Thoughts About Some Mathematical Practices

Here are some of my thoughts about some miscellaneous mathematical practices – some words of advice, warning, interesting insights, contentious disagreements, or whatever else comes to mind. This might become a multipart series as more comments come to mind in the future…

Notation of Domains of Functions

The domain of a function is a set. Notate it as such.

Here’s a simple question: write down the natural domain of the real valued function \(f(x) = \sqrt{x}\).

Many teachers say to their students to write \( x \geq 0\). While the intention of the solution is correct and shows how a square root function works – and hence if I saw that in an assessment task I would not deduct marks – the notation is actually incorrect.

Another incorrect notation is to write \( x \in [0, \infty)\) when using interval notation.

Why?

As hinted in the italicised text above, the domain of a function is a set and hence should be written using set theoretical notation.

Although \( x \geq 0\) is a relation and is therefore a subset of \(\mathbb{R} \times \mathbb{R}\), it’s the wrong one to describe the domain of the function. Instead, if one wishes to use the inequality relation in their presentation of the solution, it needs to be written with set builder notation as: \( \{ x \in \mathbb{R} : x \geq 0\}\).

If one wants to use the interval notation, then it is necessary and sufficient to just write \([0, \infty)\) as it is already representing a set. Having the \( x \in\) part indicates set membership, not the set itself.

Another correct answer to the natural domain of the function \(f(x) = x\) would simply be \(\mathbb{R}\) rather than \( x \in \mathbb{R}\).

Continuous Functions

A continuous function is not something that you can draw without lifting your pen off the page.

A classic example is \(f(x) = \frac{1}{x}\) which has a natural domain \((-\infty, 0) \cup (0, \infty)\). This is a continuous function in its domain.

Continuity of a function depends on the continuity of each point in the domain of the function. It was sad to see a trial paper from a school that I will not name insinuate the idea that \(\frac{1}{x}\) was discontinuous.

In fact, the whole notion of limits and continuity in the HSC, and hence many high schools, is confused as they’re treated too informally. To get a real understanding of limits and continuity, one needs to study Metric Spaces (or at the very least some Real Analysis) and/or Topology at university, or from a good maths textbook like Gamelin & Greene, where they are properly defined.

Implication, Or, De Morgan’s Laws, Negation

\[p \Rightarrow q \equiv \neg p \lor q\]

\[\neg (p \lor q) \equiv \neg p \land \neg q\]

\[\neg (p \land q) \equiv \neg p \lor \neg q\]

Hence: \( \neg(p \Rightarrow q) \equiv p \land \neg q\).

Please teach logic using truth tables. I write more about this here: The Language of Proof in HSC

Completing The Square

Completing the square is an underrated technique.

Completing the square of quadratic expressions exposes their inherent symmetry and can be used to simplify many expressions quickly and effectively. Noticing this in Q10 of the 2018 Extension 1 paper would have made it an almost trivial question, by transforming \(2kx – x^2\) into \(k^2 – (x-k)^2\).

Here’s another personal favourite of mine:

Factorise \(x^4 + 64\).

Solution: \(x^4 + 64 = x^4 + 16x^2 + 64 – 16x^2\) by completing the square, but this time it’s the middle term that is completed, not the constant term. Then we get \( (x^2 + 8)^2 – 16x^2 \) which is a difference of two squares, and we factorise it to two quadratic factors: \( (x^2 + 8 – 4x)(x^2 + 8 + 4x) \).

I once overheard someone say that they thought it was a useless technique outside of proving the quadratic formula – I was not very happy.