WyzGuys Tech Talk

ven though “x” is one of the least-used letters in the English alphabet, it appears throughout American culture, from Stan Lee’s X-Men superheroes to the television series “The X-Files.” The letter “x” often symbolizes something unknown, with an air of mystery that can be appealing. Just look at Elon Musk with SpaceX, Tesla’s Model X and now X as the new name of Twitter.

You might be most familiar with “x” from math class. Many algebra problems use “x” as a variable to stand in for an unknown quantity. But why is “x” the letter chosen for this role? When and where did this convention begin?

Math enthusiasts have proposed a few different explanations, with some citing translation and others pointing to a more typographic origin. Each theory has some merit, but historians of mathematics (like me) know it’s difficult to say for sure how “x” got its role in modern algebra.

Ancient unknowns

Today, algebra is a branch of math in which abstract symbols are manipulated, using arithmetic, to solve different kinds of equations. But many ancient societies had well-developed mathematical systems and knowledge with no symbolic notation.

All ancient algebra was rhetorical. Mathematical problems and solutions were completely written out in words as part of a little story, much like the word problems you might see today in elementary school.

The ancient Babylonians of Mesopotamia used many different words for unknowns in their algebraic system—typically words meaning length, width, area or volume, even if the problem itself wasn’t geometric in nature. One ancient problem involved two unknowns termed the “first silver thing” and the “second silver thing.”

Mathematical know-how developed somewhat independently in many lands and in many languages. Limitations in communication prevented any immediate standardization of notation. But over time, some abbreviations crept in.

In a transitional syncopated phase, authors used some symbolic notation, but algebraic ideas were still presented mainly rhetorically. Diophantus of Alexandria used a syncopated algebra in his great work Arithmetica. He called the unknown “arithmos” and used an archaic Greek letter similar to “s” for the unknown.

Indian mathematicians made additional algebraic discoveries and developed what are essentially the modern symbols for each of the decimal digits. One especially influential Indian mathematician was Brahmagupta, whose algebraic techniques could handle any quadratic equation. Brahmagupta’s name for the unknown variable was yavattavat. When additional variables were required, he instead used the initial syllable of color names, like “ka” from kalaka (black), “ni” from nilaka (blue) and so on.

Islamic scholars translated and preserved a great deal of both Greek and Indian scholarship that has contributed immensely to the world’s mathematical, scientific and technical knowledge. The most famous Islamic mathematician was al-Khowarizmi, whose foundational text, The Compendious Book on Calculation by Completion and Balancing, is at the root of the modern word “algebra.” (Its Arabic name is abbreviated as “al-Jabr.”)

So, what about “x”?

One theory of the genesis of “x” as the unknown in modern algebra points to these Islamic roots. The theory contends that the Arabic word used for the quantity being sought was al-shayun, meaning “something,” which was shortened to the symbol for its first “sh” sound. When Spanish scholars translated the Arabic mathematical treatises, they lacked a letter for the “sh” sound and instead chose the “k” sound. They represented this sound by the Greek letter “χ,” which later became the Latin “x.”

It’s not unusual for a mathematical expression to come about through convoluted translations. The trigonometric word “sine” started as a Hindu word for a half-chord but, through a series of translations, ended up coming from the Latin word sinus, meaning bay. Still, some evidence casts doubt upon the theory that using “x” as an unknown is an artifact of Spanish translation.

The Spanish alphabet includes the letter “x,” and early Catalan involved several pronunciations of it depending on context, including a pronunciation akin to the modern “sh.” Though the sound’s pronunciation changed over time, there are still vestiges of the “sh” sound for “x” in Portuguese, as well as in Mexican Spanish and native place names. By this reasoning, Spanish translators conceivably could have used “x” without needing to resort first to the Greek “χ” and then to the Latin “x”.

The Spanish alphabet includes the letter “x,” and early Catalan involved several pronunciations of it depending on context, including a pronunciation akin to the modern “sh.” Though the sound’s pronunciation changed over time, there are still vestiges of the “sh” sound for “x” in Portuguese, as well as in Mexican Spanish and native place names. By this reasoning, Spanish translators conceivably could have used “x” without needing to resort first to the Greek “χ” and then to the Latin “x”.

Moreover, although the letter “x” may have been used in mathematics during the Middle Ages sporadically, there is no consistent use of it dating back that far. Western mathematical texts over the next several centuries still used a variety of words, abbreviations and letters to represent the unknown.

For instance, a typical problem in the algebra book Sumario Compendioso, published by Juan Diez in Mexico in 1556, uses the word cosa, meaning “stuff” or “thing,” to stand in for the unknown.

The most plausible explanation is to credit the influential French scholar René Descartes for the modern use of “x.” In an appendix to his major 1637 treatise, Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences, Descartes introduced a version of analytic geometry, in which algebra is used to solve geometric problems. For unspecified constants, he chose the first few letters of the alphabet, and for variables, he chose the last letters in reverse order.

Scholars may never know for sure, but some theorize that Descartes chose to make the letter “x” appear often because his printer had a large cache of x’s in stock, as the letter is scarce in the French language. Whatever his reasons for choosing “x,” Descartes greatly influenced the development of mathematics, and his mathematical writings were widely circulated.

Xtending beyond algebra