The Colorful History of Binary Arithmetic
By Keith Devlin @profkeithdevlin
Performer, satirist, and mathematician Tom Lehrer in the 1950s
As the musical satirist (and mathematician) Tom Lehrer quipped in one of his songs (“New Math”), “Base-eight arithmetic is just like base-ten—if you are missing two fingers.”
It was a funny line that always got a laugh, but it highlights a major feature of the Hindu-Arabic positional number system used around the world to perform integer arithmetic. We group in tens for the simple reason that we learn to count and perform elementary calculations using our fingers. That’s why our number system is base-ten. Indeed, we use the Roman word digit, meaning finger, to refer to the basic numerical entities, each denoted by its own special symbol.
Less well known is that there is solid neurological evidence that counting is intrinsically connected to our use of fingers – the person who visibly uses their fingers when counting (or adding) has simply not (yet) gotten to a stage where they can perform the task virtually.
Gerstmann’s Syndrome, a neurological disorder named after the early Twentieth Century American neurologist Josef Gerstmann who first identified it, shows the degree to which counting is tied to our use of our fingers. Caused by brain lesions (which could be present at birth, or caused by a stroke later in life), the syndrome exhibits four symptoms:
Dysgraphia – a reduced ability to write
Dyscalculia – difficulty in understanding or learning mathematics
Finger agnosia – inability to distinguish the fingers on the hand
Left-right disorientation
That’s a tight connection for brain lesions in one particular area of the brain. The pathway to learning mathematics begins, it appears, with using our fingers to count things at an early age.
I came across Gerstmann’s Syndrome back in the 1990s when I was doing research for my book The Math Gene, published in 2001.
The base-8 arithmetic that Lehrer referred to in his song, sometimes referred to as octal arithmetic, is for the most part an artifact of the math class, but it is not just a theoretical concept. Some Native American societies used it as a result of counting using not with the fingers but the spaces between the fingers, and Mexican societies used it because they counted on the knuckles of a closed fist.
Far more prevalent today is the binary system, which is the fundamental arithmetic of digital computers. As with base-8, the binary system has its share of jokes, such as “There are 11 types of people; those who understand binary arithmetic and those who don’t.” (I like that one because it actually packs several mathematical issues into one short sentence.)
But while it is tempting to present students with binary arithmetic in the context of digital circuits (switches, or gates, that have just two states, on and off), it has a rich history of human use. One of my favorites is its common use by 13th century English brewers and wine merchants. The system of measurement used in the wine and brewing trade in England from the 13th century onwards had the following units:
2 gills = 1 chopin, 2 chopins = 1 pint, 2 pints = 1 quart, 2 quarts = 1 pottle, 2 pottles = 1 gallon, 2 gallons = 1 peck, 2 pecks = 1 demibushel, 2 demibushels = 1 bushel or firkin, 2 firkins = 1 kilderkin, 2 kilderkins = 1 barrel, 2 barrels = 1 hogshead, 2 hogsheads = 1 pipe, 2 pipes = 1 tun.
No, my interest in this system of units is independent of my adult taste in beverages; it predates it by a decade or more. The soft covered math notebooks we were issued with in elementary school had those measurements (and some other historical tables of units) printed on the outside of the back cover. I suspect I was not alone in being intrigued by this quaint-sounding system of units that my ancestors made daily use of. It’s an ideal way to arouse students’ interest in number systems. (So credit to whoever in the UK decided to leave those tables on the outer covers of math notebooks long after they had any use, even within particular professional communities.)
I am familiar only with the measuring systems used in former times in my own childhood country (England), but I am sure there are excellent examples in others.
Some of the old English terms remain in common use today in colloquial English, though in many cases they are used without an awareness of their origin or their definition.
For instance, you often hear people refer to “a ton of information/examples/cases/…” Such use of a unit of weight measurement is metaphorical, of course, but the origin of such uses in spoken English is not clear. In addition to ton as unit of weight, there is also tun, a centuries old English brewers’ unit of volume to measure a large cask, equal to 2 butts (see presently) or 252 gallons. Since volume is usually a closer analogy to the things being described by the phrase “a ton of …” than is weight, I suspect the phrase should actually be spelled “a tun of …” if we want to be aligned with its origin.
The tun is a unit of volume used by English brewers and wine makers from the 13th Century onwards
Another common example is buttload, as in “We picked a buttload of apples.” It’s generally used to indicate an unspecified large amount, but it actually has a precise historical meaning, being another English brewers’ unit of measure. A buttload is the capacity of a particular kind of heavy cart (a butt). It is equal to 6 seams, or 48 bushels, which the ten-year-old me would (with the aid of the cover of my math notebook) have been able to calculate as equal to 384 gallons.
For the record, one tun is equal to two butts. And half a butt was called a hogshead. All binary. (Though as the illustrations make clear, other fractions were in common use as well. It was very much a mixed system, though binary played a major role.)
Binary arithmetic has, in seems, a long history of practical use, going back long before digital computers came onto the scene. For many people, historical use in society may be an excellent way to arouse interest in number systems.
Just saying.