesus, the twelve gates to John the Revelator’s New Jerusalem, the twelve tribes of Israel, and many more tags.
2. Even though we now usually say that the year has 365.25 days, the correct number is really 11 minutes less than that, and so the year 2100 is not going to be a leap year, if we make it that far.
3. To be more precise, you multiply by 9/5ths and add 32, but for most normal outside temperatures, the simpler approximation is “good enough.” Ten degrees C is fifty degrees F in either calculation.

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