These questions get asked a lot in the maths education space. Here are my responses to some of them.
Frequently Asked Questions
Can I only work from LHS to RHS in identity proofs?
No, you are not required to only start from the left side to reach the right. Equality of \(a = b\) means \( b = a\).
So, you can start from RHS and work to LHS. You can also ‘meet in the middle’ by starting from both LHS and RHS and get to a common point.
Imagine proving this by LHS to RHS only:
\[2 = \frac{(\cos\theta + \sin\theta)^2 + (\cos\theta – \sin\theta)^2}{\sec^2\theta – \tan^2\theta} \]
Do I need to write an essay explaining the principle of mathematical induction in the conclusion of an induction proof?
No, HSC markers have repeatedly stated that they do not mark this part. Having it is a nice way to finish the proof like a full stop at the end of sentence, but ultimately, leaving it off does not change the substance of what has been written. You can just write, “by the principle of mathematical induction, the statement is true.”
Do I list repeats in the sample space?
A sample space is a set. So, you can freely choose to list repeats or not to.
For example, a fair six-sided dice with the numbers \(1,1,1,2,2,3\) on the faces would have a sample space of \(\{1,2,3\}\), or \(\{1,1,1,2,2,3\}\), and if you want to be even more cursed: \(\{3,3,3,2,1,1\}\) (but no one in their right mind does this).
Is \(0\) a natural number?
It depends – if you care more about addition (so start at \(0\)) or multiplication (so start at \(1\)). At the IMO 2025, my friend Tony Wang asked Terence Tao this question. Here’s his short response 1 hour 5 minutes 31 seconds in. I’d also recommend watching the full video for his brilliant talk.
In more detail, check this old series I’ve written about what numbers are: What Are Numbers?
Is \(\frac{1}{x}\) a continuous function?
A function is called continuous if it is continuous at all points in its domain. Since the number \(0\) is not in the domain of \(\frac{1}{x}\), there is no notion of checking for continuity there, and at every other point it is indeed continuous at those points. So, yes, \(\frac{1}{x}\) is a continuous function.
In the new syllabus, there is no need to classify functions as continuous or not continuous – the focus is on points rather than entire functions.
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