Skip to content

OK Theoremus is ready for download

January 7, 2015

You can get a copy of this textbook here.
Your interest and comments will be most welcomed.

I am working to release it at Amazon, CreatSpace and in Kindle.


The Internet is the “computer”

October 27, 2014


A few days ago, I went to a seminar conducted by one of my former professors on the Internet of Things and I learned how we now have plenty of sensors which can publish data into the Internet. Name it what you will, it can be traffic cameras, or weather stations etc, they can all tap into the Net.

During the seminar, my mind wondered off and I started imagining the movie Terminator. The reality of Skynet may no longer be confined to the movie franchise. Then today I stumbled on this article of Elon Musk.

Read it and let me know what you think.

Theoremus – the first draft of my booklet, available and ready for review

July 18, 2014

Front Cover of Theoremus

Front Cover of Theoremus


The first draft of my booklet called Theoremus is ready for review – if anyone is interested. I will gratefully thank you in the acknowledgment section for any critiques you may have of it. It is not a thick booklet – it is only 70 pages long and targets for an A5 size print. So an expert can finish reviewing it in a couple of days.

It’s aim is to coach a First Year In Maths student on how to make their proofs more rigorous and thus convincing. That is the broad theme of this booklet – the idea of being rigorous.
After all, why would teachers want you to prove theorems if they do not want you to prove them a bit more rigorously? Surely that should be preferred.

The booklet is suitable for any first year university student who will likely be required to put more solid maths in their course. For example, Comp Sci and Engineering students, like those doing Software and Systems Engineering. It should also be suitable for students of Economics, Finance, Physics, Linguistics, Chemistry and perhaps Biology. Well, of course, it should be for the Maths and Stats students too 🙂

Please let me know if you have an interest in reviewing it – your suggestions will be welcomed no matter what angle they come from. Just drop me a line in the comments side of this post and I will send you the draft in PDF. Thank you in advance.


Convergence Theorem for Programming Languages

July 1, 2014

Let L = the LISP language. Let P = a programming language different from L, i.e. L \cap P = \oslash. Let t = time. Then \lim_{t \to \infty} P = L.

Trivial, just look at the extensions they are making on Python, Ruby, Java, C# and C++11.


I have seen the light…a couple of years ago

June 12, 2014

I have seen the light

Credits: xkcd

Smallest piece of code in the world?

April 24, 2014

It has been said that the smallest piece of code in the world is the C code shown below for copying strings.

while (*d++ = *s++);

That is just a whopping 17 characters long! Really?

I’ll do that in Scheme/Lisp:

(set! d s)

There 9 characters long.

It should be true for n=k+1 or why I.H. works

April 17, 2014

By I.H. we mean byInductiveLeaf induction hypothesis.

First year in maths students are baffled about mathematical induction(M.I.). It is counter intuitive and it is true, it is hard to wrap one’s head around this concept. So to prove some mathematical property holds we are given the following template:

  1. First show that the property P ( the property of being divisible, being prime or equal to a formula’s value, you name it etc.) is true for the base case. If the base case is c then we prove that the property is true for it. Thus we work on showing P(c) is true.
  2. Second, we assume it is true for k, that is P(k) is true – we take this as fact. This second step is called the induction hypothesis – I.H.
  3. Lastly, we prove that P(k+1) is true as well using I.H. Thus should we succeed in this final step, we have all the right to claim that we have proof that P is true for all n.

It is I.H. that is hard to take. Why is it that a.) we should assume it and b.) why is it that if step 3 succeeds provided we use the fact of step 2, we have the right to say – Q.E.D. or say “proven as required”! What right do we have in assuming I.H.

There are a few comments that we can make about this:

  • Firstly, M.I. is about the property of numbers (in general). Numbers obey this I. H. property. As a classic example consider a number x such that x > c, for some number like 8. So if we have x > 8, can we say that the next number after x will also be greater than 8? We can guess that this should be true. So let us prove it…Consider x > 8, let us add 1 to both sides still maintaining the inequality. x > 8 \Rightarrow x + 1 > 8 + 1 = 9 > 8 \therefore x + 1 > 8.
  • Secondly, we are allowed to assume I.H. for after all we can set our n = c, i.e., to our base case and check if the property P(c+1) holds — thus by the same token we are allowed to move from the truth of P(c) to the truth that P(c+1). So we can assume I.H. because of this “domino effect”. The crucial bit is to show now that due to our utilisation of I.H. for an arbitrary n = k, we get the truth of P(k+1).
  • Lastly and strongly, we can take M.I. to be an axiom! Meaning, a proposition which is self-evidently (if I can use that word) true! Indeed Wikipaedia has it like this – \forall P[P(0) \wedge \forall k \in \mathbb{N}[P(k) \Rightarrow P(k+1)]] \bold{\Rightarrow} \forall n \in \mathbb{N}[P(n)].
  • Take a good look at this and consider the statement before the last \Rightarrow. If you look we have this form A \Rightarrow B. Remember modus ponens? It says if we have A \Rightarrow B and we have A, deduce B.

Well when we are doing M.I. what we are actually doing is that we are establishing the truth of A and when we succeed – voila, use this with the axiom and so conclude the property P holds for all n.