Welcome to a four part series on ‘What Are Numbers?’. In the previous part, we constructed the Integers by using the equivalence classes of Natural Number ordered pairs that represent equations in the form \(x + b = a\). For example, the Integer we write down in the usual way as \(-2\) describes the set… Continue reading What Are Numbers? Pt. 3: The Rational Numbers

## What Are Numbers? Pt. 2: The Integers

In Part 1, we see that the building blocks of numbers start with the Natural Numbers defined through the five Peano Axioms. In this post, we ponder the invention of the Integers. Welcome to a 4 part series (this is part 2) of ‘What Are Numbers?’. Part 1: The Natural Numbers Part 2: The Integers… Continue reading What Are Numbers? Pt. 2: The Integers

## What Are Numbers? Pt. 1: The Natural Numbers

In the last few days in the recent Covid Sydney lockdown period, I had a chance to read and revise on some abstract Algebra concepts such Group Theory, Rings, Fields and Galois Theory. I was reading mainly from the book “Abstract Algebra and Solution by Radicals” by John E. Maxfield and Margaret W. Maxfield amongst… Continue reading What Are Numbers? Pt. 1: The Natural Numbers

## The Language of Proof in HSC

Related Content Outcome MEX-P1 The Nature of Proof This post will primarily look at the following dot-point from page 28 of the Extension II syllabus: use the formal language of proof, including the terms statement, implication, converse, negation and contrapositive (ACMSM024) – use the symbols for implication (⇒), equivalence (⇔) and equality (=) , demonstrating a clear understanding of the… Continue reading The Language of Proof in HSC

## Comments on the 2020 New HSC Syllabus Exams

It’s been a few weeks since the HSC 2020 Examinations concluded. Mathematics teachers and the students who sat the exam have had time to ruminate over what went into each paper. The Sydney Morning Herald printed a story about the Standard 2 Mathematics examination with a title that can only further scare future students from… Continue reading Comments on the 2020 New HSC Syllabus Exams

## Dangerous Moivres

Related Content Outcomes: MEX-N2 Using Complex Numbers N2.1: Solving Equations with Complex Numbers Terrible Puns, Terrible Maths Excuse the terrible pun on “dangerous moves” in the title (I thought it was brilliant) but it’s to highlight the all too common mistake made by students (and teachers) of elementary high school mathematics of overlooking necessary assumptions… Continue reading Dangerous Moivres

## The Incomplete Treatment of Functions in HSC Mathematics

Related Content Outcome: MA-F1 Working with Functions F1.2: Introduction to Functions What Does The Syllabus Say? The first port of call for all students and teachers when learning or teaching HSC Mathematics is of course the Syllabus. Here is the excerpt from the Syllabus (Page 31) about how a function is defined: define and use… Continue reading The Incomplete Treatment of Functions in HSC Mathematics