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Unless you have taught highschool algebra in Iran, you could not make sense of the phrase: Elephant and Teacup Identity! This is what teachers use to refer to the following identities:

$ (a+b)(a^2-ab+b^2)=a^3+b^3$ and $ (a-b)(a^2+ab+b^2)=a^3-b^3$

Such reference is so common that today a colleague of mine (in a discussion about students' algebraic difficulty) referred to it assuming that I know what she is referring to. Whether or not such references would be of any help to students is an important question, but not my question now. For this post, the question is:

Do you know any of these linguistic references for communicating mathematics? It could be something that you use in your own class, or you have heard that someone else uses. Thus, it doesn't matter whether its usage is limited to just one class, or is as popular as the one I gave.

Edit. The first attempt to clarify the question. The question is looking for "non-mathematical" terms or phrases that are used to refer to mathematical objects (of any kind) mainly for educational purposes.

Edit. The second attempt to clarify the question. Admittedly the question is a bit vague. Do examples like "continuity", "saddle point", "horseshoe map", or "hairy ball theorem" count? I guess not. They are now formal terms belonging to Mathematics culture here, there and everywhere. What if we call what this question is looking for "mathematical slang". Here is a dictionary definition of slang:

A type of language consisting of words and phrases that are regarded as very informal, are more common in speech than writing, and are typically restricted to a particular context or group of people.

Interesting, after coming with the term, I found this paper "The blight of mathematical slang", that gives the expression "cross-multiply" as an example.

Edit. Following a number of suggestions for using a more informative title (see comments below), I changed it in a way that also better reflects the final version of the question (previous edit).

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    $\begingroup$ For instance, the "Socks-Shoes Property" for $(AB)^{-1} = B^{-1}A^{-1}$? $\endgroup$ – J W Jan 7 '16 at 14:50
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    $\begingroup$ Are you looking for a collection of these terms, or just individual examples? E.g., does FOIL qualify? As to your Edit, I wonder whether that would be a more helpful title for the question? As you remark in the first sentence of your post, the title is not sensible to many. $\endgroup$ – Benjamin Dickman Jan 7 '16 at 19:52
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    $\begingroup$ Personally, I dislike and avoid these "cutesy" identities; proper mathematical names are more universal, descriptive, and extensible. For example, the given formulas are better described as the "sum of cubes" and "difference of cubes". Now we can build on these patterns: Is a "sum of 4th powers" factorable in real numbers? Is a "sum of 5th powers" so factorable? Etc. $\endgroup$ – Daniel R. Collins Jan 7 '16 at 21:00
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    $\begingroup$ Does "binomial" expansion count, since it tells you how to expand the power of a sum of two terms? But more seriously, I am curious to know the etymology of "elephant and teacup." Maybe I am missing a good joke, or maybe it is a cultural gap that I just won't get. $\endgroup$ – user52817 Jan 7 '16 at 21:10
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    $\begingroup$ @user52817 Elephant refers to the "bigger" parenthesis and Teacup to the "smaller" one. It is also called "fat and slim identity"! Both expressions are considered to be funny. Thus, you are right in both of your "maybes", you are missing a good joke because a cultural gap :) $\endgroup$ – Amir Asghari Jan 8 '16 at 9:57

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One of the most colorful names I have heard is the Chicken Mc Nugget theorem:

for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$.

link1, link2.

From the links:

The story goes that the Chicken McNugget Theorem got its name because in McDonalds, people bought Chicken McNuggets in 9 and 20 piece packages. Somebody wondered what the largest amount you could never buy was, assuming that you did not eat or take away any McNuggets. They found the answer to be 151 McNuggets, thus creating the Chicken McNugget Theorem.

The McNuggets version of the coin problem was introduced by Henri Picciotto, who included it in his algebra textbook co-authored with Anita Wah. Picciotto thought of the application in the 1980s while dining with his son at McDonald's, working the problem out on a napkin.

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  • $\begingroup$ That's usually called Sylvester's theorem, which is the base case of the Frobenius coin problem. The McNugget number is actually 43 and can't be directly computed using the theorem. (There is a small size of 6 nuggets as well) en.wikipedia.org/wiki/Coin_problem $\endgroup$ – Adam Aug 8 '16 at 14:13
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In Central Mexico, the expression \begin{equation} x_{\pm} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{equation} that solves quadratic equations of the form $ax^2 + bx + c = 0$ is called "fórmula del chicharronero" (formula of the chicharronero).

The chicharronero is the guy who sells salty snacks made of wheat (called chicharrones). Outside most schools there is always a chicharronero selling snacks to kids (here is a photo of a chicharronero; to the right of the picture there are chips; to the left the famous chicharrones).

enter image description here

It is said that the formula is so famous that even the chicharronero knows about it; thus the name.

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    $\begingroup$ Amazing example. By the way, do you generally write x (plus/minus) or you just wrote it that way? Also, shouldn't -b be +b in the equation? $\endgroup$ – Amir Asghari Jan 9 '16 at 1:20
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    $\begingroup$ In Uruguay it's called Bhaskara after the Indian mathematician who wrote about it. $\endgroup$ – ncr Jan 11 '16 at 6:12
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    $\begingroup$ When I was 16, my classmates put this formula on my birthday cake. They knew I was a math geek and this was the only formula they knew. $\endgroup$ – Amy B Jan 13 '16 at 13:42
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    $\begingroup$ In some parts of Germany, it's called the "Mitternachtsformel" ("midnight's formula"). One explanation for the name is that the formula is so important, that you need to know it even if your teacher wakes you up in the middle of the night. $\endgroup$ – Ingo Blechschmidt Feb 7 '16 at 14:48
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    $\begingroup$ We have both: the pork skins (chicharrón de cerdo) and this wheat thing (chicharron de harina) which according to Wikipedia are known as Duros in other places $\endgroup$ – Rodrigo Zepeda Mar 10 '17 at 0:27
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How about the shoelace formula for the area of an arbitrary simple polygon?


          enter image description here
          (Image from Wikipedia.)
The formula computes the area from the coordinates of the vertices, essentially by a cross product to compute (signed) areas of triangles.

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    $\begingroup$ Neat. I have derived this via Green's Theorem as an example in Calculus III in multiple semesters, but, I never had a name for it or this neat mnemonic to remember the pattern of the formula. Nice. $\endgroup$ – James S. Cook Jan 15 '16 at 7:19
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    $\begingroup$ How many kids nowadays have never seen shoelaces? $\endgroup$ – Gerald Edgar Oct 24 '16 at 12:56
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I often refer to the identities $(AB)^{-1} = B^{-1}A^{-1}$ or $(AB)^T = B^TA^T$ as the socks-shoes identity. I'm not sure how wide-spread this is, I certainly did not invent it and I'm pretty sure I've read at least one of these in at least one text.

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If you simplify a term by adding and subtracting something you call this a "nahrhafte Null" in German (probably translates to "nutritious null"?).

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    $\begingroup$ It's called "adding a well-chosen zero" in English. $\endgroup$ – Daniel McLaury Jan 11 '16 at 7:48
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    $\begingroup$ Not to be confused with a narrhafte Null. $\endgroup$ – s.harp Nov 4 '16 at 22:02
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My elementary students always wanted to know the name of the symbol shown here: enter image description here

We called it the division house as did many of my colleagues, but my students wanted a mathematical name. We therefore wrote to Dr. Math at Drexel. We were told there is no name and were referred to this paper.

I subsequently held a contest (on election day) and the winner was a sixth grade girl, who named it the "parenticulum" because it is a contraction of parentheses and vinculm which is what the symbol is name for. After the contest, we all called it the "parenticulum"

For more about the origin of our name, see the following from Dr. Math:

"You might be able to call the horizontal line in the division symbol a vinculum, but I don't think there is a name for the whole thing. In fact, in the following page about the history of symbols, it is not named, but drawn, and the alternate text in the HTML calls it "a close parenthesis attached to a vinculum"See Jeff Miller's "Earliest Uses of Symbols of Operation""

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    $\begingroup$ The left-hand part of the radical sign is different than this. √ compared to ) $\endgroup$ – Gerald Edgar Jan 7 '16 at 22:11
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    $\begingroup$ @SixWingedSeraph The division house is the name that most people I know in elementary ed use. No need to invent the divided-into-symbol. My students often would want to know how to know which number belonged in the house for a given word problem. Asking which number belongs inside the divided into symbol is more cumbersome. Furthermore there are many division symbols and this is the only one without a name. $\endgroup$ – Amy B Jan 8 '16 at 1:37
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    $\begingroup$ @SixWingedSeraph My students wanted a more mathematical name since they were taught great respect for math vocabulary. We created one which suited our purposes and was used in the classroom repeatedly. It doesn't matter that it was invented since it was really just for us. I shared it here to answer the question with a personal anecdote. $\endgroup$ – Amy B Jan 8 '16 at 1:40
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    $\begingroup$ I have heard this symbol called (slangily, but that seems appropriate for this thread) a "gozinta" -- as in "5 gozinta 15", a phonetic representation of what was one actually says when reading such expressions (i.e. "5 goes into 15"). $\endgroup$ – mweiss Jan 8 '16 at 15:20
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    $\begingroup$ I would like to mention that the division symbol above is not universal. I learned about it several years after my Ph.D. $\endgroup$ – Martin Argerami Jan 9 '16 at 12:12
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I've just remembered that "Donkey Theorem" is used to refer to triangle inequality in geometry textbooks in Iran. The name implies that even a donkey which is on one corner of a triangle chooses the straight path (rather than the broken one) to get to the other corner where there is some hay to eat.

I checked to see if it is used elsewhere and I learned from this MSE post that it is also used in Turkey.

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In Russian, the Squeeze Theorem (a.k.a. The Pinching Theorem) is called "Теорема о двух милиционерах" — "Two Policemen Theorem". The idea is that if two policemen are holding a criminal between them, the bad guy is going to the same place, probably jail or precinct, where the policemen are going.

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  • $\begingroup$ Thank you for sharing this fun interpretation of the Squeeze Theorem :) $\endgroup$ – Amir Asghari Oct 24 '16 at 8:28
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    $\begingroup$ In Soviet Russia limit squeeze you. $\endgroup$ – James S. Cook Oct 25 '16 at 0:03
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    $\begingroup$ Same in French and Italian: theorème des gendarmes, teorema dei carabinieri. And Spanish uses the equally colorful "teorema del sándwich". $\endgroup$ – Federico Poloni Mar 11 '18 at 19:41
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I first heard the term "stars and bars" a few years ago in Mathematics Stack Exchange. From Wikipedia:

In the context of combinatorial mathematics, stars and bars is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put $n$ indistinguishable balls into $k$ distinguishable bins.

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  • $\begingroup$ See the Wikipedia article I linked to for more information. $\endgroup$ – Joel Reyes Noche Jan 8 '16 at 23:08
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What is called "fórmula del chicharronero" in Central Mexico (see the answer by Rodrigo Zepeda) is called "Mitternachtsformel" ("midnight formula") in middle school in some parts of Germany. This is because, if someone wakes you up at midnight and asks what are the roots of a parabola you have to know this in a second.

After second thought, I think that the Mitternachtformel is $$ x_{1/2} = -\tfrac{p}2\pm\sqrt{\tfrac{p^2}{4}-q} $$ for the roots of $$ x^2+px+q=0. $$

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  • $\begingroup$ Is it your personal way to write $x_{1/2}$ or a common way in those parts of Germany? $\endgroup$ – Amir Asghari Jan 11 '16 at 17:13
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    $\begingroup$ Writing $x_{1/2}$ is common in middle school in Germany (at least it was when I was in school). $\endgroup$ – Dirk Jan 11 '16 at 17:57
  • $\begingroup$ When I went to school in Germany (in the 70s), we wrote $x_{1,2}$ instead. $\endgroup$ – Frunobulax Feb 8 '16 at 13:19
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I was about to write a comment to @Amy B division house, saying that in Iran we use enter image description here

to denote $a$ divided by $b$, that a colleague entered the room asking me what I am doing. I explained and she told me that in their primary school (somewhere in Leicestershire in England) they called it (the symbol drawn by Amy B) "bus stop" where the bigger number needs to be covered and the smaller number remains outside! I thought it is worth mentioning as a separate answer.

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Reading a paper from an English student in England I came across this sentence:

To expand brackets in Algebra, they were taught the "crab claw" method, a method that I was used to.

Since I had personally never heard of the term "crab claw" in Algebra, I thought it would be beneficial to add it here. enter image description here

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The identities $(a + b) (a - b) = a^2 - b^2$ and $(a + b)^2 = a^2 + 2 a b + b^2$ (and sometimes $(x - a) (x - b) = x^2 - (a + b) x + a b$) are called "productos notables" (notable products) in Spanish. This is sometimes extended to the products mentioned in the question, and even higher order ones.

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    $\begingroup$ In Dutch, they are known as "merkwaardige producten" (also notable/noteworthy products). $\endgroup$ – J W Jan 8 '16 at 14:57
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    $\begingroup$ This would be the category of "special products" for most English texts (which includes the given "difference of squares" and "binomial square", as well as the OP's "difference of cubes" and "sum of cubes"). I don't think that's really "slang", as it's properly-descriptive, and used widely in the literature. $\endgroup$ – Daniel R. Collins Jan 8 '16 at 17:42
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    $\begingroup$ @vonbrand: you might want to add a geographical restriction to your statement. I did all my schooling (from kindergarten to PhD) in Spanish and I never heard that expression. $\endgroup$ – Martin Argerami Jan 9 '16 at 12:15
  • $\begingroup$ In Russian they are called "Формулы сокращенного умножения" -- "formulas for reduced/condensed multiplication" (not sure which is a better translation). $\endgroup$ – zipirovich Mar 10 '17 at 3:10
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My school talked about sausages and cocktail-sticks, meaning questions where you have to find the formula for the $n$th term in the sequence. I never worked out what the point was.

enter image description here

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  • $\begingroup$ I guess it somehow refers to using natural numbers (cocktail sticks) one to one to pick the numbers (sausages) of the second set. Funny :) $\endgroup$ – Amir Asghari Jan 8 '16 at 15:26
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    $\begingroup$ The sausages are the two lists of numbers in ovals, the cocktail stick is the line between, that somehow represents the rule for getting from one to the other. But I have no idea why drawing it that way helps. ... Actually, looking at it now, I wonder whether it's a corruption of a standard function picture, with an arrow from the domain to the range. $\endgroup$ – Jessica B Jan 8 '16 at 16:20
  • $\begingroup$ Aha! Silly me. That is why you included the picture in your answer $\endgroup$ – Amir Asghari Jan 8 '16 at 17:13
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The phrase chain rule in Calculus is a mathematical slang.

The rule goes like this: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x))h'(x)$.

When I learned the rule at school, my teachers just called it "the function of a function rule", which describes precisely the situation in which you use it. When I got to university in a different state, they kept mentioning this "chain rule" and I had not the slightest clue what they were talking about.

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    $\begingroup$ I do not agree that this is slang. It may have German roots: Composition of two functions is called "Verkettung" in German, hence "Kettenregel" for the derivative of a "Verkettung" and this translates to "chain rule". $\endgroup$ – Dirk Jan 12 '16 at 8:22
  • $\begingroup$ But to an English-speaking student with no knowledge of German, it is completely unrelated to the rule it describes, so it might as well be slang. $\endgroup$ – DavidButlerUofA Jan 12 '16 at 9:06
  • $\begingroup$ Since the English Wikipedia does not mention any other name for the chain rule I guess that this is the common name (I see the same from my limited knowledge of English textbooks). On a different matter: Would "Nullstellensatz" count as slang? $\endgroup$ – Dirk Jan 12 '16 at 9:42
  • $\begingroup$ Hmm. My interpretation of the question was that it was seeking terminology that doesn't seem to relate to the thing it names. So I thought my answer fit. Rereading the definition of slang given in the question, maybe my answer doesn't fit after all. $\endgroup$ – DavidButlerUofA Jan 12 '16 at 13:25
  • $\begingroup$ Yeah, it seems quite hard to pin down what "slang" is… Anyway, interesting to see that "chain rule" does appear strange without some background in German! $\endgroup$ – Dirk Jan 12 '16 at 14:09
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In Uruguay (at least) famous formulas are often referred to using the name of a person who has been associated to their initial discovery. For instance (see my comment above), the quadratic formula is called Bhaskara and the corollary to the first part of the Fundamental Theorem of Calculus $\int_a^b f(t) dt = F(b)-F(a)$ is called Barrow. Mathematicians there say "by Barrow" or "by Bhaskara" which I find much more interesting than "by the Corollary to the first part of the Fundamental Theorem of Calculus" or "by the quadratic formula". It feels more human to me.

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  • $\begingroup$ While an interesting fact, this does not really sound like slang but like normal mathematical language. $\endgroup$ – Dirk Jan 11 '16 at 8:28
  • $\begingroup$ As an Iranian, I certainly wish you used "by al-Khwārizmī" instead of "by Bhaskara" :) $\endgroup$ – Amir Asghari Jan 11 '16 at 17:18

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