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That which we call an identity

I’m grateful to Jemma Sherwood and Rob Low for reading an early draft of this and for their comments thereon. All opinions are, of course, my own.

This post is inspired by something that I see crop up now and again in discussions with other Maths teachers. It usually manifests itself as a rallying cry to use ≡ in place of = in identities and reserve = for equations. My standard response is to mutter something about identities being equations and leave it at that. But in the latest round, Jemma Sherwood challenged me, in the nicest possible way, to explain a bit further. This is that explanation.

Although I’m going to state my case here, I’m well aware that there are different opinions. In matters of opinion, such as this, agreement and disagreement is less important than that all sides think. So if what I write seems to you wrong, that’s fine so long as it makes you think about why you think that it is wrong.

I’m actually going to give two answers to the question “Should we use ≡ for identities?”. Both are “No”, but for different reasons:

  1. No, because it is trying to solve the wrong problem.
  2. No, because, in the words of Inigo Montoya: “You keep using that word. I don’t think it means what you think it means.”

The second answer is the one that I usually mutter about when I come across this idea of using ≡ but it’s the first that is the important one.

In preparation for writing this I posted a poll on Twitter with four mathematical statements and asked which of them were identities. The four statements were:

  1. $\sin (180 n) = 0$,
  2. $a^{2} + b^{2} = c^{2}$,
  3. $ (x + y)^{2} = x^{2} + 2 x y + y^{2}$,
  4. if $2 x + 6 = 10$ then $x = 2$.

You may wish to ponder what your answer would be before continuing.

For Some Values of True

From the discussion that ensues whenever anyone posts about ≡, the rationale for insisting on it would seem to be that students find it difficult to distinguish between identities and equations so using notation to clarify the difference would be a good idea.

Seems reasonable. But to my mind, it’s trying to solve the wrong problem.

In the comments around my twitter poll, someone linked to the Wikipedia entry on Mathematical Identity which starts (emphasis mine):

In mathematics an identity is an equality relation $A = B$, such that $A$ and $B$ contain some variables and $A$ and $B$ produce the same value as each other regardless of what values (usually numbers) are substituted for the variables.

Another person gave a similar criterion for an identity which involved, as I understood it, putting “$\forall x$” at the start (or whatever unbound variables existed in the expressions).

The poll wasn’t long published before someone made a comment that slightly let the cat out of the bag. They queried the $\sin (180 n) = 0$ and said that it would be okay if $n$ was an integer but that I hadn’t made that clear. (Actually, they also queried the fact that I’d written $180$ rather than $180^{\circ }$; I must confess that one was due to me not being bothered to hunt down a unicode degree symbol but it really just underlines my point.) After that, some others remarked that they wanted to change their vote as they hadn’t noticed that.

So just putting $\forall x$ or $\forall n$ in front of an expression and seeing if it is still true isn’t a valid test of anything. We have to provide a context for the variables, and that allows me the freedom to make any of my equations into an identity or not.

  1. $\sin (180^{\circ }n) = 0$ is an identity with $\forall n \in \mathbb{N}$ but not with $\forall n \in \mathbb{R}$.
  2. $a^{2} + b^{2} = c^{2}$ is an identity with “$\forall a,b,c \in \mathbb{R}$ where $a$, $b$, $c$ are the sides of a right-angled triangle with $c$ the hypotenuse”, but is not an identity with just $\forall a,b,c \in \mathbb{R}$.
  3. $(x + y)^{2} = x^{2} + 2 x y + y^{2}$ is an identity with $\forall x,y \in \mathbb{R}$, but is not an identity with $\forall x, y \in M_{2}(\mathbb{R})$, the space of $2 \times 2$–matrices.
  4. “If $2 x + 6 = 10$ then $x = 2$” might surprise you: it is actually an identity with $\forall x \in \mathbb{R}$ since it then asserts that for any real number $x$, if $x$ satisfies $2 x + 6 = 10$ then $x = 2$. However, it is not an identity in $\mathbb{Z}/12\mathbb{Z}$ since both $2$ and $8$ satisfy $2 x + 6 = 10$.

To be a valid mathematical sentence, an identity requires a context. My contention is that the real problem behind the equation vs identity debate is that students are filling in the missing context for themselves and often getting it wrong. And once the context is made explicit, we no longer think of the identity as anything special and no longer need special notation for it.

I would also contend that the distinction between a double and triple line is not sufficient. If someone is having difficulty with the difference between an equation and an identity then an extra horizontal line will not make it clear.

None other than the great Don Knuth once said that in a mathematical document it should be possible to replace all the bits of maths by “blah” and for it to still make grammatical sense. I strongly suspect that my students do the opposite and replace all non-maths by “blah”. For example, fill in the “blah”s in these two questions and consider how the different possibilities would lead you down different routes to an answer:

  1. Blah $x^{2} + 5 x + 6 = 0$
  2. Blah $x^{2} + 5 x + 6$

Then add in the fact that a novice learner is likely to overlook the fact that the second doesn’t have an “$= 0$” in it and try to “solve” that quadratic.

If we make the context clearer, we are lessening the work that the student has to do to understand what they are being asked to do. And this is not an artificial weakening: context becomes more and more important the deeper one goes into mathematics. In school, certainly pre-16, it is a safe assumption that the context is “numbers”. It is only later that students learn that the context could be vectors, functions, matrices, sets, objects, morphisms, groups, rings, fields, manifolds, sheaves, schemes, … if I missed your favourite, I apologise.

But even a context of “numbers” can be misconstrued. How many students look at an answer with extreme puzzlement when it turns out to be a fraction? They were expecting a whole number.

And wouldn’t it set up expectations for quadratics and trigonometry much better if we consistently said “Find all (real) numbers $x$ for which …” instead of just “Solve”? And “Show that for all real numbers $x$ …” instead of just “Show that”?

The language doesn’t even have to be that formal, we don’t need $\forall $ or $\exists $ in Y7, but it should make clear the context. It can even be something like “I’m thinking of a real number, call it $x$; it satisfies $2 x + 6 = 10$. What is it?”

So What, Exactly, is an Identity?

I have very few memories of my own time at school, but one that I do recall very vividly is my A-level Chemistry teacher announcing at the start of the course that everything we’d been told up to then had been a lie. “Sodium,” he declared, “doesn’t want to lose an electron. It doesn’t want anything.”

It was dramatic, I’ll give him that, but it did make me lose a bit of faith in Chemistry. For all I knew, everything I was going to be told in A-level would also be a lie (spoiler: it was).

I try my utmost not to do the same in my own teaching.

Of course, I can’t tell my students the whole truth. When teaching about negative integers, for example, I don’t set up an equivalence relation on pairs of positive integers and prove that the operations of arithmetic descend through the relation. What I aim for is the following thought experiment: suppose that one of my students did go on to do a mathematics degree, possibly even further, and encountered some fancy part of mathematics that recast something that they’d learnt in school. What I would hope is that they would feel that the recasting fitted in with the story that they already knew. That if they ever came back to visit, they’d say, “Now I understand why you told the story that way.”

So when I consider something like identities, I think about how the concept is used later on and try to use that to inform how I talk about it in school.

And that’s a bit tricky with identities because, in my mathematical experience, they all but disappear. The Wikipedia page does rather give the game away when it says (emphasis mine):

In other words, $A = B$ is an identity if $A$ and $B$ define the same functions. This means that an identity is an equality between functions that are differently defined.

Thus once we are happy talking about functions, the need for the word identity disappears.

When I think of the word identity, the first concept that springs to mind is the identity function (or, rather, the identity functions since there are rather a lot of them), which might happen to be representable by the identity matrix. There’s also the identity element in a group or ring.

The closest I get to the concept of identity under discussion here is in a topic called universal algebra. Very briefly, this is the area of mathematics that studies operations like $+$ or $\times $ in the abstract. Such operations satisfy relations which are sometimes called identities. These are things like $x + y = y + x$. The catch is that in this area, the identities are imposed. They don’t occur by accident but by design.

This idea of imposing identities also chimes with where I see the ≡ sign used. I don’t think of it as “is identically equal to” but as “is equivalent to in this context”. The classic situation is in modular arithmetic, where I will happily write things like $4 \equiv 1 \mod 3$, by which I mean that in the context where I ignore multiples of $3$ then I can view $4$ as equivalent to $1$. In the wider context of integers then I know that $4$ and $1$ are different, but in the smaller context of modular arithmetic then I can consider them equivalent.

So I feel that I should exercise caution in using the term “identity” to refer to what is an equality of functions, and where the term is used differently later on. Particularly because, as I argue above, using ≡ is unlikely to solve the underlying issue of establishing context.

Small Sets of Arc-Sided Tiles

Tim has previously written guest posts here about tiling by tricurves, and is now looking at ways of tiling with other shapes.

In an earlier post elsewhere I covered some basic arc-sided shapes that tile by themselves. Lately I’ve been playing with groups of curved tiling shapes, asking a question common for me: how to get the most play value as an open-ended puzzle? This means getting the most interesting possibilities from the simplest set. “Interesting” includes variety, complexity, challenge and aesthetic appeal. “Simplest” covers not only size of set and the shapes, but also the least total information needed to describe or construct the shapes.

My simple approach here is to start out with one interesting main shape and see what other (minor) shapes are needed to fill in the gaps, by trial and error; then try to refine and optimize that set to make it, in a sense, efficient.

For this post I’ve avoided the frameworks of the self-tiling regular triangles, squares and hexagons. Let’s look at two main shapes: the first is based on the pentagon; the second is a tricurve.

Using Pentagons

The regular pentagon of course can’t tile by itself. The set of tiles needed to help tile the plane with regular pentagons is well known. But let’s replace the sides of a regular pentagon with concave arcs of 72°.  We can lay these out in various ways to get different types of gaps, as shown here:

Note in many cases a point is hitting midpoint on a neighboring arc. Many of these gaps can be filled with simple lens shapes of 36°, 72° and 108°:

The remaining gaps need to be filled with partial lenses: 72° or 108° lens with one or more chunks gone. To fill the remaining gaps as is would require at least four more shapes. But we can reduce this number by backing up and combining the smaller tiles. If we start with the 36° lens and add a 4-side concave diamond (with corners of 36° and 144°, and 36° concave arcs) we can get the 72° lens and any partial 72° lens. In order to make the 108° lens we need to use another concave diamond, with corner angles of 72° and 108°.  This also lets us fill out the end of the elongated 108° lens shape.

So now the part count is four shapes (above): one major and three minor, and these let us fill the gaps:

If this were a real puzzle we would probably complain about the large number of little 36° lens pieces. Can we use less of these? The 36° lens only needs to be separate from the 4-concave diamonds in the cases where the lenses would overlap. We can permanently attach two of the 36° lenses to the thin diamond; and attach three 36° lenses to the wide diamond. So now our minor shapes look like this:

 and the tiling looks like this:

This set of tiles seems a reasonable solution (although other similar sets are possible). Now rather than simply filling gaps, we can start exploring various tilings:

Using Tricurves

The second main shape is a 36°-72°-108° tricurve, which is quite different. The tricurve already has great flexibility for tiling by itself periodically, non-periodically, and radially (as shown in previous post). So any additional parts should add to the possibilities – and it doesn’t take much. Even adding a single 36° or 72° lens at the center of a radial tiling opens many possibilities:

Since the underlying geometry is similar let’s start out with our original three minor shapes: the 36° lens and the two 4-sided concave diamonds. These let us create a very wide range of tilings:

Some of these patterns are of course not sustainable for tiling the plane. The additional complexity allowed comes partly from a means to fill gaps between adjoining convex sides or concave sides. Each of the three minor shapes by itself can add to tricurve tiling complexity, as can the use of any two of the minor shapes. Also the minor shapes can tile without the main shape –which the pentagon minor shapes can’t do (Why not?).  

Because of the ways a tricurve can tile with itself, there are many more opportunities for odd-shaped gaps that can’t be covered with the three minor shapes. With the tricurve the tiling is much more open-ended that with the pentagon above. There are no doubt various other minor shapes that could be added to fill gaps, but we’ll stick with these three for now. This whole set is interesting since it consists partly of nested lens shapes:

Also the tricurves—or either of the concave-diamond shapes for that matter—can make a circular hole, which can be filled with a circle made of the set or just the minor pieces:

Thoughts on tiling set design

Designing a small tiling set involves making tradeoffs between shape complexity, part count, and aesthetic appeal. In both shape sets, part of the complexity of the final tiling is in the use of the arcs. There is a pattern of arcs interwoven with the pattern of shapes; this may be seen as full or partial circles, or in the patterns of the arcs as they branch and connect. Also we can choose shapes to make tiling (as a puzzle) more challenging; for instance, if we modify the concave-sided pentagon so one of its sides is a convex arc, tiling will require more thought and thus be more interesting.

Both main shapes above are of course compatible with the minor shapes. This is not surprising since all the shapes incorporated 36 and 72 angles. The underlying diamonds with corners of 36° and 144°, or 72° and 108°, are two rhombus shapes used in a version of the Penrose tiles.

We could of course reduce these sets and their tilings by replacing all 36° arcs with straight lines (facets). The 36° lens shape disappears, reducing the set part count and the count of the lens pieces in the tiling.

Surprisingly, this reduction by faceting makes some things a little more complex. The larger arcs of the two main shapes would now be more complex to describe and construct. Since we sometimes connected at the midpoint of pentagon’s concave side, we’ll need to describe the shape as having ten faceted sides. Likewise, to keep the effect of the concavity of the smallest arc, the faceted equivalent of the tricurve needs 12 sides and four unique angles –whereas the much simpler tricurve can be described with two angles (36° and 72° – the 108° is the simple sum and redundant).

Compared to structurally equivalent tilings with faceted tile shapes, the above arc-sided sets:

  • have the additional part count of the 36° lens shape
  • have more complex diamond shapes, due to their 36° arcs
  • have main shapes that are simpler to describe and construct
  • have the aesthetic appeal and interest of connected arcs; and
  • overall provide more challenge and play value.

Further possible investigations:

What happens when we use both the concave pentagon and the tricurve as main shapes in the same set?

What other main shape would you try as a starting point?

The Maths of Life and Death – The God Equation

Aperiodicolleague Kit Yates has recently had a new book out: The Maths of Life and Death. He’s kindly agreed to share a sample chapter with us, explaining the God Equation: it’s used by NICE to decide whether to fund new drugs.

In my new book, The Maths of Life and Death, I explore the true stories of life-changing events in which the application (or misapplication) of mathematics has played a critical role: patients crippled by faulty genes and entrepreneurs bankrupt by faulty algorithms; innocent victims of miscarriages of justice and the unwitting victims of software glitches. I follow stories of investors who have lost fortunes and parents who have lost children, all because of mathematical misunderstanding. I wrestle with ethical dilemmas from screening to statistical subterfuge and examine pertinent societal issues such as political referenda, disease prevention, criminal justice and artificial intelligence. I aim to demonstrate that mathematics has something profound or significant to say on all of these subjects, and more.

High definition

We asked #bigmathoff competitor Lucy Rycroft-Smith to tell us a little about her latest project – CM Define It, an app aiming to collect and define mathematical vocabulary, which launches today.

When you teach mathematical vocabulary, how do you define its meaning?

Are you exact, choosing your words specifically?  Do you give a written definition?  Do you give multiple explanations?  Do you use diagrams?  Metaphors? Connect to previous vocabulary?

As part of our work at Cambridge, creating a Framework for mathematics learning, we are creating a network of semantic links across nodes in our mathematical layer – and we initially thought we could just import a mathematical glossary from somewhere else to populate this.  But we found so many inconsistencies, technical errors, and definition loops in many existing glossaries that we decided to make an app to ask the mathematical community what they thought, with the aim of developing a crowdsourced, multi-layered collage which takes into account different layers of mathematical experience.

MathsJam’s “Back of an Envelope” Fermi Challenge

Aperiodipal and MathsJam regular Rob Eastaway organised an inter-MathsJam competition for last month’s events, challenging Jams to make Fermi estimates on the back of an envelope. The prize was a copy of his new book, Maths on the Back of an Envelope. Here Rob gives a summary of the entries he received, and shares his favourites.

Regular attendees of MathsJam will know that in September, Katie Steckles kindly allowed me to hijack the evening (in the nicest possible sense) by posing some envelope-related challenges, in celebration of the publication of my new book Maths On The Back Of An Envelope. In addition to some envelope-related puzzles, there was also an open challenge to Maths Jam groups to come up with their own back-of-an-envelope problems, with the chance to win the book as a prize.

Alan Turing is the face of the new £50 note

The Bank of England has announced, following a public poll, that the new £50 note will feature mathematician, cryptographer and computer science pioneer Alan Turing. While this might seem like unambiguous good news, the issues it raises are more complicated than they first appear. Here’s a guest post from LGBT+ mathematician Calvin Smith with his thoughts on the decision.

Image result for £50 note

It’s obviously fabulous that Alan Turing is being recognised on the new £50 note, but this joy at seeing a gay mathematician given this recognition is tainted with the memory of his cruel treatment by the society of the day and the ongoing persecution of queer and trans people today.

It’s absolutely right that we celebrate the achievements of Turing the mathematician, and I’m hoping that this also creates an opportunity to talk more about the amazing work he did away from his well-known code-breaking in the field of mathematical biology, where he developed models which help us to understand the formation of shapes and patterns in biology. As a gay man working in mathematics I’m also hoping that the additional prominence given to Turing’s work acts as a further catalyst to drive inclusion in STEM subjects and that more young queer people consider study and careers in these areas.

If we are committed to doing the best science and solving modern problems then we need the most diverse set of thinkers available and for each of them to be able to bring their whole authentic selves to work. There are some wonderful organisations like Pride In Stem and events like the LGBT STEMinar which bring together and give voice to a diverse mix of queer scientists who can otherwise feel invisible or over-exposed in their local work environment. Stonewall’s work in making workplaces (especially universities) more LGBT+ inclusive cannot be underestimated as a driver for positive change. Hopefully the greater awareness and visibility afforded by Turing can drive improvements: the future of science is fabulously queer and intersectional. Fire the glitter cannon!

However, setting aside my maths-joy at seeing another mathematician celebrated I’m also filled with a cocktail of bitterness and seething rage at this announcement. Turing and society’s treatment of him is hugely symbolic and cannot be underestimated. Turing’s appearance on a bank note does not excuse society its historic treatment of him of other victims of homophobia. Just because we now have same-sex marriage does not mean the fight for queer inclusion is over. I grew up under Section 28 telling me that homosexual relationships were “pretend families” and while I am now a proud gay dad to two lovely scamps it is worth reflecting that Section 28 was only repealed in 2003, same-sex adoption introduced in 2005. Many of our recent victories seem fragile and culturally we still have a long way to go to eradicate the homophobia, lesbophobia, biphobia and transphobia which continues to lead to assaults on queer people to this day. Turing is also a call to action and constant vigilance so that we don’t turn the clock back on acceptance without exception, on inclusion for all, and for no outsiders.

July is LGBT wrath month: a Turing banknote cannot be allowed to become a pinkwashed sticking plaster over the underlying issues of intolerance, and we need LGBT+ people and their allies to continue the fight so that it remains a call to action – and not the undeserved pat on the back which leads to complacency and the further loss of queer and trans lives.