Being in quarantine had me wondering about math-y words starting with Q. Here is my quest to find out their meaning and history.

**Quarantine** refers to a forty-day period in which ships were required to stay isolated before passengers and crew could go ashore during the Black Death epidemic in the 1300s. The word is related to the Venetian/Italian words *quarantena* or *quarantino,* meaning “forty days”, derived from the Italian word *quaranta*, and similar to the words for 40 in French and other languages.

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Let’s next look at the number 4, and the Latin *quadri-* gives us **quadrilateral** (4-sided figure), **quadrangle** (4-angled figure), **quadrillion** (4^{th} power of 1 million, or 10^{24}, see below), and **quadruple** (to multiply by four). Notice that quadratic is missing from this list; more on that below.

**10**^{24} = 1,000,000,000,000,000,000,000,000

^{24}= 1,000,000,000,000,000,000,000,000

A **quadrangle** is a plane figure in which segments connect 4 non-collinear points, and it has some interesting math properties to explore. If the points are connected in cyclical order, a convex or concave quadrilateral is the result, otherwise the figure is called a crossed, non-simple, butterfly, or bow-tie quadrilateral (and many high school geometry texts do not consider this to be a quadrilateral at all).

If we construct the 6 lines connecting the 4 points in all possible pairs, we create a “complete quadrangle”. The 3 extra points of intersection (that are not vertices) are called diagonal points. The midpoints of the sides, along with the 3 diagonal points, all lie on a conic called the **Nine-point conic**.¹

Check out this **GeoGebra visualization**; the nine-point conic seems to be an ellipse when the quadrangle ABCD is concave and a hyperbola when the quadrangle ABCD is convex or a non-quadrilateral “bow-tie” shape. What else do you notice?

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A related Latin root is *quartus* or fourth. Taking this to mean one-fourth (¼) gives us **quarter**, **quartile**, and **quart**, whereas a fourth degree polynomial is a **quartic** function. Similarly, the Latin *quintus* means fifth, and yields the words **quintic**, **quintile**, **quintillion**, and **quintuple**.

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So, what about **quadratic**, which feels like it should have to do with four, but instead is a polynomial of degree two? It comes from the underlying Latin word *quadratum* which means “square”. The Greeks and Romans understood the abstract quantity x^{2} as a square with side x. That’s why something raised to the second power is said to be squared or quadratic. The related word **quadrant**, from *quadrare* (“to make square”) is one of the four “square” regions of the Cartesian plane. And graph paper is sometimes called **quadrille** ruled, based on the French word for “small square”.

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Two last Q words relating to math are **quantity** (from the Latin *quantus* meaning “how much” or “how great”) and **quotient** (Latin *quotiens* = how often, how many times). So the quotient is the quantity that tells how many times one number fits into another number.

**Q.E.D.**²

**Notes & Resources:**

I’ve had the idea for this post rattling around in my brain for more than forty days, and thankfully, Ed Southall’s (@edsouthall) presentation for MathsConf23 **“****From Abacus to Zero: The Etymology of the Words of Mathematics****“** has spurred me on to write it. The full virtual conference recordings are at **this link**.

- Ed helpfully suggested a few books that detail the meaning of mathematical words. Much of my information is from
**The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English**by Steven Schwartzman (1994), The Mathematical Association of America. - The ebook is available
**here**[on 50% sale through the summer!]

¹ This image of the Nine-Point Conic is from Weisstein, Eric W., “Nine-Point Conic.” From *MathWorld*–A Wolfram Web Resource. **https://mathworld.wolfram.com/Nine-PointConic.html**. More about the Nine-Point Conic and Complete Quadrangles can be found on Wikipedia **here** and **here**.

Note that a Complete Quadrilateral (right below) is a different figure (and is the dual of the complete quadrangle, left below); read more about this at Cut The Knot **here** and Wolfram MathWorld **here**.

² Latin *quod erat demonstrandum* “which was to be shown”. Typically used at the end of a proof to show that the proposition in question had been proved.