Partition of real numbers – Improving upon the concept of division


The 42 partitions of 10
The 42 partitions of 10 (Source: Wikipedia)

I believe the concept of division is inadequate and that entropy in the universe screws with our ability to do division.

So by using “partition of real numbers” I intend to improve upon the concept of division. And yes I am rather confident that this is a clever bit of mathematical artistry, a token of art that proves how we shouldn’t complaisantly accept convention, not even centuries old mathematical conventions. Of course, I would be ridiculously embarrassed if it were shown that I was merely hallucinating a new ternary operator and making up silly conjectures that simply weren’t tenable to begin with. So please do comment and let me know if I ought to be embarrassed about this article. To my knowledge, other people have not used my kind of approach to partitioning real numbers and it is a claim I am making for the sake of bragging rights and being complacent.

Lets first start with the motivation for trying to partition real numbers and a quick discussion of few commonly accepted binary math operators.

Skip to “The mathematical operation called partition” to avoid the rambling hackneyed stuff about my motivations. 

Motivation and background

The concept of division must be fairly familiar to all people who have been through primary school but the process of division might also be very intuitive as we encounter daily situations where we must  take a whole entity and break it into a number of parts.

As an experiment, let’s take a sheet of paper and try to tear it up into 5 smaller parts. Yes, please try this at home. This process of dividing a sheet into 5 parts might result in pieces of paper that are 1/5th the size of the orignal sheet. However, look more closely at those pieces. Are they truly 1/5th the size of the original? Is each piece identical? Is the difference between each part truly zero?

In the practical scenario of trying to tear a sheet of paper by hand, you will see some discrepancies or errors… the pieces aren’t perfect replicas of each other. You will see that there are rough edges to each piece because of which some pieces are dissimilar when we try to overlap them. The fact is, the human mind attempts division for the sake of convenience even though it’s almost never practicable to do division. And I claim that division is only feasible in number theory while practically dividing physical artifacts results in a fundamentally different operation.

So if we don’t do the division operation when dealing with physical stuff what do we do? And would we be wrong to say that 1/5 = 0.2?

Well the process of division only tells us what the average value of each part ought to be. It doesn’t tell us the measured value of each part nor the disparity between those parts, it just tells us the expected value of a particular part. The equation 1/5 = 0.20 only tells us that each of the five parts of a whole would be expected to have the size of 0.2

This might be more clear if you think of division as an inverse operation to multiplication. Multiplication is also a binary operator which takes two arguments and applies a rule upon those arguments. The first argument indicates the instance of an entity and the second one says the number of replicas of that entity. The rule tells us that we ought to summate each of those replicas. Here the assumption is that those replicas will indeed superpose and merge into each other to create yet another entity. This is certainly true in the case of numbers and various physical phenomena. So inverting this should involve finding the number of replicas if the summated entity and the instance were known or finding the instance if the summated entity and the number of replicas were known.

If we write 4 x 3 = 12 we are saying that an instance of an entity with a value of 4 replicated 3 times and summated throughout gives an instance of an entity with a value of 12. The notion that an instance of 3 taken 4 times gives 12 is another way of reading that equation. In a more practical scenario think of it as as 4 droplets of water each with 3 units of volume (or 3 droplets each with 4 units volume) when merged together give a single droplet of water of 12 units in volume. So if we divide the big droplet into 4 parts we should get back the original instances, each of 3 unit volume. Sure that should totally work. But! It actually doesn’t work too well in practice because we simply don’t have an instrument which would ever manage to precisely measure that value at a given temperature. We would always have errors in measurement and we would merely have to accept a convention where we ignore errors after some degree of precision. Entropy in the universe seems to screw with our ability to do division.

Sometimes in the case of multiplication we don’t actually do the summation operation on a physical quantity because in our minds we are merely trying to enumerate instances to find their total count and we are using multiplication as a short hand linguistic expression. Here is what I mean… if we have 4 cars each in 3 garages we have a total number of 12 cars and we could use the notation 4×3 = 12 to express that enumeration. We wouldn’t take multiplication in this kind of a scenario to mean that a car has the value of 4 units, take three such cars and combine them to give a car with a value of 12 units.

When considering whole number counts signifying physical entities the division operation may be absolutely precise with no discrepancies because we are only dealing with abstract numbers and not actual physical phenomenon of breaking up a whole into some parts. Of course, there are many occasions where division even in abstract numerical constructs is perfectly imprecise.

What would be your answer if I were to say that there were 12 cars in total within 3 garages and asked you how many cars were in each garage? Would you really say 4?  Figuring out an answer to that problem would be probabilistic even if I told you: the maximum capacity of each garage and if having at least a single car in a garage was a limiting condition. There really could have been a number of ways in which 12 cars could be distributed within 3 garages including the arrangement [12,0,0].

You might say that distribution of entities into classes or sets is not a division operation. I guess so but I think the mental or cognitive process of distribution or breaking appart or differentiation (linguistic or mathematical) or inverting multiplication are similar… perhaps… they are the same notion at a deeper level in the mind.

So this ridiculously verbose discussion could seem very trite and patronizing perhaps more like something directed at first grade students. Well if something was being discussed in our very first grade in school then it must have been really essential and foundational, right? But things of this sort weren’t discussed back then. I am revisiting the foundations of mathematics and logic for my sake and perhaps for the sake of regearing primary education. I remember that there was the concept of “plus”, “minus”, “times” and “divide” being taught in primary school but I wish we had instead discussed the concepts of whole and part and merging and arranging and segregating during those school days. I wish we had the opportunity to deal with mathematical notions in primary school using common words rather than overtly dwelling into mathematical syntax.

In fact I used to memorize the symbols as chucks of images during primary school and if I were to see the image “2 x 3 = ?” I was supposed to draw the image “2 x 3 = 6” to avoid getting beaten or humiliated by the teacher. I really had no idea that the symbols “2” or “3” or “6” or the “x” or the “=” had individual meaning. You see, I was really not bright at all. But that is how I got through till secondary school, I mostly memorized images and associated them in my mind with other images for the sake of reproduction in exams without an understanding of how symbols within those images could possibly be associated with concepts or intuitions. I wish people had told me that mathematics is merely a language and I also wish that people stopped using corporal punishment in schools and repetitive  schemes for examinations but, I digress.

I’ve been thinking about “partition” because I do think like a primary school kid quite often and I ended up asking a very puerile question… in how many ways can I possibly divide that droplet of water or arrange those cars within those garages? Or if I take those 12 cars and chop one in half by some means and throw away one of those halves, would I then really have 11.5 cars? Wouldn’t it be 11 cars and some junk that couldn’t be defined as a car? And then because I have the habit of really comming up with convoluted ideas appart from purile ones I thought to myself… can I have a single math operator that elegantly describes a process of breaking things into its components?

Oh wait, ins’t that math operator called division? Hmmm… no… division only spits out an answer that is based on an underlying assumption of uniform distribution of identical parts.

Well… it turns out that I had run into a problem that Euler and Ramanujan were medling with. And over the ages most probably every mathematician has run into this problem at some point in their life but till date the speculations and explorations regarding partition have been confined to integer numbers.

The mathematical operation called partition

Just go ahead and read all this stuff: List of Partition Topics

Ok, maybe reading all that will be painful so instead brush through these two things so that you get a reasonable idea about the concept of partition:

  1. Partition of integers
  2. Composition of integers

Partitioning an integer basically involves finding all possible ways of adding up to that number using integers. The ways can include repetition of integers. For example, partitions of the number 4 would be like this

List 1: p(4) = 5
  1. 4 = 4 (whole taken as is; lets think of this is as discrete uniform distribution)
  2. 4 = 3 + 1 (whole taken in two dissimilar parts and there is something like a gama distribution amongst parts)
  3. 4 = 2 + 1 + 1 (whole taken in three parts where there is something like a normal distribution amongst parts if we look at it as 1,2,1 instead of 2,1,1)
  4. 4 = 2 + 2 (whole taken in two equivalent parts; discrete uniform distribution amongst parts)
  5. 4 = 1+1+1+1 (whole taken in 4 equivalent parts; discrete uniform distribution amongst parts)

So there are a total of 5 ways of adding up to 4 using integers. Counting the number of ways in which an integer can be partitioned is called the number’s “partition function” and its notation is p(4) = 5.

But partition of integer only deals with problems like: number of ways of distributing cars into a garage. What about problems like: number of ways of breaking appart a droplet of water?

Partition of real numbers and a ternary operator

Well, intuitively one would say that there are infinitely many ways of partitioning a given real number into a list of other real numbers. This is where my mathematical artistry come in. If I consider an assumption that constrains the number of ways in which the partitions can be generated then I could have a practical means to partition a real number.  So I have this notation:

p(m,n,'distribution')

Where one attempts to partition a real quantity m into n parts conforming to a given probability distribution. This notation is merely a “partition function” taking two constraining conditions or arguments: one is a simple number but the other is a function in itself.

I am also thinking about an operator called the partitioner… I would write it like this

d
m |- n

And that would be read as, “m partitioned into n given d distribution”

Comparing partition of integers and of real numbers

Let’s do this by revisiting the example of integer-partitioning of 4 i.e. p(4) = 5 as shown in List 1

If I were to ask the question, “How many ways are there to integer-partition 4 yielding two parts?” the answer would be…

p(4,2) = 2

In the above notation I’m passing an argument to the integer-partition function which sorts out specific ways amongst all the possible ways of integer-partitioning 4. So p(4,2) = 2 because amongst the  5 ways in List 1, two of them (2nd and 4th) conform to our condition.

So what would p(4,100) mean?

Well it wouldn’t have meaning in integer-partitioning because of the way it is defied which merely returns the maximum number of ways of adding up to an integer using integers. Perhaps if we considered real numbers… would then a notation like p(4,100) have any meaning? Most probably it would have little useful meaning because there could be countless sets of 100 parts (real numbers) that could add up to 4.

So let’s think of another way of constraining integer-partition function. What if I asked the question, “How many ways are there to integer-partition 4 such that the parts have discrete uniform distribution?” The answer would be…

p(4,’discrete uniform’) = 3

This is because the 1st, 4th and 5th ways in List 1 conform to our condition. If you disagree with the idea that taking a whole as itself  is an example of discrete uniform distribution (the 1st way in List 1) then…. uh… fine… p(4,discrete uniform) = 2.

However, I don’t think something like p(4,exponential) would work out because I don’t think an exponential distribution would be a “best fit” on any of the ways described in List 1. Maybe for some other integer k, p(k,exponential) would exist. And certainly, when considering a real number m, a countless number of sets – each with arbitrary number of elements  summating to m – would exist such that those elements conform to the given distribution.

But if we combine the two constraints we get something more reasonable. And it turns out that division of quantity m into n parts is a special case of partitioning:

(m / n) is equivalent to p(m,n,discrete uniform)

One could explore other ways of partitioning a real number such as p(m,n,normal) or p(m,n,binomial) and so on.

Why is all this stuff significant?

As a behavioral and cognitive scientist I wonder how a certain brain function exists and how come a human behavior arrises. We do seem to take pride in our cranial functions that yield so called logical behavior. Whenever I have asked adults to tear a sheet of paper into certain parts I see them going to excruciating lengths to make sure that the individual parts are all equal. Essentially they are attempting to divide the sheet of paper. When I give the same experiment to very young kids, they simply tear it up into arbitrary portions of the required number of parts and don’t bother to do division unless explicitly asked to do so. The kids are very fast and they fulfill the given instructions exhibiting no amount of frustration. The kids seem to be much more logical I’d say.

I have shown this to university professors at Central University of Hyderabad, done the experiment of tearing a sheet of paper with them and asked them whether the sheet divided into equivalent portions of 5 parts or partitioned into arbitrary portions of 5 parts are the same thing? Yes, the consensus was that these are indeed different things, different processes for that matter which most probably evoke different brain functions. But I think it is the same brain function that exhibits itself in different behavior. And when I asked them why they assumed that there had to be an equivalence amongst the parts when they had to tear the sheet into some parts they couldn’t provide an answer. Even I don’t know why I used to do division when I could have simply done partition of a sheet of paper. I think the desire to preserve constant proportions isn’t directed by a conscious decision.

My guess is that this unconscious desire is a force of habit; an example of how going through schooling has made us think in limited or so called standardized ways. We were forced to learn arithmetic in school through limited prototypical examples rather than limitless thought experiments. So now we know how to use just those few ways of dealing with things that conform to those prototypical examples. The option to learn something through independent speculation is typically available only in adult life where it might be too difficult to learn new things… too difficult because of interference caused by all the things that were hardcoded into us in our formative years.

My experience is that in all these years I did not notice how division in practical life is merely an approximation where we ignore the discrepancies between the parts obtained through arbitrarily fixed partitioning of real values. The discrepancies also arise because of the way we have come to represent quantities. Whether it is base 10, base 2, base 5, base 60 or any other numeral system the numbers that aren’t a prime factor of the base cause problems during division. For instance 1/3 is 0.3333… (recurring) in base 10 system. We deal with such problems by assuming a degree of precision that is suitable for the physical scenario. I am suggesting one more way which involves assuming a probability distribution that is suitable for the physical scenario.

How we make use of division effects our daily transactions as it impacts our notion of fairness and justice. If we want to be more deeply concerned with things like distribution of resources or evaluation of credit ratings we should adopt a perspective involving partition rather than division.

I suspect that there might be cognitive underpinnings that lead us to choose division over partition and approximations or thumb rules or heuristics that eventually impact our notion of fairness, equity and justice. I haven’t yet figured out a good way to investigate this speculation but I am confident that I will need to use partition of real numbers while trying to explain how someone would cut up a pie to serve it to guests composed of friends and mere acquaintances. I think the friends would end up getting bigger pieces if the pie was supposed to be good.

Partitioning numbers and looking for patterns amongst the ways in which numbers can be partitioned has lead to some interesting discoveries: Scientific American article about Ono et al.’s work

I’d say partitioning has practical applications. I started dealing with partition while attempting natural language processing where “1/5th of a word” has no sensible meaning. However, I found that by asserting some assumptions a better way of breaking a word into its “components” was feasible. And then I tried to do partitioning with general quantities but kept getting the same trivial answer of “infinite ways to partition” till I managed to adopt probabilistic methods of partitioning.

I hope that people will use partition of real quantities to arrive at fractal relations as opposed to using division to arrive at  fractional relations. This way they can see the intricate symmetries and beauties linking artifacts of the real world; see the fractal links convolving through countless dimensions.

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