Square of Integer: A Different Perspective

[sujoykroy]

Square of integer is quite easy, childlike stuff. But there is no harm in seeing a known thing in different lights. Experimenting on any thing is always fun, at least initially. So, while reading some stuff on semiconductor physics I came to think about viewing square of integer in different perspective. Following is the description what went in my head.

Assume, x and y represents occurrence of two distinct actions. Also, assume that x and y are related in such a way that if x occurs 2 times, y occurs 4 times, if x occurs 3 times, y occurs 9 times. In other words, y = x²

Square is a special case of multiplication. We can say or write that, y = x * x.

Now, if we forget about Mathematics in its present form, it is quite easy to understand a fact that occurrence of some action A can cause occurrence of another action B in such a way that B will occur as many times as A occurs. Or in simple Mathematical term, we can write (occurrence of B) = (occurrence of A) or simply B = A.

But it is bit hard to grasp a fact like y = x * x. How it may happen that occurrence of A causes occurrence of B as many times of occurrence of A as A actually had occurred itself. It seems like some other thing is also happening apart from A and that unknown thing is influencing the occurrence of B, too.

If we write y = x * z and z = x, it also yields same result as y = x * x does if we assume z as the occurrence of a third thing in such way that its occurrence is influenced by x, and z itself influences occurrence of y.

So, y = x² may signify the following fact.
Occurrence of x causes occurrence of z in such a way that z occurs as many times as x occurs.
And occurrences of y is dependent on x and z in such a way that y happens as many times as x occurs if z was not there at all or y happens as many times as z occurs if x was not there.

So, if we find a situation in measurement where we find a relation involving square of something, we can think (as there is no harm in thinking) that may be we are missing one influential factor. And unknown factors drive human crazy :)

Since square of integer is discussed, its worth mentioning viewing square of 1 with different eye. For y = x * x = x * z, it means if x happens 1 time, z also happens 1 time and y happens 1 time. Well, it is more weird than to understand 3² =9 with the light of above way of thinking. Though I am not entirely convinced myself, but I think we can view 1²=1 fact in the following way.

Making relation between occurrence of x and y will involve some sort of measurement. Measurement is just comparing something with some another thing or simply called unit. But before measuring something you need to first observe the occurrence of something. Without first observing something it is hard to be enthusiastic to go for measuring that thing. 1 may represent the very start or basis or unit of measuring of that observed thing. So, we can think that “if x=1 then y=1” may represent that very beginning of measurement.

 

[Gib Z]

Indeed if we think of x and y to be the occurences of actions, the number of which times these actions are performed as you described, it seems remarkable that any sort of machine that performs these actions would be able to somehow "know" how many times to perform the required action. But it is indeed possible without the machine having any special mind of its own, or without "It seems like some other thing is also happening apart from A and that unknown thing is influencing the occurrence of B, too." happening as well.

How this is done though is not very easy to explain to hasnt learned about Mathematical Logic, or Logical Systems. In Logic we have imaginary machines called Turing Machines which, from some valid inputs, can give some output that we can read and interpret. It produces this output by reading the inputs and performing a basic set of predefined instructions.

Basically you feed a turing machine a long tape of paper, which is blank except, for example, n 1's written on it. Many Turing machines can be constructed that, following a predefined set of simple instructions, can replace the n 1's on the tape with n^2 1's.

And the machine has no mind of its own, but since you don't know actually how simple the instructions Turing machines perform are, they go something like this: Read the first letter, if its A replace with B, if its B replace with A, move right to next letter. If its A replace with B, if its B replace with A etc etc.

I won't be able to tell you more about how exactly a Turing machine could manage to perform n^2 actions after seeing n of them without confusing you unfortunately, but I hope this sort of answers your question and perhaps interests you enough to read up on mathematical logic.

 

[sujoykroy]

I won't be able to tell you more about how exactly a Turing machine could manage to perform n^2 actions after seeing n of them without confusing you unfortunately, but I hope this sort of answers your question and perhaps interests you enough to read up on mathematical logic.

“But what is a Turing machine? It is, indeed, a mathematically idealized computer (the theoretical forerunner of modern general-purpose computer) – idealized so that it never makes any mistake and can run on for as long as is necessary, and so that it has an unlimited storage space” - Roger Penrose, “shadow of the mind”, page 17.

Thanks for raising the the point about Turing machine and formal logic. If you place “n” number of similar coins in a vendor machine and get “n*n” number of cold drinks bottles, it indicates that the vendor had put some logic into it by the help of some professional programmers.

Machines are generally made to perform a known or desired action repeatedly. You put the logic into it to get a desired equation be working.

But things go somewhat difficult when you are trying to understand a physical phenomenon and trying to find the correlation between different parameters which are involved in that phenomenon.

It is kind of stupid to think about querying for an unknown influential factor in the formula of PI*r* r for area of a circle with radius r. If anybody tries make a couple of measurement of area of circle for some values of radius, he can find out those measurement are related by that formula. And nobody is interested why a square is involved there. But you can think of the fact of the area enclosed by circle in a following manner,

Take a rope and fix one end of it on the ground and hold the other end and go far as you can so that the rope is straight enough. Now, start walking from that location in such a way that the other end of rope does not get loose and the rope remains straight until you reach the location from where you have started. This is very basic way of drawing a circle. While walking you put some grains of white powder in a small can along the length of the rope onto the ground. At the end of the walking a full circle, you get your circle filled with white powder. And you count how many times you put the grain in the small can and write them down Then you try same action with double of the size of the initial rope. Then you apply the $$\Pi$$*r*r to correlate the values of rope size and can numbers and you find wow! the formula is quite right. But I would say I am missing something :). After scratching my head few times, I will find that circumference of the circle of 2*$$\Pi$$*r . So, may be the area covered (the number of cans to fill the circle) is a multiplication combinational influence of radius and circumference. Even if I don't think this way or may be this way of thinking is stupid, the fact $$\Pi$$*r* r doesn't get disturbed.

Now, if you can machine where you give the value of rope length and it will give as much cans as necessary to fill the circle enclosed by it.

Logic can be made out of anything and no formal logic will ever be self-complete (Gödel agreed with me).

Moreover everybody is free to himself to think anything in any direction. And experimenting yields exciting stuff, which is more true in the field of Mathematics itself without even thinking about if a Mathematical operation has any meaning in real day-to-day life at all. But if somebody finds that an apparently imaginary stuff of Mathematical operation can actually explain some real world problem, life becomes interesting then :)