An integral expression for n!

I gave a challenge question at the end of class a week or so ago. Here I will give the solution and show that it gives us something rather strange and surprisingly useful.

I wrote down the following, and asked you to prove it:

 

\int_0^\infty e^{-t} t^N dt=N!

 

For N\ge 0, N\in \mathbb{Z}. Now, N! can be thought of as the number of different orderings of pulling N objects out of a bag (without replacement) when they are all different. If you have N things in a bag, then there are N possible things that you can pull out first. There are then N-1 ways of pulling out the next object, N-2 ways of pulling out the next, etc. and finally, when you’ve pulled out N-1 objects there’s only a single possibility of pulling out the last. So:

 

N!=N(N-1)(N-2)(N-3)...3.2.1

 

And the number of ways of pulling no objects out of a bag is 1, because you just don’t pull anything out.…

A general formula for partial fractions

Out of the blue I wrote down a rather confusing mass of indices and summations on the board a few days ago. Writing this down at the last minute was perhaps a bad idea, but I wanted to show what the general form for expanding a fraction into partial fractions was. Here I’m just motivating it a little more. It’s not something that you will need to use, but it’s often good to write things down in as general a form as you can.

Let’s say that we have an expression of the form:

 

\frac{P(x)}{(x+3)(x-2)^2}

 

Where P(x) is some polynomial of degree less than 3 (because the denominator is degree 3). We can write this as:

 

\frac{A}{x+3}+\frac{B}{(x-2)^2}+\frac{C}{x-2}

 

To find A,  B and C, you cross-multiply, and then match coefficients of powers of x with those in P(x).  If you have an irreducible quadratic in the denominator you will have terms of the form:

 

\frac{Ax+B}{ax^2+bx+c}

 

in your partial fraction, and of course if it’s an irreducible quadratic to an integer power greater than one, you will have multiple terms, just as you do for the (x-2) expression in the example above.…

By | July 30th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Where did that substitution come from?

If you want to understand maths, you really have to do it. I recommend going through these examples and using the substitutions given here as hints. Get a blank piece of paper, put your notes away and try to do these examples and see if you get the same answers as in class. If you don’t, write in the comments, and we can see where things may have gone astray.

 

I’ve been teaching integration by substitution, including by trig substitutions over the last few days, and a frequent question which a newbie substituter will ask is “how did you know to make that substitution?”. It’s a very reasonable question, and one that takes practice to build the correct intuition, but I’ll do my best to give some motivation now as to why we made some of the substitutions we made. We won’t solve the integrals, but we will motivate here why we make particular choices for substitutions.…

By | July 22nd, 2016|Courses, English, First year, MAM1000, Undergraduate|2 Comments

Down the rabbit hole

The following has been a rather interesting journey – from a test question which seemed fine, to a subtlety which seemed easy, to a discussion with a number of different mathematicians about the nature of distributions, measure theory and regularisation. I will try and make it as clear as possible in the post below. Note, as mentioned in the comments, we have actually only found the solution to this problem for a constrained range of x, and not x \in \mathbb{R}. I didn’t want to complicate things any more than necessary here for first year students, but the comments are very important too.

 

In a recent class test, there was a question, written by me, which was not quite the question that I wanted to ask. It turns out that it does have an answer, but it’s not an answer that can yet be found by the means at the class’s disposal.…

The Newton-Raphson Method

How would you go about finding the value of \sqrt{3} if you didn’t have a square root button on your calculator? Well, the most obvious thing might be to try some values, based on your knowledge of the square root function. You are being asked to find that x for which:

 

x=\sqrt{3}

 

or, in other words, that x, which, when squared  gives 3. We have to be a little careful here because we know that there will actually be two numbers which satisfy this (one positive, one negative), and we are interested in the positive one only.

 

So, we try some values, but we don’t do it randomly, we can see that because 1^2=1 and 2^2=4 that whatever number squared gives 3 must be between 1 and 2. We can try something called a binary intersection. This just means taking the values which we know bound the right answer (ie. we know that 1<x<2), and trying the number in the middle.…

Proof by induction winner: Gianluca Truda

With many congratulations for the winning entry, as voted for by mostly MAM1000W students!

At the end of every year, many families celebrate the holiday season by decorating a Christmas tree, and almost all of them will use some form of lights. The kind of cheap lights that sit on a long wire that gets wrapped around a tree, and then sparkle in a whole lot of awesome colours when you plug them into the power socket.

 

In my family, we have a whole lot of these kinds of cheap, fragile lights in a big box. Every year, when we decorate the tree, the box gets opened and is full of hopelessly tangled wires which were hastily shoved in the year before. It’s always a bit of a pain untangling the wires and it can get really
boring testing each string of lights to see if they are still working.…

By | April 23rd, 2016|Courses, First year, MAM1000, Undergraduate|2 Comments

Logical implications and the structure of if and only if statements

We had a homework assignment a couple of weeks back. It was looking at mathematics in a very different way from how many had seen it before, and it caused a lot of confusion. I would like to try and add some clarity to what we were doing. My thought was, rather than going through the questions themselves, I would like to add annotations to the proof itself. Let’s see how this works. The proof that you were given is in black, the annotations are in blue, and after I’ve been through the proof, I will expand on it in a simplified form.

 

Theorem: The function f is differentiable at x=a if and only if there is a constant m and a function E of x, defined for all x \not = a, such that

 

f(x)=f(a)+m(x-a)+E(x)(x-a) for all x \not = a      – (eq 1)

 

and,

 

\lim\limits_{x\to a}E(x)=0.

 

(If both these conditions are satisfied, then f'(a)=m.)

 

What we are doing here is giving another definition of differentiability (at a particular point, a).

By | April 22nd, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

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

Lecture 2: Sets

List of the things I learnt today


  • Definition of a set
  • Different ways of representing a set
  • Different kinds of sets
  • Intervals
  • Combinations of sets
  • Set notation in functions

 

Definition of a set

A set is a collection of objects

Different ways of representing a set

Sets of objects are usually denoted by uppercase letters e.g.

let \mathit{A} be the set of all odd numbers.The objects contained in a set are called elements which are denoted by lowercase letters e.g. \mathit{a} is an element of \mathit{A}.

A set can be represented in two ways:

1.List

\mathit{A}=\{a, b, c, d\}

The set above is an finite list, you can count the number of elements.

\mathit{B}=\{a, b, c, d,... t\}

Is also a finite list. It’s important to present enough elements to produce a sequence if the use of ellipses is implemented for shorthand purposes.

\mathit{C}=\{1, 2, 3, 5, 8,... \}

This is an infinite set, you cannot count the number of elements in the set.

2. Set Builder Notation

\mathit{D}=\{e \in \mathbb{R} : \mathit{C} (e) \}

In the example above e is an element in the set \mathbb{R} where \mathit{C} (e) is a characteristic of the elements which are specific to set \mathit{D} or you can replace “:” with”|”.…

By | February 17th, 2016|Uncategorized|3 Comments

First year resources – part 4: revision

This is a continuation of the previous posts, essentially collecting thoughts for first year students. I am asking you, the reader to suggest what might be wrong, or missing from this, and anything else which will be helpful for a new first year who is just arriving at university to study maths…

The following sections in the resource book are about mentors and the whiteboard workshop. They are really quite specific to the course, and more about the details than the philosophy of it, so I am not including them here.

The next section on the other hand is vital, as most students are never given much guidance in how best to revise, and it is one of the most important skills they can gain. I have written my thoughts from my own experience, but I am not trained specifically in education, so I am grateful for any additional thoughts.

 

How to revise

Revision is a bit like comedy: Timing is everything!

By | January 26th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments