# Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples particularly enlightening, and would love to have a list of them. To make the question well-defined, my (arbitrary) cutoff for "elementary" is USA K-12. By "sophisticated" I mean 20th-21st century research mathematics, and the applications should be nontrivial enough to have appeared in research-level publications.

I list two examples I know.

• The carrying operation in base 10 addition involves the computation of a group 2-cocycle, as explained in this MO answer and in this ncatlab page following (Daniel C. Isaksen, A cohomological viewpoint on elementary school arithmetic, Amer. Math. Monthly (109), no. 9 (2002), p. 796--805).

• The passage from the poset of non-negative integers to the monoid of their differences (i.e. working with expressions of the form $x-y$ where $x$ and $y$ are numbers) is the forgetful functor from a comma category $0/\mathcal{C}$ to $\mathcal{C}$ where $\mathcal{C}$ is a monoid with unique object $0$, as described by Lawvere in Section 4 of Taking Categories Seriously.

Question: What are other examples of sophisticated mathematics elucidating elementary mathematical ideas and concepts?

Edit: One example per answer, please. Also, if you have examples which are not research-paper content but are close then that's still interesting for me, but less so as it gets further from research level. The most interesting examples for me are those with citations to research papers, as in the listed examples.

• Do you want us to give separate examples in separate answers? – Qiaochu Yuan Dec 7 '15 at 8:59
• I just feel like your examples are fine without the references, and that the references don't add much. Your first example is just making explicit the observation that $\mathbb{Z}_{100}$ is a nontrivial extension of $\mathbb{Z}_{10}$ by $\mathbb{Z}_{10}$. Anyone with a decent understanding of group cohomology and/or homological algebra could've made it. – Qiaochu Yuan Dec 7 '15 at 9:37
• Could you exand on "USA K-12" please? Does this mean the last class in high school? What do you teach there? Please excuse my ignorance. – Sebastian Goette Dec 7 '15 at 19:20
• I edited K-12 to elementary in the title since K-12 is not universal, but US inner language. The OP claims that elementary is subjective which is possible, but anyway the title is supposed to give a vague information, and K12 is not vague information, but empty information to any user not acquainted with US gibberish. – YCor Dec 7 '15 at 22:12
• "school mathematics" sounds fine! – YCor Dec 8 '15 at 11:19

The angle addition formula $\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \tan(\beta)}$ for tangent gives one of the simplest nontrivial examples of a formal group law, namely $F(x, y) = \frac{x + y}{1 - xy}$. A variation corresponding to the hyperbolic tangent governs the addition of velocities in special relativity, and a further variation is related, via the dictionary between formal group laws and genera, to the Hirzebruch $\chi_y$ genus. See, for example, this paper.

• Interesting example. The formal group law you describe even defines an algebraic group, a torus splitting over $\mathbb{Q}(i)$. – Julian Rosen Dec 7 '15 at 20:31

I wouldn't really call this sophisticated, but maybe not so far from the 20th century.

There are several half-angle formulas for tangent, among them: $$\tan \frac{x}{2}=\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x}=\sqrt{\frac{1-\cos x}{1+\cos x}}.$$ The first two fractions look essentially different, but both appear to be reduced in the sense of having no common factor in the numerator and denominator. If we square both fractions, they simplify the same way, and this gives the third form.

An explanation for this is that the ring $\mathbb{R}[\sin x,\cos x]$ is a Dedekind domain of class number $2$, and $(\sin x,1+\cos x)$ is a non-principal ideal. This means $\sin x/(1+\cos x)$ cannot be written as a ratio of comaximal elements, but its square can.

If we work over the complex numbers instead, $\mathbb{C}[\sin x,\cos x]$ is a PID, so we can write the tangent half-angle formula in "lowest terms": $$\tan \frac{x}{2}=-i\frac{\cos x+i\sin x-1}{\cos x+i\sin x+1}.$$

• And by the Serre-Swan theorem it is somehow related to the fact that there are two line bundles over the circle: the trivial one and the Mobius strip. – Fan Zheng Dec 10 '15 at 21:16
• +1: I mention this in a research paper alpha.math.uga.edu/~pete/ellipticded.pdf as a consequence of a theorem of...Rosen. I like the comment about line bundles on the circle too (and think that I've written about it somewhere, but I can't lay hands on that right now). – Pete L. Clark Dec 11 '15 at 14:26

It's common in calculus classes and textbooks to state that the antiderivative of $\frac{1}{x}$ is $\log |x| + C$, where $C$ is a constant. This is incorrect: $C$ need only be a locally constant function on $\mathbb{R} \setminus \{ 0 \}$ (so it can take different values on the negative and positive reals). This reflects the fact that $\mathbb{R} \setminus \{ 0 \}$ has nontrivial $H^0$, which is being detected via de Rham cohomology.

For a reference I only have this nCafe post.

• Thanks! I've used this example in teaching and I like it a lot (although I haven't taught the de-Rham cohomology connection). It seems a bit borderline though, because integration is borderline K-12 and de-Rham cohomology is borderline sophisticated, I think. – Daniel Moskovich Dec 7 '15 at 10:12
• Yeah, I thought it would be borderline, but I quite like it too. – Qiaochu Yuan Dec 7 '15 at 10:13
• A riddle for freshman calculus classes: if $f(x)=\arctan(x)+\arctan(1/x)$, compute $f(1)$, $f(-1)$ and $f'(x)$, and explain. – Pietro Majer Dec 7 '15 at 10:59
• @PietroMajer On the surface this seems to be a relatively straightforward application of the chain rule — what is the insight/connection I am failing to make? – 1110101001 Dec 9 '15 at 6:26
• @1110101001 If you straightforwardly take the derivative, you will see that it always equals $0$, so logically it should be constant. But upon computing values, you can see that it hits more than one different value – ASKASK Dec 9 '15 at 6:30

Wilkie's solution to the so-called Tarski high-school algebra problem shows that not all identities involving addition, multiplication, and exponentiation that are true for all positive integers are provable from the usual rules given in high school.

Linderholm's book, Mathematics Made Difficult, is full of this sort of thing. These samples might give you the idea:

1. Proposition. If you can add, you can count.

Proof. In counting with the additive monoid $N_0$, we start at $0$; after saying $n$, we say $n+1$. Thus, we already have a counting system. $$n\mapsto n+1\qquad\{0\}\hookrightarrow N_0\rightarrow N_0.$$ But having a counting system is not enough. What you must have in order to assure success in all your counting endeavours is a real, true, initial counting system. So let $$\{x\}\hookrightarrow X\rightarrow^{\!\!\!\!\!\!f}X$$ be any counting system. The set of all functions $$X\rightarrow X$$ is easily verified to be a monoid $\cal X$. Hence there is a unique monoid homomorphism $$N_0\rightarrow{\cal X}$$ sending $$1\mapsto f;$$ which is written $$n\mapsto f^n.$$ Now it is possible to define a mapping $$N_0\rightarrow X$$ by writing $$n\mapsto f^n(x)$$

1. I assert that every number other than $1$ and $-1$ has indeed got a prime factor. Since the number $n$ in question is not a unit, the set of its multiples $${\frak a}=\{xn:x\in z\}$$ is not all of $Z$. Consider the class $\cal I$ of all proper ideals of $Z$ containing $\frak a$ as a subset; the set-inclusion relation $\subset$ makes $\cal I$ a partially ordered set. Now consider any subclass of $\cal I$ with the property that if $\frak b$, ${\frak c}\in{\cal C}$ then either ${\frak b}\subset{\frak c}$ or ${\frak c}\subset{\frak b}$; the union of $\cal C$ is trivially a proper ideal of $Z$ containing as a subset every ideal of $\cal C$ and also containing $\frak a$ as a subset. By Zorn's Lemma, a proper ideal $\frak m$ of $Z$ exists that is maximal with respect to the property of being a proper ideal of $Z$ containing $\frak a$ as a subset. Hence $\frak m$ is maximal with respect to the property of being a proper ideal; and hence is a prime ideal.

Now, in the ring $Z$ every ideal has a generator. The generator of a prime ideal is prime; since $n$ is in this ideal, we are done.

• Do you have a good example from there? – Daniel Moskovich Dec 7 '15 at 10:08
• I'll get back to you when I'm in my office and have the book to hand. – Gerry Myerson Dec 7 '15 at 11:52
• Natural number objects are no joke. – Steven Gubkin Dec 8 '15 at 4:09

I like scissors congruences. Dehn found an interesting necessary condition for two three-dimensional polyhedra to be scissors congruent. The notion of a tensor product gives a nice conceptual explanation. It shows that Dehn's invariant indeed takes values in $\mathbb R\otimes_{\mathbb Z}(\mathbb R/\mathbb Z)$. To see that two polyhedra are scissors congruent if their volumes and Dehn invariants agree, one needs methods from group cohomology.

• Note that Dehn invented it a long time before abstract tensor products were defined! – Allen Knutson Dec 12 '15 at 17:45
• Thanks for the hint - so it fits even better into this thread. – Sebastian Goette Dec 12 '15 at 19:34

Another categorical example: the laws of arithmetic, as applied to the arithmetic of finite integers, are ultimately explicable by the fact that the category of finite sets $\mathbf{Fin}$ is a cartesian closed category with coproducts.

• $a + b = b + a$: the canonical symmetry isomorphism $A + B \cong B + A$ follows directly from the universal property of coproducts. Similarly for $a b = b a$, from the universal property of products. And similarly for the associativity laws: they follow from universal properties.

• $a(b + c) = a b + a c$: the functor $A \times (-): \mathbf{Fin} \to \mathbf{Fin}$, being left adjoint to the functor $A^{(-)}: \mathbf{Fin} \to \mathbf{Fin}$, preserves any colimits which exist and in particular finite coproducts (sums).

• $(ab)^c = a^c \times b^c$: the exponential functor $(-)^C: \mathbf{Fin} \to \mathbf{Fin}$, being right adjoint to $C \times (-)$, preserves any limits which exist and in particular finite products.

• $a^{b + c} = a^b \times a^c$: the functor $A^{(-)}: \mathbf{Fin}^{op} \to \mathbf{Fin}$ is right adjoint to its opposite $(A^{(-)})^{op}: \mathbf{Fin} \to \mathbf{Fin}^{op}$ (this in turn follows from the symmetry of products), and so it takes products in $\mathbf{Fin}^{op}$ (which are coproducts in $\mathbf{Fin}$) to products in $\mathbf{Fin}$.

• $(a^b)^c = a^{b c}$: the fact that $(A^B)^C$ satisfies the same universal property as $A^{ B \times C}$ may be proven as a consequence of the associativity of products.

All of these arithmetic laws apply also to logical operations where $+$ is interpreted as $\vee$, $\times$ as $\wedge$, and $a^b$ as $b \Rightarrow a$. They can be explained in exactly the same way, by the fact that the truth values form a cartesian closed category with coproducts, $\mathbf{2}$.

Many such questions are dealt with in Klein's book "Elementary mathematics from an advanced standpoint" originally published in German in 1908 and translated into English in the 1920s as I recall. There have been many editions of this since.

To respond to Daniel's comment, one example from Klein's book that immediately comes to mind is that of small oscillations of the pendulum. The "elementary" treatment of this is in terms of circular motion and a superposition principle (and that's the way I was taught small oscillations in a physics course in high school). Klein recommends treating this via calculus and (linear) differential equations. That's a nice simple illustration of the power of general methods (ODE) over ad hoc "elementary" solutions (involving a somewhat mysterious superposition principle).

• Thanks! Do you have a good example from there? – Daniel Moskovich Dec 11 '15 at 5:28
• We were taught it in terms of linear ODEs in the maths and physics curricula followed at my school. – Peter Taylor Dec 11 '15 at 23:33

This might not be exactly what you're looking for, but it has always struck me that the irrationality of $\pi$ is one of the few mathematical facts that is taught very early on, yet one can easily graduate with an advanced degree in mathematics without having the slightest idea why it's true.

• I guess that's a consequence of teaching calculus with Stewart instead of Spivak ;) – Martin Argerami Dec 9 '15 at 10:14

Not sure if these would be directly relevant, but here's what I've found helps make them feel really "happy" about math by showing them the advanced concepts lurking underneath (or using advanced approaches to solve elementary problems). Although you did suggest, one example per answer, I hope this is not too much.

1. Solving simultaneous equations: There are commonly 2 ways of solving them - purely algebraic and geometric. The latter being the "intersection" of 2 lines in the cartesian space. However, an uncommon "3rd" way is to convert the equations into vectors (i.e., matrices) and solve them in vector space (uncommon to be taught at K-12) They're still geometric in terms of visualization, but radically different in how we approach the problems. Besides, the 'vector' way can easily scale up to manual computations by hand, but the algebraic solution, not so much. This tends to fascinate some of them and opens their minds to advanced linear algebra and its applications and teaches them about 'vectors' along the way.

2. Introductions to "lines": This is more applicable problems of the type: If 3 apples cost X then how many apples can be bought for Y - these problems can be twisted in various ways, confusing many students. This can be easily converted to a geometrical problem, with "apples" on one axis and "cost" on the other and "lines" can be plotted to compute the correct answer. No matter how the wordings are twisted, this has a higher probability of being solved correctly. I use this to help them understand "equations" of a line $y = mx + c$ and elucidate how useful knowing about "lines" is and introduce them to the notion of "linearity" and its importance to math.

3. The connection between integration and numerical methods: This is not initially obvious given that most calculus questions asked in K-12 can be easily solved using pen/paper. However, if one were to "code them up" using a computer, you'd realize, it's not as easy as you thought. Computers are discrete machines and thus have difficulty in computing in the continuous space as "assumed" when solving questions on pen/paper. Showing the reason and necessity of using numerical methods tends to open up their minds to why one should learn such topics. The entire field of computer graphics, pretty much relies on this.

4. Introduction to computational complexity: multiply two $n$-digit numbers - the classic grade school algorithm, will have $n^2$ steps. Can this be done any faster? In most cases, no one has really thought about this, but asking this question can open up their mind to the notion of "computability" and what it means to "compute" showing them how math and computing are rather intertwined.

5. Introduction to NP-completeness: Given a boolean 3-SAT formula, can we find an assignment of truth values that satisfies the formula? Given that we know from the above multiplication example, we can solve it in a finite number of steps. However, this satisfiability problem is a different beast and opens a new can of worms. One can then begin to understand that there are various "classes" of computable problems that we may not have thought about.

There are several examples related to high-school geometry.

The Tarski–Seidenberg theorem essentially shows that (much of) elementary geometry is decidable. If you want synthetic proofs, then he area method is a sophisticated modern technique for finding them.

Adventitious angles were not classified until the 20th century.

This might not be "sophisticated" enough but there is an approach to the butterfly theorem using projective geometry.

I'm surprised no one has mentioned Doyle and Conway's Division by Three yet. Though maybe it doesn't count as sophisticated...

• Thanks for the nice answer! Rather than not being sophisticated though, I'm not sure it's school material to prove bijections between sets, although it is surely elementary. – Daniel Moskovich Dec 8 '15 at 6:21

I would say that behind homology/cohomology business there is an elementary intuitive idea that you cannot for example speak about the lenght of three dimensional shapes (in other words: the boundary of the boundary of such shape is empty). Another example which striked me also comes from homology and cohomology: the fundamental theorem of calculus is the simplest instance of more general Stokes theorem and the Stokes theorem can be reinterpreted in cohomological terms: one can pair forms (cycles in de Rham cohomology) together with chains (cycles in singular homology) and we have a formula $\langle \omega, \partial c \rangle =\langle d \omega, c \rangle$ where $d$ is exterior differential and $\partial$ is a boundary operator in homology.

• Thanks for the answer! This answers a different question though, as these are not topics taught in primary or secondary school. – Daniel Moskovich Dec 8 '15 at 11:26
• Well, the fact that a solid has volume and not length is taught in primary school, and the FTC is taught in secondary school. – Daniel McLaury Dec 8 '15 at 17:04
• "you cannot for example speak about the lenght of three dimensional shapes." Oh? What about Steve Schanuel's paper, What is the length of a potato? maths.ed.ac.uk/~tl/docs/Schanuel_Length_of_potato.pdf – Gerry Myerson May 23 '17 at 12:24

Category Theory may furnish an insightful point of view on elementary mathematics. Why the product of topological spaces is constructed by topologizing the cartesian product of the underlying sets? Because the forgetful functor from Top to Set is a right adjoint. Mc Lane's Category for the working mathematician has several beautiful examples on these lines.

• ... I apologise for this. By "elementary" I meant "grade school"; this was obscured by an edit. So I was specifically ruling out this type of example, which I personally find less interesting (and of course is common). Thanks for the answer! – Daniel Moskovich Dec 7 '15 at 11:21
• Ops, sorry… To me "elementary" means "concerning principles", "foundational", like in Euclid's "Elements". – Pietro Majer Dec 7 '15 at 13:05
• I know! This is the first interpretation that comes to mind if I don't word my question precisely. Sorry for the confusion! – Daniel Moskovich Dec 7 '15 at 14:31

The elementary school definition of "prime numbers" is really the definition of "irreducible element of a ring" but they are equivalent because $\mathbb{Z}$ is a PID.

The infinitude of prime numbers is a finicky example, as there's one proof that can be explained to elementary school students, one that can be explained to college students, and the rest are hard.

• I'm not sure I agree with this one... In elementary school, the meaning of "prime number" is that it is divisible only by itself and by 1. The fact that we may abstract its properties in various different ways isn't really a sophisticated treatment of this. – Daniel Moskovich Dec 8 '15 at 8:24