# Why are X and Y commonly used as mathematical placeholders?

I realize that X and Y are relatively popular terms when wanting to use a placeholder for an unknown English or math term. What is the origin of this term, and why was it X and Y; why not the other letters?

• May I ask why this is off-topic or unclear? Dec 30 '18 at 18:14
• I was somewhat put off by the migration from ELU to here, but the word mathematical was finally added. However, after looking up placeholder it seems to only have two general definitions. It has the obvious math one that we all know. But there's also some (academic?) linguistic definition that I don't quite understand. Regardless, what I cannot find is what I thought it meant, which is what one might assume place holder means. IMO: a stand-in (which is about people; n/a) - tl;dr: I can't find a definition of placeholder in reference to a physical object, which is surprising. Dec 30 '18 at 23:23
• @Mazura Originally I posted it on ELU because most people were using X and Y in their questions. I was aware that it was used in math (obviously), but I wasn't aware of this S.E. By placeholder, I mean an unknown number/word. Dec 30 '18 at 23:26
• With the addition of mathematical I don't see how it could be anymore on-topic or anymore clear. And unfortunately, any less the question that I actually wanted answered: When was X first used as a substitute for a given object or person, with an unknown quantity or quality? Dec 30 '18 at 23:28
• Because a and b are busy being a) used elsewhere and b) to denote alternatives Dec 31 '18 at 10:27

They became popular because of René Descartes’ usage in his La Géométrie. The letters at the end of the alphabet are chosen as the variables, while those at the beginning are constants. There is speculation about why this might have been done. It is likely to allow the largest number of sequential letters without overlap between the two sets.

Why x became the most common is unknown. Some sources attempt to draw a line from the Arabic word for unknown through the Greek letter chi (which resembles a capital X), but the claims are unsubstantiated (Arabic being the source of our numerals and Greek being the common letter set for variables).

The link from mathematics to common speech is likely just a simple repurposing of known concepts.

• It seems the obvious choice as, X is what you make your mark with. But when's that from, cuneiform?
– Mazura
Dec 28 '18 at 0:32
• Oh possibly. As crossed lines are the simplest way to mark an exact point on a map, this was also likely used far before cuneiform, even. However, the usage of “x” in the place of an arbitrary or unknown didn’t rise before Descartes popularized it.
– Ian MacDonald
Dec 28 '18 at 0:46
• What is the origin of the phrase “Leave your Mark”? - Well, I think X already was the most common, 'variable' so to speak, but why he chose it is speculative. (Just how common it was to draw a picture instead of signing an X, IDK...)
– Mazura
Dec 28 '18 at 1:02
• Have you watched this Ted talk given by Terry Moore on why X is the unknown? ted.com/talks/terry_moore_why_is_x_the_unknown/…
– Equinox
Dec 28 '18 at 8:22
• I have read the transcript and the proposal made is weak at best.
– Ian MacDonald
Dec 28 '18 at 13:22

See Earliest uses of mathematical symbols, which quotes F. Cajori, A History of Mathematical Notations, 2 volumes (1928-29)

The use of z, y, x ... to represent unknowns is due to René Descartes, in his La géometrie (1637). Without comment, he introduces the use of the first letters of the alphabet to signify known quantities and the use of the last letters to signify unknown quantities.

• It's fascinating how much influence one person can have. I wonder what René would say today if he saw x as variable name splattered all over classrooms worldwide, journals, whiteboards, labs, computer code...
– Lightness Races in Orbit
Dec 28 '18 at 1:17
• @LightnessRacesinOrbit he'd likely be happy that people agreed on a consistent naming convention and flattered that it was his that they chose. Dec 28 '18 at 4:40

Here is the original source : René Descartes, La Géométrie (1637), I, page 299, for $$a,b$$ used to denote parameters.

And see I, page 301 for $$z$$ and I, page 303 for $$x,y$$ respectively, used to refer to an unknown quantity.

Letters was already used by François Viète (but the use of alphabetical variables to represent magnitudes is due to euclidean geometry).

See In artem analyticem isagoge (1591), Rule III :

Sunto duae magnitudines $$A$$ & $$B$$. [Let there be two magnitudes, $$A$$ and $$B$$.]

And also :

Oportet $$A \dfrac {\text { plano }}{B}$$ addere $$Z$$ [Suppose $$Z$$ is to be added to $$A^p / B$$].

But obviously the success of Descartes' "new geometry" explains the success of the new algebraic notation.