The Evolution of Arithmetic

This post is the second in a series; if you haven’t read the first post, on the evolution of English counting words, I’d recommend reading that one first.

As promised, this post looks at the origins of the English words for arithmetic operations. Read on, friend!

  • Plus and Minus. These two are fairly straightforward—they’re the Latin words for “more” and “less”, respectively. The symbols, though, are less clear. It appears that the letters p and m were used (sometimes appearing as p and m) during the 1400s—Wikipedia claims that these first appeared in Luca Pacioli’s Summa de Arithmetica, though I’ve been unable to find a satisfactory example. In the 1500s, the modern + and – signs began to appear; Schwartzman attributes the + to an abbreviation of the Latin “et” (taking the t only) and the – to the bar from m.
  • Multiply. This word comes from the Latin multiplicare, meaning “to increase.” Breaking it down a little further, we have the prefix multi– (“many”) and the suffix -plex (“fold”) so that the compound word multiplex means “many folds.” (We still use “fold” language today—when we speak of a “threefold increase,” we mean that something had been multiplied by three.) The x symbol for multiplication is attributed to William Oughtred, while Schwartzman gives credit for the dot • to Gottfried Wilhelm Leibniz.
  • Divide. This word comes from Latin as well, with the origin being dividere, meaning “to separate.” (As a side note, the root videre means “to see” and gives us the modern word video, which means “I see”.) Putting di– and videre together, I suppose this means that division is literally “to see in two.”

Notice that all four of these words originate in a description of the operation itself. It turns out that exponents and roots are a little more metaphorical in their meaning:

  • Exponent. Once again, we have a Latin origin: the prefix ex– and the verb ponere, roughly meaning “to put out.” Unlike the four arithmetic operations, though, the original meaning is typographical—the exponent is the number that is “put out” above and to the right of the base. In part, it’s because the exponent is a relatively new development; Schwartzman attributes the notation to Descartes, specifically La Géométrie (1637).
  • Root. Finally, a non-Latin word! The word rot means “cause” or “origin”, which makes sense when you consider that since 8 = 23, its “origin” is 2. If you trace the word further back, the Proto-Indo-European root (see what I did there?) is wrad-. Thus, the Latin-based words radical and radish come from a source similar to root.

And there you have it! In the next installment, I’ll get a little more geometric and explore some words we’ve come to use for algebraic curves.

The Evolution of Numbers

I’ve always loved word origins. Often, knowing where a word comes from can provide you with insight on what it means today. At the very least, it makes you look smart at parties (e.g., “Did you know that the word apocalypse shares the same Greek root as calypso…”).

This post is the first in a series on mathematical word origins. Since math(s) is an old subject, many mathematical terms have ancient roots (for English, this usually means Greek and Latin). For today, we’ll explore the origins of the English words for counting and arithmetic. For all posts in this series, my primary source is Steven Schwartzman’s The Words of Mathematics, with an occasional assist from the Online Etymology Dictionary.

  1. One through Ten. I’m actually going to skip these; most Indo-European languages have a base 10 system whose words are in rough correspondence with each other (for instance, the word for 6 is six, sechsseiseis, and seks, in French, German, Italian, Spanish, and Norwegian, respectively). If you’re interested in exactly how those consonants correspond, go read up on Grimm’s law.
  2. Eleven and Twelve. These words have a particularly Germanic origin: for example, in French you’d use onze and douze while in German it’s elf and zwölf. The English word eleven comes from the Old English endleofon, which basically translates as “one left over.” If you were counting up 11 items, you’d get to ten, and then say “and there’s one left over.” Eventually, this got condensed down into our modern eleven. (It makes a certain amount of sense, no?) The same thing goes for twelve: the Old English twelf comes from the Proto-Germanic twa-lif, which means “two left.”
  3. Thousand and Million. The word thousand comes from the Germanic thus (thick) and hund (hundred), making a thousand a “thick hundred.” In the Romance languages, though, the word thousand comes from the Latin mille. The English word million originally meant “a great thousand.” Interestingly, there appears to be a connection between the words thousand and dozen (e.g., the Dutch word for a thousand is duizend), leading some scholars to speculate that some Germanic cultures had a mixed base-10 and base-12 system. This may also be seen in the fact that in the UK, a “hundredweight” is 112 lbs.
  4. Zero. This one’s a relatively new addition to English—according to Schwartzman, the first appearance in print of the word zero was in Philippi Calandri’s De Arithmetica Opusculum (1491). The word itself was borrowed from the Arabic صفر (sifr) which means “empty.” Interestingly, the word cipher has the same origin. One other point: in most European languages, zero is treated a plural (“I have zero apples” instead of “I have zero apple”). I’d be interested if this is the case in Arabic and other semitic languages.

Up next: the operations of arithmetic…

Internet Tendencies

While I prepare for my next full post, here’s a surprising and random thing that I discovered in my research today.

You may be familiar with McSweeney’s (, an online compendium of hilarious short essays, lists, and other creative forms of Internet writing. The part that I usually go to is McSweeney’s Internet Tendency.

Well, by random chance, I’m reading Tokaty’s A History and Philosophy of Fluid Mechanics, and I see an image of Hero of Alexandria’s “Reactive Motor” (link is here)—it’s the same as the McSweeney’s Internet Tendency icon (go check here).

A weird, and incredibly geeky, coincidence.