Well, there was something(know not what) that kept my thoughts inversed today (Perhaps, I suspect the involvement of my Algorithm Teacher's "preachings" on the complexities and the treasures that are hidden in these problems!or on the flip-side it might also be due to lack of tests and assignments this week!).So, today I felt that I had to get some thoughts in and extract some really serious inverse algorithms,if not for algorithms that are worthy enough for a patent;). And guess what?!, the thoughts did flow and the thought flow did work!- and what followed that is this post which forms the basis function to an extrapolation of these little seemingly brilliant myriad "thought flows"...

Imagine how much easier it would be to succeed and be happy in life if you were constantly expecting the world to support you and bring you opportunity. Successful people do just that. This sixth step for creating the life you want, Become an "Inverse Paranoid", isn't about being self-centered, it's about being self-confident.

Too often people who believe in themselves and their purpose (see yesterday's post for more on finding your life's purpose) are sometimes labeled as arrogant or just plain cocky and stuck-up. The opposite couldn't be truer. Those of us who expect great things out of the world, each other, and ourselves realize the importance to focus inwards to build and create the necessary energy to tell the world what we want and to help others do the same. As a result, Inverse Paranoids are not afraid to speak their minds or stand up for their beliefs and for what they want out of life.

Typically the word "paranoid" means suspicious or fearful. In this context, however, the word takes out a whole different meaning. Truly paranoid people, in the sense we usually think of, are always expecting something bad to happen to them. In terms of creating the life you want, Inverse Paranoids are expecting good things to happen to them. They expect to have strong relationships and nurturing families; they expect rewarding careers; they expect to be happy and successful. Why? Because they have told the Universe this is what they want and they are taking the necessary steps and doing the necessary things to make it happen.

You can easily spot Inverse Paranoids. These are the people who take the time to improve themselves. They read, take courses, and attend lectures and seminars. They reach out to others who have achieved a particular success in order to soak in all they can from the people who are accomplishing the same things they want to achieve.

The encouraging point here is you can learn to become an Inverse Paranoid. Start by believing you deserve to have good things happen to you. Put yourself out there and begin to do the things successful and happy people do. It's about taking time for you and investing time in yourself. However, it can be hard to do. Early on it may feel like you are being selfish and self-absorbed. But, as you begin to make progress and learn to balance all of the demands for your time in a more productive manner, it will become more natural.

After all, by taking the time to improve yourself will have positive, residual effects for everybody in your life. From an inverse focus, you can tap into your life's desire and passion, expect it to happen, and then share it with any one you choose. That, my friend, is not selfish at all. It is a gift.

I happened to have a trade-off rant today in my Algorithms Programming Application Software! class (*Now aint that exotic?!*), I got a metaphor involving the undescribable numbers for my teacher. An interesting confusion came up from him in the comments about just what that meant. Instead of answering it with a comment, I decided that it justified a post of its own. It's a fascinating topic which is incredibly counter-intuitive. To me, it's one of the great examples of how utterly wrong our intuitions can be.

Numbers are, obviously, very important. And so, over the ages, we've invented lots of notations that allow us to write those numbers down: the familiar arabic notation, roman numerals, fractions, decimals, continued fractions, algebraic series, etc. I could easily spend months on this blog just writing about different notations that we use to write numbers, and the benefits and weaknesses of each notation.

But the fact is, the vast, overwhelming majority of numbers cannot be written down in any form.

That statement seems bizarre at best. But it does actually make sense. But for it to make sense, we have to start at the very beginning: What does it mean for a number to be describable?

A describable number is a number for which there is some finite representation. An indescribable number is a number for which there is no finite notation. To be clear, things like repeating decimals are not indescribable: a repeating decimal has a finite notation. (It can be represented as a rational number; it can be represented in decimal notation by adding extra symbols to the representation to denote repetition.) Irrational numbers like π, which can be computed by an algorithm, are not indescribable. By indescribable, I mean that they really have no finite representation.

As a computer science guy, I naturally come at this from a computational perspective. One way of defining a describable number is to say that there is some finite computer program which will generate the representation of the number in some form. In other words, a number is describable if you can describe how to generate its representation using a finite description. It doesn't matter what notation the program generates it in, as long as the end result is uniquely identifiable as that one specific number. So you could use programs that generate decimal expansions; you could use programs that generate either fractions or decimal expansions, but in the latter case, you'd need the program to identify the notation that it was generating.

So - if you can write a finite program that will generate a representation of the number, it's describable. It doesn't matter whether that program ever finishes or not - so if it takes it an infinite amount of time to compute the number, that's fine - so long as the program is finite. So π is describable: it's notation in decimal form is infinite, but the program to generate that representation is finite.

An indescribable number is, therefore, a number for which there is no notation, and no algorithm which can uniquely identify that number in a finite amount of space. In theory, any number can be represented by a summation series of rational numbers - the indescribable ones are numbers for which not only is the length of that series of rational numbers infinite, but given the first K numbers in that series, there is no algorithm that can tell you the value of the K+1th rational.

So, take an arbitrary computing device, φ, where φ(x) denotes the result of running φ on program x. The total number of describable numbers can be no larger than the size of the set of programs x that can be run using φ. The number of programs for any effective computing device is countably infinite - so there are, at most, a countably infinite number of describable numbers. But there are uncountably many real numbers - so the set of numbers that can't be generated by any finite program is uncountably large.

Most numbers cannot be described in a finite amount of space. We can't compute with them, we can't describe them, we can't identify them. We know that they're there; we can prove that they're there. All sorts of things that we count on as properties of real numbers wouldn't work if the indescribable numbers weren't there. But they're totally inaccessible.