Blog – Page 18 – Mathemafrica

Riemann sums to definite integral conversion

In the most recent tutorial there is a question about converting a Riemann sum to a definite integral, and it seems to be tripping up quite a few students. I wanted to run through one of the calculations in detail so you can see how to answer such a question.

Let’s look at the example:

$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(9\left(4+(i-1)\frac{6}{n}\right)^2-8\left(4+(i-1)\frac{6}{n}\right)+7\right)\frac{13}{n}$

There are many ways to tackle such a question but let’s take one particular path. Let’s start by the fact that when the limit is defined, the limit of a sum is the sum of the limits. We can split up our expression into 3, which looks like:

$\lim_{n\rightarrow\infty}\sum_{i=1}^n9\left(4+(i-1)\frac{6}{n}\right)^2\frac{13}{n}-\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(8\left(4+(i-1)\frac{6}{n}\right)\right)\frac{13}{n}+\lim_{n\rightarrow\infty}\sum_{i=1}^n7\frac{13}{n}$

Let’s tackle each of these separately. Let’s look at the first term:

$\lim_{n\rightarrow\infty}\sum_{i=1}^n9\left(4+(i-1)\frac{6}{n}\right)^2\frac{13}{n}$

Well, we can take the factor of 13 outside the front of the whole thing to start with, along with the factor of 9, and this will give

$13\times 9\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(4+(i-1)\frac{6}{n}\right)^2\frac{1}{n}$

We see here that we have a sum of terms, and a factor which looks like $\frac{1}{n}$ in each term.…

By Jonathan Shock| August 23rd, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|8 Comments

From http://press.princeton.edu/titles/11024.html

This book was sent to me by the publisher as a review copy.

PHILOSOPHY OF MATHEMATICS OR PHILOSOPHY FOR MATHEMATICS? By Henri Laurie.

Review of Øystein Linnebo’s “Philosophy of Mathematics”, Princeton University Press, 2017. (This one is impressionistic; I hope to present a more conventional summary-of-contents review in due course).

I’ve just read Øystein Linnebo’s superb book on the philosophy of mathematics. It is very, very good. Superbly clear, concise, well organised, it gives not only a very accessible introduction but also takes the reader all the way to the cutting edge of what philosophers are doing in the philosophy of mathematics. Above all, Linnebo writes as a fully engaged philosopher and makes his preferred choice of philosophical position clear. But this is no mere polemic: I felt he clearly and forcefully presents the strengths and weaknesses of all the philosophical positions he discusses.

That said, even an introductory text in philosophy these days is not always easy reading.…

By Jonathan Shock| August 21st, 2017|Book reviews, Reviews|4 Comments

Some sum identities

During tutorials last week, a number of students asked how to understand identities that are used in the calculation of various Riemann sums and their limits.

These identities are:

$\sum_{i=1}^n 1=n$

$\sum_{i=1}^n i=\frac{n(n+1)}{2}$

$\sum_{i=1}^n i^2=\frac{n(n+1)(2n+1)}{6}$

$\sum_{i=1}^n i^3=\left(\frac{n(n+1)}{2}\right)^2$

Let’s go through these one by one. We must first remember what the sigma notation means. If we have:

$\sum_{i=1}^n f(i)$

It means the sum of terms of the forms f(i) for i starting with 1 and going up to i=n. Sometimes n will actually be an integer, and sometimes it will be left arbitrary. So, the above sum can be written as:

$\sum_{i=1}^n f(i)=f(1)+f(2)+f(3)+f(4)+....+f(n-2)+f(n-1)+f(n)$

We haven’t specified what f is, but that’s because this statement is general and applies for any time of function of i. In the first of the identities above, the function is simply f(i)=1, which isn’t a very interesting function, but it still is one. It says, whatever i we put in, output 1. So this sum can be written as:

$\sum_{i=1}^n 1=1+1+1+1+....+1$

Where there are n terms.…

By Jonathan Shock| August 20th, 2017|Courses, First year, MAM1000, Uncategorized, Undergraduate|3 Comments

MAM1000W 2017 semester 2, lecture 1 (part ii)

The distance problem

If I want to know how far I walked during an hour, I can ask how far I walked in the first five minutes, and how far I walked in the second five minutes, and how far I walked in the third five minutes, etc. and add them all together. ie. I could write:

$d=d_1+d_2+d_3+d_4+...d_{12}$

Where $d_i$ is the distance walked in the $i^{th}$ five minutes. To calculate a distance, we need to know how fast we are going, and for how long. In fact:

$distance=velocity \times time$

where you can think of velocity as the same thing as speed (though there are subtle differences which you will find out about later). This formula works if the velocity is constant, but what if it is changing. Well, if we have a graph of velocity against time, then we can think about splitting the graph into intervals (like the five minute intervals above), and approximating that during a small interval of time, the velocity is roughly constant.…

By Jonathan Shock| August 13th, 2017|Courses, English, First year, Level: Simple, MAM1000, Uncategorized, Undergraduate|1 Comment

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MAM1000W 2017 semester 2, lecture 1 (part i)

Courses, English, First year, Level: Simple, MAM1000, Uncategorized, Undergraduate

MAM1000W 2017 semester 2, lecture 1 (part i)

I wanted to put up a little summary of some of the most important things to remember from the end of last semester. There was a sudden input of new concepts, so let’s put some of them down here to get a clear reminder of what we need to know. A few things in this post:

The antiderivative
Sigma notation
Areas under curves

Antiderivatives

An antiderivative of a function $f$ on an open interval $I$ is a function $F$ such that:

$F'(x)=f(x)$ for every $x\in I$

Note that we say an antiderivative, not the antiderivative. There can be many functions whose derivatives give the same thing. While we know that:

$\frac{d}{dx}\sin x=\cos x$

and therefore $\sin x$ is an antiderivative of $\cos x$ , we can also say that:

$\frac{d}{dx}(\sin x+3)=\cos x$

So $\sin x+3$ is also an antiderivative of $\cos x$ . In fact for any constant $c$ it is true that $\sin x+c$ is an antiderivative of $\cos x$ . We will come up with some clever notation for the antiderivative soon.…

By Jonathan Shock| August 13th, 2017|Courses, English, First year, Level: Simple, MAM1000, Uncategorized, Undergraduate|1 Comment

from Princeton University Press.

This book was sent to me by the publisher as a review copy.

This is a book of some impressive magnitude, both in terms of the time span that it covers (being millennia), as well as the ways in which it discusses the context and content of the ciphers, most of which, as the title suggests, are unsolved. The book starts with perhaps the most mysterious of all unbroken ciphers: The Voynich Manuscript (the entirety of which can be found here). This story in itself is perhaps the most fascinating in the history of all encrypted documents, and that we still don’t know if it truly contains anything of interest, or is just a cleverly constructed (though several hundred year old) hoax makes it all the more intriguing.

The writing rather effortlessly weaves between the potential origin stories, the history of the ownership of the manuscript and the attempts to decode it.…

By Jonathan Shock| July 29th, 2017|Book reviews, English, Level: Simple, Reviews|1 Comment

I have decided to share something which I found interesting while reading up some Mathematics this holiday.

The idea I am going to talk about is that of the cardinality of a set. In simple terms,

Definition 1: Cardinality is a “measure of the size” of a set.

Example:
Suppose that we have a set, $A$ , such that $A=\{a,b,c,d,e\}$ . The cardinality of the set, denoted by $|A|$ , is $5$ , because there are $5$ elements.
It is indeed worth noting that unlike ‘lists’, in Mathematics, order and number of elements doesn’t determine much concerning the identity of a set. This means that

$\{a,b,c\}=\{a,a,a,b,c\}=\{a,b,b,b,c,c\}=\{a,...,b,...,c, ...\}$

as long as you use the same elements, they are all equal. Of course, cardinalities would be the same, because all of them are a representation of the same mathematical entity.

Let’s talk about something slightly more interesting. Suppose that you have another set with infinitely many elements, like the set of natural numbers, $N$ , or real numbers $\mathbb{R}$ .…

By Siphelele Danisa| July 28th, 2017|Uncategorized|0 Comments

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Quantum Computing (Part 0): A Brief Introduction

Level: Simple

Quantum Computing (Part 0): A Brief Introduction

These days, the vogue in science and technology is all things quantum, especially in the continuously-advancing frontiers of computation and data analysis. Applications such as AI and crunching through Big Data require ever-faster processing of ever-larger datasets. The amount of data generated by companies such as Google and Facebook is already mind-boggling and is only set to increase at an exponential rate. However, it is clear that in many contexts, simply adding datacenters and processors is not going to be enough: a complete paradigm shift is required if these kinds of technologies are to become effective in the lives of billions of people around the world.
This is where the quantum realm comes in. At the smallest scales of our universe, the behaviours of the “classical” mechanics governing our everyday lives is replaced by something entirely different; the laws of quantum mechanics are unintuitive and confusing, hiding layers of complexity in ways that simply cannot exist at our macroscopic scales.

…

By Sam Wolski| July 10th, 2017|Level: Simple|0 Comments

Getting ideas into action: Statway and Quantway networked improvement communities

Siyaphumelela Conference 2017, The Wanders Club, Johannesburg 28 June 2017

Getting ideas into action: Statway and Quantway networked improvement communities

Bernadine Chuck Fong, Carnegie Math Pathways

Andre Freedman, Capital Community College

It’s an exciting time to be tackling problems that appear to be worldwide. The Carnegie Foundation is in California, home to many disruptive changes / technologies, e.g. Uber, Google.

An example of the problem: When asked which is bigger, a/5 or a/8 many students respond saying 8a = 5a so 8 = 5. For many, mathematics is about following algorithms and although they may pass algebra courses they can’t apply the knowledge.

Diagnostic tests can leave students feeling depressed, like they do not belong in higher education. A principle in the development of a disruptive transformation of mathematics in American colleges was to meet students where they are.

15% of students needing developmental maths complete the required college maths or stats course after 2 semesters.…

By Anita Campbell| June 30th, 2017|Conference, Uncategorized|0 Comments

Radically transforming mathematics learning experiences: Lessons from the Carnegie Math Pathways

Siyaphumelela Conference 2017, The Wanders Club, Johannesburg

http://www.siyaphumelela.org.za/conf/2017/#!

Carnegie Foundation Math Pathways Workshop 27 June 2017

Andre Freedman, Capital Community College, Connecticut

Bernadine Chuck Fong, Carnegie Math Pathways

Workshop goals:

Learn about the design, goals, implementations of Carnegie Math Pathways
Experience Pathways lessons
Engage in design tasks to improve student success in maths and college
Engage in conversations about professional learning to address issues and concerns that are specific to local contexts

Staff had to learn new ways to teaching maths, there had to be ‘buy in’ for it to be successful. This is a great challenge.

How to radically transform outcomes for all mathematics students?

aim to get more student to complete maths courses and degrees
persistence (students don’t easily give up when challenged)
quality of learning (e.g. students can explain what a function is years after taking a maths course)
identities of learning (students see themselves as someone who can do maths).

…

By Anita Campbell| June 28th, 2017|Uncategorized|0 Comments

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