The Fundamentals Of Rational Numbers

I was never really taught the fundamentals of rational numbers in school. The more that I think about it, the more convinced I get that this is the sort of stuff that one cannot be taught.

It is something that one ponders upon; something that one arrives on one’s own mathematical journey; something that one seeks and not the other way around. And here you are!

In this essay, I will cover the fundamental properties of what we call rational numbers in mathematics. I will do so using fundamental arithmetic reasoning as well as fundamental geometric intuition. None of this is beyond any average human being.

This will also serve as a strong basis for the concept of the Dedekind cut. Why don’t we begin by discussing the fundamentals of numbers?

This essay is supported by Generatebg

The Fundamentals of Numbers

The entire business of arithmetic or even that of numbers arises from our need for counting. Counting is so fundamental to life, that it is not even exclusive to human beings; several other species are capable of basic counting. It is just that humans take it to the next level.

Well, as it turns out, there are several levels to ‘counting’. But only some human beings reach the upper levels.

One such human being was Richard Dedekind. Throughout this essay, I will be building upon some of the upper levels established by Dedekind. Now, back to counting.

What is Counting, Really?

Dedekind defines counting as follows:

Counting is “…nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding.”

— Richard Dedekind.

It is safe to say that Dedekind was a strong abstract thinker. But alas, those words might be too cumbersome for many. So, let me try and unpack what he means.

Among other things, what Dedekind means is, counting is an act in which you start with some arbitrary starting number (say, zero) and keep adding 1 for every time you come across a unique occurrence of the element you are tracking.

The type of element that you are counting is sometimes materialistic and sometimes abstract.

We have already established that counting is not exclusive to human beings. What, then, is the first arithmetic concept that is exclusive to human beings?

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Arithmetic Beyond Counting – The Fundamentals of Rational Numbers

The concept of counting serves as a remarkable tool for human beings to create an inexhaustible set of abstract laws that are exclusive to them. Central to this abstract set of arithmetic laws are the four fundamental arithmetic operations.

Addition is quite simple. If you have already counted the number of elements in two separate groups, you may combine these two numbers to arrive at a collective count. Multiplication is an advanced form of addition.

Subtraction does the opposite of addition. Let us say that you have already counted the number of elements in two separate groups.

Then, the subtraction operation tells you how much bigger or how much smaller one group is in comparison to the other. Division is an advanced form of subtraction.

So far, so good, right? Well, this is where things start getting weird. Addition and multiplication are always possible. However, subtraction and division are not always possible. Human beings seem to be the sole species to have arrived at this level of abstraction.

So, what is special about subtraction and division that renders them inaccessible to other species? You see, as it turns out, to overcome the limitations of subtraction and division, we needed to create further abstract concepts such as negative numbers and fractional numbers!

Since we have come this far, we might as well just go one step further. Why don’t we create a group of numbers that includes all of the types of numbers we have so far: the counting numbers, the negative numbers, and the fractional numbers?

Let us call this group of numbers Rational Numbers and denote it using ‘Q’.

The Fundamentals of Rational Numbers — Basic Properties

As sharply noted by Dedekind, this system of rational numbers possesses the properties of completeness and self-containedness. If those words don’t make sense to you, don’t worry. I’ll unpack them now.

The first point to note is that it is always possible to perform any of the four fundamental arithmetic operations with any two members of ‘Q’. Furthermore, the result of any such arithmetic operation results in a number that is also a member of ‘Q’.

There is, however, one operation that is a strong exception. Can you guess what this is? Take your time, if you need to. The answer is: division by zero. I have written an essay that covers this topic in detail. Check it out if you are interested.

Another interesting feature of this system of ‘Q’ becomes obvious when we use fundamental geometric intuition. When we stack all the members of ‘Q’ in a row in the order of smallest to greatest, we arrive at a well-arranged one dimensional domain that extends to infinity on two opposite directions.

Furthermore, we say that two members of of ‘Q’, say ‘a’ and ‘b’, are unique if their difference has either a positive value or a negative value. If (a − b) is positive, then we say that ‘a’ is greater than ‘b’ (symbolically denoted as a > b). If (a − b) is negative, then we say that ‘a’ is lesser than ‘b’ (symbolically denoted as a < b).

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The Fundamentals of Rational Numbers — Extended Properties

The first extended property of rational numbers is as follows:

1. If a > b, and b > c, then a > c.

If you think that this is an intuitive realisation, then be warned that it is not. There are other areas in mathematics where generalisation by logical extension leads to errors and/or contradictions (perhaps a topic for another day).

For now, let us resort (again) to fundamental geometric intuition to make sense of what this property conveys. As long as ‘a’ and ‘c’ are two uniquely different numbers, if we were to arrange them in a row in the order of smallest to greatest, then ‘b’ would lie between the two numbers ‘a’ and ‘c’.

The next extended property of rational numbers, as opposed to the first one, is a logical leap in terms of complexity. It is as follows:

2. If ‘a’ and ‘c’ are two uniquely different numbers, then there are infinitely many numbers lying between ‘a’ and ‘c’.

To make sense of this, imagine two subsequent points on a ruler, and keep dividing the distance between them by two. The number would get smaller and smaller, but you would never stop dividing.

You would keep going beyond the realm of finite small numbers into the realm of infinitely small numbers, which would lead us straight to the notion of infinitesimals.

This is what I meant when I said that this property is a logical leap in terms of complexity; it requires further abstractions to make sense of.

To round up this essay, I will present one more extended property of rational numbers. This property is highly interesting and will serve as a precursor for the notion of the Dedekind cut.

A Precursor to the Dedekind Cut

Consider a number ‘a’ that belongs to ‘Q’. If this is the case, all other numbers in ‘Q’ fall into one of two classes: A1 and A2, each of which has infinitely many members.

Every number in A1 is smaller than ‘a’, whereas every number in A2 is greater than ‘a’. So far, so good, right? Now, here comes the most interesting question. If there are only two groups with the aforementioned properties, which group does ‘a’ belong to?

You see, the answer to that question is not so simple. What Dedekind proposed is that you may pick and choose as per your convenience. If you say that ‘a’ belongs to A1, then it is the greatest number in that group.

Conversely, if you say that ‘a’ belongs to A2, then it is the smallest number in that group. Regardless of which option you choose, one property remains valid: every member of A1 is smaller than every member of A2.

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Final Comments

Dedekind would later go on to extend this property to define a class of numbers that perfectly cuts the number line. This became famously known as the Dedekind cut, and in the process, Dedekind invented the notion of real numbers.

I will cover these concepts in a follow-up essay in the future. For now, I hope that you enjoyed this discussion on the fundamentals of rational numbers.

I conclude this essay with the same thoughts with which which I began. The more I ponder upon the fundamentals of rational numbers, the more I am glad that I did not learn this stuff in school.

I am not sure if I was mentally prepared back then. I am not even sure if I am mentally prepared now!

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Reference and credit: Richard Dedekind.

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Further reading that might interest you:

• Does Division By Zero Really Lead To Infinity?
• Infinite Regress: How To Really Understand It?
• The Lonely Search For Deus Ex Machina In Real Life.

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