# Mnemonics for some properties in mathematics

I am looking for various mnemonics which help students to remember some important properties or theorems. Very often students confuse signs or relations such as $\leq$ and $\geq$ in some expressions. I wonder if there are some mnemonics that can be used to help the students not to confuse some particular expressions, theorems and the like. One example of a particular mnemonic which is concerned with the Min-Max-Inequality, is given in my answer below so that you can see what I am after.

In my opinion such mnemonics can help students a lot and make learning much easier.

• I think that there are two different kind of answers below: pure mnemonics, and ways to help remember that rely on the meaning or the reason for the fact to be remembered. I find that pure mnemonics are fine to remember pure conventions (e.g. the signs $<$, $>$), but that whenever possible one should favor link with meaning over pure mnemonics. One of the biggest challenge in teaching mathematics is to have student relate a formalism to a meaning, and pure mnemonics tend to enforce the idea that maths are just a bunch of senseless formulas to be remembered. This idea is enemy nb 1. Apr 28 '18 at 9:47
• @BenoîtKloeckner: In my experience, the answer is always "you need both". A careful explanation/proof, and separately a way to achieve automatic recall of the fact. There's a trap that only the former is required; similar to students who resist memorizing multiplication tables, expecting to rely only on repeated addition forever, and wind up arithmetically crippled as a result. Apr 28 '18 at 11:05
• I'd like to point out the worst possible answer (before someone posts it): "Ours is not to reason why, just invert and multiply." This is a mnemonic for how to divide fractions, but it succinctly encodes a lot of destructive thinking about mathematics. Not only does it imply that students should not learn why mathematics works, it is also a reference to the Charge of the Light Brigade! Apr 28 '18 at 13:17

Recently, a student in my beginning algebra course offered the following to the class, regarding signed number multiplication:

Assuming positivity is like love, and negativity is like hate, then...

• "If you love love, that's love." $\Rightarrow$ positive $\times$ positive = positive
• "If you love hate, that's hate." $\Rightarrow$ positive $\times$ negative = negative
• "If you hate love, that's hate." $\Rightarrow$ negative $\times$ positive = negative
• "But if you hate hate, that's love." $\Rightarrow$ negative $\times$ negative = positive

[Read this by treating the first instance of love or hate as a verb.]

• I heard a joke recently: A professor of linguistics was explaining that in some languages a double negative is a positive, while in others a double negative is just added negative emphasis. Then he observed that there was no culture in which a double positive was a negative, to which one of the students responded "Yeah, right." Apr 27 '18 at 15:32
• @StevenGubkin: I've heard the same joke in Dutch, with the similar response "Ja, ja."
– J W
Apr 30 '18 at 5:31
• @StevenGubkin: That actually happened! The linguist was J. L. Austin, and the guy who said "Yeah, right" was the philosopher Sidney Morgenbesser. en.wikiquote.org/wiki/Sidney_Morgenbesser Apr 30 '18 at 12:12

The minimax theorem states the following:

Let $X\subset \mathbb{R}^{n}$ and $Y\subset \mathbb {R} ^{m}$ be compact convex sets. If $f:X\times Y\rightarrow \mathbb {R}$ is a continuous function that is convex-concave, i.e.

$$f(\cdot ,y):X\rightarrow \mathbb {R} \text{ is convex for fixed } y, \text{and}$$ $$f(x,\cdot ):Y\rightarrow \mathbb {R} \text{ is concave for fixed } x.$$

Then we have that

$$\min _{x\in X}\max _{y\in Y}f(x,y)=\max _{y\in Y}\min _{x\in > X}f(x,y).$$

For arbitrary functions $f$ the equality does not hold in general. However, the Max-Min-Inequality is always satisfied.

For any function $f: Z \times W \to \mathbb{R}$ we have

$$\inf _{w\in W}\sup _{z\in Z}f(z,w) \geq \sup _{z\in Z}\inf_{w\in W}f(z,w) .$$

Since this property always holds for arbitrary functions $f$, it is well worth to be kept in mind by students. Naturally, many students will tend to confuse the order of the max and min operations as well as the direction of the inequality. In a lecture on nonlinear optimization our professor told us the following mnemonic to remember the property, which, as I think, makes it really easy to remember:

The shortest giant is at least as tall as the tallest dwarf.

Here, the shortest giant refers to the inf sup on the left hand side, which is at least as tall (i.e. $\geq$) as the tallest dwarf, which corresponds to sup inf.

I haven't forgotten the Max-Min-Inequality since I learned the above mnemonic which is why I posted the question whether you are aware of other such neat mnemonics which make students' learning easier.

• I love it! :-) Thanks! Apr 27 '18 at 13:55
• The way I usually remember that inequality is: $\lim\inf f \leq \lim\sup f$ clearly holds. The same inequality remains true if you change the first word: $\sup \inf f \leq \inf \sup f$. Apr 28 '18 at 8:56
• I don't really get why I should remember this one. Why should "shortest giant $\geq$ tallest dwarf" make more sense than "$shortest giant$\leq$tallest dwarf"? Is it more intuitive, or easier to remember? Apr 28 '18 at 8:58 I tell students to visualize$<$and$>$as mouths. They always want to eat the bigger number. • As long as the variable is always on the left, the < or > sign will point to which direction should be shaded on the number line. Apr 27 '18 at 17:18 • @DanielR.Collins It is hard to remember whether it is pointing to the first number or the second. The mouth analogy resolves this. Apr 29 '18 at 2:09 • The whole idea of it pointing at one of the numbers misses the point. More clarifying to translate to the phrase "is to the left of" or "is to the right of". Apr 29 '18 at 4:10 • @ruferd There is not always a variable, and when there is, it’s not always on the left. This mnemonic has served me since I was 8 or 9 years old, so IMHO it can’t be that bad. Apr 29 '18 at 14:49 • @NiloCK The inequality sign is not the mouth of either number. It is an independent animal which is deciding which number to eat. Apr 29 '18 at 19:51 Fatou's Lemma states: for nonnegative measurable functions$f_n$, $$\int_E \liminf_{n\to\infty} f_n\;d\mu \le \liminf_{n \to \infty}\int_E f_n\;d\mu$$ The mnemonic is $$\text{ILLLLLI},$$ meaning "the Integral of the Lower Limit is Less than the Lower Limit of the Integral". • This is precisely what I meant! Thank you very much! :) Apr 27 '18 at 10:57 • I am surprised that the sign here is a problem to remember. In my head, it descends from the property that$\int \min_{i\in S} f_i \leq \min_{i\in S} \int f_i$, which is clear to visualize: the lower hull of a family of functions has a smaller integral than any function of the family. Apr 28 '18 at 9:02 • It would take longer to decipher that mnemonic than it would take to check on a trivial example like$f_n = n \chi_{[0,1/n]}$. – user1362 Apr 29 '18 at 13:11 • This is easier for me to remember if I group the letters: IL-LL-LL-I. Apr 29 '18 at 14:48 • @ToddWilcox... I group them I-LL-L-LL-I of course Apr 29 '18 at 16:34 One I recently learned -- for the order of the signs in factoring a sum or difference of cubes, remember SOAP: Same sign, Opposite sign, Always a Plus. Sum of Cubes: $$x^3 + a^3 = (x + a)(x^2 - ax + a^2)$$ Difference of Cubes: $$x^3 - a^3 = (x - a)(x^2 + ax + a^2)$$ Credit: OpenStax College Algebra • I would rather retain the bulk of the formula, and explain how to choose the sign in a way that makes terms cancel out. Apr 28 '18 at 9:58 • Nice! Even with a degree in pure mathematics and many years of tutoring experience, I can never remember this one. Apr 29 '18 at 14:51 This only makes sense in Spanish but it's pretty fun. For integration by parts, $$\int u dv = u v - \int v du$$ Si un día vi una vaca menos sexy vestida de uniforme Which translates roughly to: I saw one day a not-so-sexy cow wearing a uniform. This mnemonics has generated awesome memes. Here is my way of memorizing the three main trigonometric functions. An angle$\theta$is in standard position locating a point$(x,y)$on a circle with a radius$r$centered at the origin. There is a convertible (a car with the roof removed) being driven down a road during the daytime. The sun (sine) is above (vertical$y$) the road ($r$):$\sin\theta=y/r$. The car (cosine*) is moving horizontally ($x$) over the road ($r$):$\cos\theta=x/r$. When the sun is above ($y$) the car ($x$), the driver gets a tan:$\tan\theta=y/x\$.

*In Filipino (my language), "car" is "kotse" which is pretty close to "cos."

• hahah funny stuff Apr 27 '18 at 13:57
• I find this quite heavier to remember than the picture of the trigonometric circle with cos sin tan and the main angle marked. Then converting this image to the ratios takes a bit of thinking, but I think it is the kind of thinking to encourage rather than try to avoid. Apr 28 '18 at 9:57

First heard it from a former classmate of mine, might be her own invention:

When the second derivative is positive, the function is happy (i.e., its graph looks like a smile). When the second derivative is negative, the function is sad (i.e., its graph looks like a frown).

added (I hope Federico doesn't mind ... Gerald Edgar)

Pictorially,
second derivative positve, second derivative negative:

Two plus signs signifies second derivative

First derivative may be added, if you somehow remember they guy faces to our left):

First derivative positive, first derivative negative :

• A similar one for concavity is concave up like a cup and concave down like a frown.
– J W
Apr 28 '18 at 9:12
• I think it is much more relevant to insist on the reason why this is so: when the second derivative is positive [negative], the derivative is increasing [decreasing]; this can be pictured mentally to distinguish between convex and concave. Apr 28 '18 at 9:49
• @GeraldEdgar Thanks, those pictures are great! Apr 28 '18 at 13:35

For the 4 quadrants of a Cartesian graph I say "All Students Take Calculus" counterclockwise (in order) to remember which trig fxns are positive in which quadrants.

• ? It’s “All strippers take cash”, we really want to avoid shaming high school students who don’t take calculus. And it’s a fact that most strippers don’t have a credit card reader handy. Apr 27 '18 at 19:46
• I think here visualizing the definition of trig functions is preferable to such a pure mnemonics. If these definitions are not understood and learned, then there is little point to know where each has which sign anyway. Apr 28 '18 at 9:54
• Ben, I tend to agree will you on this. I tell students that the unit circle reflects the standard Cartesian coordinates, and if by the time they are taking trig, they don’t know where X and Y ate positive and where negative, the mnemonic might not be enough to help. Apr 29 '18 at 0:06
• My high school calculus teacher also taught us All Suckers Take Chemistry. Apr 29 '18 at 16:53
• All Students Take Courses @JoeTaxpayer we don't have to take calculus May 2 '18 at 0:30

SOH-CAH-TOA

How to remember the ratios for the 3 main trig functions.

• xkcd.com/809 Apr 28 '18 at 12:21
• My maths teacher when I was in year 8 taught us the useful advice "Socks On Holiday Can Always Have Tons Of Advantages" ... and then mentioned he wasn't really thinking about "socks". Always stuck with me. Probably no longer permissible. Apr 28 '18 at 23:49
• We were taught the following for this: Silly old hags, chatter all hours, 'til old age Apr 29 '18 at 0:55

Last year I heard of

$$\text{Lo De Hi Mi Hi De Lo}$$

$$\text{(sing: "Low Dee High my High Dee Low!")}$$

as a mnemonic for the numerator in the quotient rule:

$$\left(\frac{f}{g}\right)' = \frac{g\cdot Df - f \cdot Dg}{g^2}$$

(of course Lo(w) = denominator, De = derivative, Hi(gh) = numerator, Mi = minus). Make sure you emphasize you have to divide the whole thing

$$\text{over LoLo}$$

though.

• I am not a huge fan, as I think that relating this formula to the formulas for a product and for an inverse carries more meaning. However in this case it might be needed to be quick to apply this, and a pure mnemonics is not out of the question. Apr 28 '18 at 9:50
• My version of this is "Low dee high minus high dee low, all over the square of whats below". Apr 29 '18 at 2:11
• Although I agree with @BenoîtKloeckner that students should be able to easily derive this using product rule and chain rule. They should also be able to reason about the order of the numerator by thinking about how small changes in the numerator or denominator alone would effect the ratio. Apr 29 '18 at 2:13

To remember concave Up (vs concave down), I remember that the U shape is concave up. Similarly, in conVex, the V is convex.

(If you like, v is the only letter in the word which is the graph of a function. Sadly, concave also has a v, but this mneumonic seems to work anyway. You just have to remember which word (convex) you've assigned the mneumonic to, which seems easier than remembering which word means which.)

In the 11th grade our math teacher taught us following:

was the maiden brave

the belly stays concave

but unprotected sex

makes the belly convex

• This seems like an almost unbelievably bad idea. Apr 30 '18 at 12:17
• @ChrisCunningham like i care. but i haven't mixed those two up since then May 1 '18 at 20:21
• also pointing out "dangers" to 17 years old, does not sound like a bad idea to me. anyway whatever May 1 '18 at 20:30