## Reverse Mathematics – By John Stillwell, a review

NB. I was sent this book as a review copy.

I’m not sure I’ve read a mathematics book which was so hard to review, not because of the quality of the book (which is superb), but because the way of thinking is in some senses so different to the way we normally think about mathematics. This, indeed, is also the book’s best feature. This book gets you thinking about mathematics in ways which I have never explored before, and which have definitely given me a new, and I think, improved perspective on formal mathematics.

In general in mathematics we start with a set of assumptions (axioms), and explore the consequences of them. Within Euclidean geometry we start with ideas about lines, and points, and circles, and then see what other theorems can be proved from these. Within set theory too, we start with a set of ideas about equalities of sets, existence, pairings, unions etc which we hold to be true and then see what can be said of other properties of sets, which are not straightforwardly stated in the axioms.…

## Singalakha’s guide to plotting rational functions

To sketch the graph of a function $k(x)=\frac{f(x)}{g(x)}$:

1. Find the intercepts:
1. X-intercepts, set y=0 (there can be multiple)
2. Y-intercept, set x=0 (there can be only one)
2. Factorise the numerator and denominator if possible:
1. Sign table: determine where the function is negative and where it is positive
3. Find the Vertical asymptotes:
1. This occur if the function in the denominator is equal to zero, i.e $g(x) = 0$, AND that in the numerator must not be zero, i.e $f(x)\ne 0$.
4. Find any Horizontal asymptote:
1. If the degree of the function in the numerator, i.e $f(x)$, is less than the degree of the function in the denominator, i.e $g(x)$, then the horizontal asymptote is the line $y = 0$.
2. If the degree of the function in the numerator, i.e $f(x)$, is equal to the degree of the function in the denominator, i.e $g(x)$, say for example, the degree of $f(x)$ and $g(x)$ is $n$ for some non-negative $n$ element of integers, then there is a horizontal asymptote.

## My vlogging channel

Hi all, I’m not sure if it counts as vlogging, or making maths videos regularly fits into a slightly more niche category, but anyway, I wanted to advertise some videos that I’ve been putting up recently. I’m doing this in an attempt to find a different communication channel with my first year maths class, and so far the videos are getting reasonable feedback. I have a long way to go in terms of making them slick, and I goof up from time to time, but it’s an interesting experience. If you have specific questions that you would like me to discuss in a video. Let me know.

In this video I talk about a method for solving inequalities involving absolute values:

 How clear is this post?

## 0.1 Sets

If like me, you’ve spent most of your mathematical high school years introduced to basic sets at the beginning of the year from Grades 8 to 12, then I think you’d agree that sets was one of the quickest and easiest sections we traditionally did. We would quickly recap the same fundamental properties of sets before moving onto more interesting topics, usually algebra. The section would go a little bit like this:

• define the differences between whole and natural numbers, integers, rational numbers and real numbers
• define the differences between unions, intersections and complements, usually through the understanding of Venn-diagrams
• use set builder notation (introducing algebra through this)

If like myself, you truly believed that this was as complicated as sets could ever get, then you, dear reader, like my former-myself, are in for a treat. In university, we build on these basic ideas and have a more in depth understanding about the importance of sets and their greater role in the scheme of mathematics.…