Blog – Page 14 – Mathemafrica

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.…

By Jonathan Shock| March 18th, 2018|Uncategorized|2 Comments

Singalakha’s guide to plotting rational functions

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

Find the intercepts:
1. X-intercepts, set y=0 (there can be multiple)
2. Y-intercept, set x=0 (there can be only one)
Factorise the numerator and denominator if possible:
1. Sign table: determine where the function is negative and where it is positive
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 $f(x)\ne 0$ .
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.

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By Jonathan Shock| March 15th, 2018|Uncategorized|1 Comment

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:

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By Jonathan Shock| March 13th, 2018|Uncategorized|2 Comments

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.…

By Yolisa| March 7th, 2018|Uncategorized|5 Comments

A quick introduction to writing mathematics in WordPress using LaTeX

Here are a couple of very useful links about writing mathematics, for new authors of this blog:

LaTeX for wordpress
Equations in LaTeX in wordpress (and links in that same page, which are very useful).

I will update this as I find more useful material.

Generally I like to use the Visual Tab on the editor here rather than the Text Tab, unless there is some sort of strange formatting in which case I will go in and alter the Text.
I usually like to put formulas centrally justified on their own on a line with blank lines above and below.
Add Media to upload pictures or gifs and use the Fusion Shortcodes button (to the left of the yellow star in the blue box), to embed Youtube content.

Please let me know if, as an author, there is anything which is unclear about posting here and I will update accordingly.…

By Jonathan Shock| February 28th, 2018|Uncategorized|0 Comments

A Whimsical Introduction to Graph Theory (1)

Part 1 – What are Graphs?

Mathematics is full of fascinating ideas and concepts. These can, however, be very challenging to tackle and make sense of, especially when you are put under pressure to answer questions about them! In this post, and those to come, I hope to share some insight into these concepts without getting too formal. Where some definitions and more technical bits are introduced, they will be explained at end of the post: look out for the dagger $\dagger$ symbols!

To begin, let’s ask what we do in mathematics. The first step in any area of maths is almost always to abstract things. We take some concept we want to be able to work with and pull out the essential ideas. From a bunch of maps we may take out just destinations and the routes between them; from 3D objects we may only need to know what ways we can rotate them and still see the same thing; from a collection of algorithms we may only care about how long they take to run on a computer, and so on.…

By Brandon Du Preez| February 12th, 2018|Level: Simple|0 Comments

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e-day – A mathematical holiday celebrated on February 7th

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e-day – A mathematical holiday celebrated on February 7th

Today, February 7th, 2018, is called e-day because e is approximately 2.718, and this date is written 2/7/18 in some parts of the world.

e, also called Euler’s Number after the Swiss mathematician Leonhard Euler, is a very important constant that comes up in many different places in mathematics. The numer e was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest where $e$ arises as the limit of $(1 + 1/ n) n$ as $n$ approaches infinity. e can also be calculated by summing:

The constant e appears naturally on the exponential function, which models growth. Hence, the same way that the constant π appears in everything that is round, the number e appears in everything that grows: size of baby animals, leaves in trees, bacteria populations, spreading of diseases, spirals in flowers and snails, radiactive decay of elements, money invested in a bank, processing power of computers… Everything that grows the faster the bigger it is follows an exponential law, and contains the number e.

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By Andreas Daniel Matt| February 7th, 2018|Uncategorized|1 Comment

Can you find a simple proof for this statement?

I thought more about the last question I added into the addendum of the Numberphile, Graph theory and Mathematica post

It can be succinctly stated as:

$(\forall m\in\mathbb{Z}, m\ge 19) (\exists p,q\in\mathbb{Z}, 1\le p,q<m, p\ne q)$ such that $\sqrt{p+m}\in\mathbb{Z}$ and $\sqrt{q+m}\in\mathbb{Z}$ .

In words:

For all integers m, greater than 19, there are two other distinct positive integers less than m such that the sum of each with m, when square rooted is an integer.

What is the shortest proof you can find for this statement?…

By Jonathan Shock| January 17th, 2018|Uncategorized|2 Comments

Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn’t ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at:

Always write in a comment if there is anything you would like to see us write about, or you would like to write about.…