Jonathan Shock – Page 12

About Jonathan Shock

I'm a lecturer at the University of Cape Town in the department of Mathematics and Applied Mathematics. I teach mathematics both at undergraduate and at honours levels and my research interests lie in the intersection of applied mathematics and many other areas of science, from biology and neuroscience to fundamental particle physics and psychology.

Proof by induction for a non-mathematician – a competition: Vote for the winner!

I set a voluntary assignment for my course a few weeks back. Students had just learned about proof by induction, and I tend to find that this is a subject which many get confused by. I think that one of the best ways to really understand a topic is to try and teach it to someone else, so I set up an exercise which was to write an explanation of Proof by Induction for a young high school student. We had around 100 entries, which took a while to read through! Of these 100 entries there were four which stood out (and many which were also very good). We (myself, and two senior tutors) have been unable to come up with an outright winner. That’s where you come in!

Please take a look at the following entries, and vote for the one that you think is best at this link: https://www.surveymonkey.com/r/9K3P8BJ.…

By Jonathan Shock| April 12th, 2016|Competition, Courses, First year, MAM1000, Undergraduate|2 Comments

The squeeze (or sandwich) theorem

Let’s say I ask you how tall Craig is going to be when he’s 15 years old, and let’s say, given his genetic information, you just can’t tell what height he will be at that age. However, you do know that he’s going to be taller than Lisa, and shorter than Khangelani up until the age of 15 (they are all the same age). With this information alone, you haven’t learnt much about the height of Craig when he’s 15. However, what if you can figure out, using your clever genetic detective work, that at the age of 15 Lisa and Khangelani will be the same height? The only way for this to be true, is if Craig (whose height is sandwiched in between that of Lisa and Khangelani) is also the same height at that age.

That’s really all the sandwich theorem is. Let’s look at a mathematical example.…

By Jonathan Shock| April 9th, 2016|Courses, First year, MAM1000, Undergraduate|4 Comments

The 2016 International Earth and Sky Photo Contest

If you’re into astrophotography, take a look at this contest.

How clear is this post?

…

By Jonathan Shock| March 31st, 2016|Advertising, Competition|0 Comments

Calculating the date of Easter, and a computational challenge

The date on which Easter Sunday falls was decided, by the Council of Nicaea in 325 AD, to be the first full Sunday after the first full moon after the Vernal Equinox. This is very easy to work out with a lunar calendar or if you have data about the lunar cycles and the equinoxes, but what if you just were given the year. How would you work out what date it was on? Well, it turns out that there isn’t a very simple formula which can tell you this, but there is a rather complicated formula. This was presented to the journal Nature in 1876, though I haven’t been able to track down the author of this particular formula. So, this is how you calculate the date of Easter Sunday:

Take the year (in the Gregorian calendar) and divide it by 19. This will give you a quotient and a remainder.

…

By Jonathan Shock| March 24th, 2016|Uncategorized|0 Comments

Proof by contradiction – part 2

So, in the last post we proved that $\sqrt{2}$ is irrational, by trying to see what the consequences would be if it were rational. We first said that if it were rational, then we should be able to write it in a simplest form $\frac{p}{q}$ where p and q had no common factors, and then showed that in fact this was impossible, so our original proposition was indeed true (as trying to prove otherwise gave us a contradiction).
Now we are going to look at another example, which looks very similar, but here our contradiction will be a little different to last time. The fact is that, unlike much of what you would have done at highschool, there isn’t such a roadmap to how to do things here – you have to figure out for yourself where the contradiction comes in. In these posts I will point them out for you, but in general, you need to build your intuition about where something looks a bit dodgy and a contradiction is raising its head.…

By Jonathan Shock| March 22nd, 2016|Courses, First year, MAM1000, Undergraduate|5 Comments

Proof by contradiction – part 1

Proof by contradiction may at first seem completely weird! I give you something to prove, and you seem to ignore me and try and prove that what I want you to prove is wrong!

Actually, this isn’t nearly as strange as it first seems, and it can work in contexts other than mathematics. The idea stems from the fact that a statement is either true, or false (well, if you listen to Gödel, then you have to be a bit careful, but it’s reasonable enough for now). The process is the following:

You want to prove that a statement is true.
You say “what would happen if the statement were actually false?”
You explore the consequences of it being false.
If you find that it gives you a contradiction (something which you claimed to be true, but which you now see isn’t true), then you know that in fact the original statement can’t be false…so it must be true, and you’ve proved it.

…

By Jonathan Shock| March 20th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|7 Comments

Can we find the inverse of a function which is not one-to-one? (part two)

So, in the last post we had seen that while the sin function is not one-to-one and thus doesn’t have an inverse, so long as we restrict it to a given domain, you will find that it is invertible. The domain that we found (indeed chose), was between $[-\frac{\pi}{2},\frac{\pi}{2}]$ . It’s inverse was a function with domain $[-1,1]$ . The name of the inverse is arcsin(x). How can we use this to help us to solve problems?

Well, what if I asked you to solve:

$sin(x)=\frac{1}{2}$

You might think that because we have found the inverse of sin, that we can simply say that the solution to this is:

$x=arcsin\frac{1}{2}$

Well, because arcsin is itself a one-to-one function, restricted to the domain $[-1,1]$ this will clearly give us a single number (the answer is about 0.52):

Is that it then? Well, let’s look at the graph of sin(x) and see if this is the only solution to $sin(x)=\frac{1}{2}$ :

In fact, clearly there are an infinite number of solutions to the equation $sin(x)=\frac{1}{2}$ and we have just caught the one within the region $[-\frac{pi}{2},\frac{\pi}{2}]$ .…

By Jonathan Shock| March 16th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Can we find the inverse of a function which is not one-to-one? (part one)

Asking what the inverse of a function is, is the same as asking what is the function that will undo this function?

What is the inverse of the function $f(x)=x+3$ ? That is asking the question, if I put a number into this function (call that number a), it will give me another number (call it b). What is the function which, whatever number I put in, when applied to the number that comes out from the first function will be the original number. That is to say:

If f(a)=b. Then what is the function g for which g(b)=a. That would give you g(f(a))=a. g Is then the inverse of f and we can write $g(x)=f^{-1}(x)$ . g(x) is the thing that undoes f. Put simply, composing the inverse of a function, with the function will, on the appropriate domain, return the identity (ie. not do anything to the number you put in).…

By Jonathan Shock| March 15th, 2016|Uncategorized|3 Comments

Domain of a composite function – part 2

This is the second part of a post, written by Muhammad Azhar Rohiman, a first year student on MAM1000W at UCT. This post came about when he asked me a question related to domains of composite functions, and it was clear that on first learning about such topics, there are some simple misunderstandings. I suggested that he write a couple of paragraphs explaining what he had learnt, and the following is, I think, a very clear explanation of some of the ideas and pitfalls of this topic. The first part of the post is here.

Consider the two functions below, from which we want to find the domain of

(a) $(f\circ g)(x)$ .

(b) $(g\circ f)(x)$ .

(d) $( g \circ g )(x)$ .

$f(x) = x + \frac{1}{x}$

$g(x) = \frac{x+8}{x+2}$

(a) The functions f(x) and g(x) cannot be defined at the values x = 0 and x = -2 respectively. Therefore, we can write this as follows: f(0) and g(-2) are not defined.…

By Jonathan Shock| March 12th, 2016|Courses, First year, MAM1000, Undergraduate|1 Comment

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