Blog – Page 13 – Mathemafrica

Advice for MAM1000W students from former MAM1000W students – part 5

While I resisted Mam1000W every single day, I even complained about how it isn’t useful to myself. Little did I know when it all finally clicked towards the end that even though I wasn’t going to be using math in my life directly, the methodology of thinking and applying helps me to this day.

Surviving Mam1000W isn’t really a miraculous thing. While everyone tends to make it seem like it’s impossible, it is challenging (Not hard) and I said that because I have seen first-hand that practice makes it better each time. Getting to know the principles by actually doing the tuts which is the most important element of the course in my opinion will make sure that even though you feel like you aren’t learning anything when the time comes (usually 2nd semester) it will all click on how you actually are linking the information together.

Another important aspect is playing the numbers game.

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By Jonathan Shock| May 14th, 2018|Courses, First year, MAM1000, Undergraduate|1 Comment

Advice for MAM1000W students from former MAM1000W students – part 4

In high school, as I believe was the case for many students, there wasn’t much incentive to work very hard regularly on math – concepts were easy to grasp first hand in class. That’s the kind of attitude I brought towards MAM1000W last year (2017). Unfortunately things didn’t turn out as anticipated…by as early as April I had already started playing “catch-up” for I hadn’t been putting in any practice on the staff done in class. Tests were nightmares. With every course demanding its share of my attention, I found myself crying the other day alone in my room, asking myself, “what went wrong?”. Well, the answer was pretty simple – EVERYTHING.

Eventually, I figured a way to potentially get back on my feet – I became a very good friend of my WebAssign voluntary quizzes. In combination with past papers (WHICH I HIGHLY RECOMMEND) and the daily uploaded ‘practice questions and solutions’, I was able to gain back some bit of confidence.

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By Jonathan Shock| May 9th, 2018|Uncategorized|0 Comments

Advice for MAM1000W students from former MAM1000W students – parts 2 and 3

Part 2:

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So one thing that really helped me was having a partner in tuts. We would do the tuts as far as we could and we would then try to help one another in the tuts and ask the tutors for help if there was a difference in opinion.

Another thing that helped studying, going through past papers and tuts were so important.

If I was ever stuck and couldn’t really understand the textbook I would go on YouTube and watch a guy named Professor Leonard. He’s videos are super long but extremely helpful and worth your time.

And last but not least, it’s important that you try your best to work everyday with maths because once you fall behind its difficult to catch up. Even if you do just one problem a day I promise it will help In the future.

and part 3:

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I would suggest to MAM1 students that they should not fall behind the maths syllabus if they have tests in other subjects because it is very difficult to catch it up and requires much more effort than one thinks.…

By Jonathan Shock| May 8th, 2018|Uncategorized|1 Comment

Advice for MAM1000W students from former MAM1000W students – part 1

This is the first in a series of posts where I will be putting up the sage words of advice of former MAM1000W. Often, these students struggled their way through the course, before making a breakthrough in their study methods. I hope that maybe it will be easier to listen to students who have been through the struggle, than the advice of lecturers who seem to know it all (though I promise you, we do not!).

Here is the first:

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As an Actuarial Science student I was aiming for 70% last year. I clearly remember that at orientation I asked some of the older ActSci students at orientation what they had done when they scored below what they needed to. I was so shocked, and a little scared when the group I asked said they never had. I wasn’t worries at this stage though because I thought I’d done well at maths at school, and I’d do well at maths here.…

By Jonathan Shock| May 8th, 2018|Uncategorized|4 Comments

Hypatia, The Life and Legend of an Ancient Philosopher – by Edward J. Watts, a review by Henri Laurie

Review written by Henri Laurie.

From Oxford University Press

This is an important book for anybody interested in the history of mathematics and in the history of women intellectuals.

To recap very briefly: Hypatia is well-known as the mathematician/philosopher who was murdered by a Christian mob in 415 CE in Alexandria. She is one of the best-attested woman philosophers in the Greek tradition.

Watts turns this on its head: he tells the story of a life, one of singular achievement, and one in which the manner of death is not the most important part. The picture he paints is of a very remarkable woman, who became the head of her father’s school at a relatively young age and came to dominate the scholarly activity of her city, at the time one of the three most important centres of learning in the Mediterranean.

It is important to realise that although women did study philosophy at the time, and therefore also mathematics, which was seen as preparation for philosophy, very few of them were able to continue well into adulthood.…

By Jonathan Shock| May 6th, 2018|Uncategorized|2 Comments

Mathematical Foundations of Quantum Mechanics – By John Von Neumann, edited by Nicholas A Wheeler, a review

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

From Princeton University Press

I have to admit that I was rather embarrassed to encounter this book, as I had never heard of it, and given the topic, and the author, it seemed that it must be one of the canonical texts in the field. However, it turns out that although Von Neumann wrote this book in 1932 (full German text here), it was not translated until 1955 (by Robert Beyer), and this edition aged quickly, particularly with the limitations of typesetting the equations. It wasn’t until now that a modern edition has been put together, by Nicholas Wheeler, and the result is lovely.

The book is really a collection and expansion of Von Neumann’s previously published works, attempting to put quantum mechanics on a firm mathematical footing. The first chapter is dedicated to the equivalence of Matrix Quantum Mechanics, and Schrodinger’s Wave Mechanics.…

By Jonathan Shock| May 6th, 2018|Book reviews, Reviews|1 Comment

An Introduction to analysis – By Robert G Gunning, a review

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

From Princeton University Press

While this book is called An Introduction to Analysis, it contains far more than one might expect from a book with such a title. Not only does it include extremely clear introductions to algebra, linear algebra, intregro-differential calculus of many variables, as well as the foundations of real analysis and beyond, building from their topological foundations, the explanations are wonderfully clear, and the way formal mathematical writing is shown will give the reader a perfect guide to the clear thinking and exposition needed to go on to further areas of mathematical study and research. I think that for an undergraduate student, taking a year to really get to grips with the content of this book would be absolutely doable and an extremely valuable investment of their time. While a very keen student would, I think, be able to go through this book by themselves, as it truly is wonderfully self-contained, if it were used as part of a one year course introducing mathematics in a formal way, I think that this really would be the ideal textbook to cover the foundations of mathematics.…

By Jonathan Shock| May 5th, 2018|Book reviews, Reviews|2 Comments

0.4. Cartesian product

We know we can use binary operations to add two numbers, x and y: $x+y, x-y, x \times y, x \div y.$ Furthermore there are other operations such as $\sqrt{x}$ or any other root and exponents. Operations can involve other mathematical objects other than numbers, such as sets.

$def^n$ Given two sets, A and B, we can define multiplication of these two sets as the Cartesian product. The new set is defined as

$A \times B = \{(a,b): a \in A, b \in B\}$

Before looking at abstract examples, consider this case:

e.g.1. Assume there is a student in a self-catering residence and they want to make food preps for the first four days in the week. They want to know how many possible combinations they can make using fruits (between grapes and apples) and meals (pasta and meatballs, chicken wrap).

To solve this, let $A = \{ \text{ grapes, apples } \} \text{ and } B = \{ \text{pasta and meatballs, chicken wrap} \}$

Then the possible meal options are: (grapes, pasta and meatballs), (grapes, chicken wrap), (apples, pasta and meat balls) and (apples, chicken wrap).

The Cartesian Product of sets A and B would be:

$\text{A x B} = \{( \text{ grapes, pasta and meatballs}), (\text{ grapes, chicken wrap }), (\text{ apples, pasta and meat balls }), (\text{ apples, chicken wrap}) \}$

We can think of the above example in more abstract terms.…

By Yolisa| April 14th, 2018|Uncategorized|3 Comments

0.3. power sets

Recall powers (or exponents) of numbers: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$

Similarly, sets have the power operation to create new sets.

$def^n$ If A is a set, then the power set of A is another set denoted as

$\mathbb{ P }(A) = \text{ set of all subsets of A } = \{ x: x \subseteq A \}$

Recall: A is a subset of B if every element in A is also in B. Furthermore, if A is a finite set with n-elements, then we can find the number of subsets in A by using this formula:

$2^n$

To find the power set of A, we write a list of all the subsets of A first – remembering that:

the empty set is a subset of every set,
and every set is a subset of itself

Let’s look at some examples:

e.g.1. $A = \{1, 2, 3 \}$

Using the formula $2^n$ , we know that there are $2^3 = 8$ possible subsets of A, namely:

$\varnothing, \{1, 2, 3 \}, \{1 \}, \{2 \}, \{3 \}, \{1, 2 \}, \{2, 3 \} \text{ and } \{1, 3 \}$

Hence the power set is the set that contains all the above subsets:

$\mathbb{ P }(A) = \{ \varnothing, \{1, 2, 3 \}, \{1 \}, \{2 \}, \{3 \}, \{1, 2 \}, \{2, 3 \}, \{1, 3 \} \}$

Note: The cardinality (size) of $\mathbb{ P }(A) = 8 = 2^3$ where size of A= 3 elements

e.g.2.

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By Yolisa| April 10th, 2018|Uncategorized|1 Comment

https://giphy.com/gifs/infinite-boxes-vG1Dgq3JRXLMc

Consider a set $A = \{2, 3, 7\} \text{ and } B = \{2, 3, 4, 5, 6, 7\}.$ Note that every element in set A is also found in set B, however, the reverse is not true (B contains elements 4, 5 and 6 which are not in A)

Consider another case, $A = \{2n: n \in \mathbb{ N }\} = \{ 0, 2, 4, 6, ... \} \text { and } B = \mathbb{Z} = \{ ..., -2, -1, 0, 1, 2, ... \}.$ Again, we can see that every element in set A is also found in set B and similarly, everything in B cannot be found in set A. B contains negative and odd integers, which are not in A.

To describe this phenomena, mathematicians defined subsets:

$def^n$ Suppose A and B are sets. If every element in A is an element of B, then A is a subset of B and we denote this as $A \subseteq B$

If B is not a subset of A, as in the above cases, then there exists at least one element, say $x \in B \text{ such that } x \notin A. \text{ We denote this as } B \subsetneq A$

e.g.1. $\{2, 3, 5, 7, ... \} \subseteq \mathbb{ N }$ but $\{\frac{1}{3}, 2, 5, 7, ... \} \subsetneq \mathbb{ N }$ since $\frac{1}{3} \in \mathbb{ Q }$

e.g.2. $\mathbb { N } \subseteq \mathbb { Z } \subseteq \mathbb{ Q } \subseteq \mathbb{ R }$

e.g.3. $(\mathbb{ R } \times \mathbb{ N }) \subseteq (\mathbb{ R } \times \mathbb{ R })$ since $(\mathbb{ R } \times \mathbb{ N }) = \{(x, y): x \in \mathbb{ R }, y \in \mathbb{ N }\}$ and $( \mathbb{ R } \times \mathbb{ R }) = \{ (x, y): x \in \mathbb{ R }, y \in \mathbb{ R } \}$ Hint: look at what sets y is in