Real Numbers Demystified

Prologue

I remember a dialogue I had with a member of the Mathematics community on Orkut about the concept of real numbers. This was around 4-5 years ago and he was one of the really intelligent members out there in that community and had just joined IIT Bombay. To protect his identity lets call him by the name X. So we have a discussion which goes something like the following:

Me: So you have been doing great solving the math problems posed by the community.
X: Sort of. I really love solving problems.
Me: Looking at the depth of your skills, I suppose you must have some idea of real numbers.
X: Oh yes, they are just below the Complex Numbers in the number system hierarchy.
Me: Yeah that's fine, but what would you tell about real numbers to a guy who does not know about the complex numbers?
X: I would say that the real numbers are "the rational numbers and the irrational numbers" taken together.
Me: And then what would say about the irrational numbers?
X: Irrationals are just those real numbers which are not rational.
Me: If you look carefully at what you said, you will notice that there is a circularity involved and you have not defined any of the terms "irrational" and "real" in context of numbers.
X: Yes I guess that you are correct, but I don't know how we can avoid that circularity. We have never been told otherwise about real numbers.

The above dialogue sums up the understanding about real numbers of (some of the brightest) students who have completed their 12th grade and are going for undergraduate education. And as you might have understood from the circularity mentioned in the dialogue, this is in fact no understanding at all of the concept of real numbers. Their extent of knowledge of irrational numbers consists of the fact that certain surds are irrationals and few isolated but important numbers like $ \pi$ or $ e$ are irrational.

In this series of posts I will try to present a rigorous conception of the real numbers in a way which avoids the above circularity. (Some readers who are already familiar with the theories of real numbers would be mad at this point: Not one more silly post on this dry subject. To pacify them all I have to say is: "Please bear with me. Almost all the exposition of real numbers found in books and online treat the topic in a very high handed manner. They expect the readers to be already familiar with the concepts of limits and calculus in general and then fill the gap about real numbers. I would be doing quite the contrary. I am expecting that my readers are not aware of any calculus concepts and they have a vague idea of reals and irrationals as expressed in the dialogue. I strongly believe that a true conception of real numbers is a must before we jump on to the machinery of calculus).

In what follows I would assume that the readers are familiar with the rational numbers and the operations defined on them. In particular I would stress on the following two properties:

1. There is no least positive rational number. In other words given any positive rational number we can find another positive rational number which is less than the one specified in the beginning.
2. Between any two given rational numbers we can find another rational number. By repeating this argument as many as times as necessary we can find as many rationals as we please between any two given rational numbers.

That I have to state above seemingly obvious properties of rational numbers needs to be justified. The discussion which follows will be of a completely different character than is usually encountered while learning rational numbers. The focus will be on the order relations (of "less than" and "greater than") rather than on the algebraic operations of addition, subtraction etc. Another point which needs some more elaboration is about the use of the phrase "as many as we please". This is a shorthand way of saying "more than any arbitrarily specified positive integer". We will be using similar phrases and we will provide a clear explanation of such a phrase when it is used for the first time.

To begin with we need to start with the rudimentary idea of an irrational number which students have in mind at 10th grade and we will proceed with the simplest example $ \sqrt{2}$.

What is the number $ \sqrt{2}$?

We assume the reader is quite familiar with the fact that there is no rational number whose square equals $ 2$. And therefore if we want to accept $ \sqrt{2}$ as a number, it would have to be quite different from a rational number. Now why do we want to have a number of this kind? There is some requirement of it from geometry if we want to express the diagonal of unit square in terms of a numerical value. This is the way historically such specific numbers like $ \sqrt{2}, \sqrt{3}$ etc. came into being.

But we still have no other properties about $ \sqrt{2}$ except $ (\sqrt{2})^{2} = 2$. So we need to focus on some of the properties we commonly attribute to numbers in general. The idea of numbers (integers and rationals alike) arose from two processes: counting and measuring. Inherent in the process of measuring a magnitude was the concept of order relations. We could compare magnitudes in terms of "larger" and "smaller" and hence the integers and rationals numbers were endowed with the concepts of $ <, >$. If we really wish to accept $ \sqrt{2}$ as some number capable of measuring a magnitude (like the diagonal of unit square), we must also be able to compare it with existing numbers which are used to measure magnitudes. Hence we must be able to compare $ \sqrt{2}$ with the existing rational numbers.

Now thats an easy task if we remember the decimal expansion of $ \sqrt{2}$ as follows: $$ \sqrt{2} = 1.4142135623730950488016887242097\cdots $$ Again this is a vague expression because till now we don't have a clear conception of $ \sqrt{2}$ as a number let alone its decimal expansion. But still this is going to help us. By the above expansion we have to understand that $ \sqrt{2}$ should be greater than the rationals $ 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, \cdots $ and at the same time it should be less than $ 2, 1.5, 1.42, 1.415, 1.4143, 1.41422, \cdots$. In fact since we wish to have $ (\sqrt{2})^{2} = 2$, we should also be ready to accept that $ \sqrt{2}$ should be greater than any positive rational whose square is less than $ 2$ and at the same time should be less than any positive rational whose square is greater than $ 2$. In other words if we have rationals $ 0 < a < b$ such that $ a^{2} < 2 < b^{2}$ then we must have $ a < \sqrt{2} < b$. This requirement comes from the intuitive fact that "square of a smaller number should be smaller than the square of a greater number". Thus we are trying to preserve this fact when introducing the new numbers.

Accordingly it makes sense to divide the set of all rational numbers $ \mathbb{Q}$ into two subsets $ L, U$ such that $ L$ consists of all the negative rationals, zero and all the positive rationals whose square is less than $ 2$ and on the other hand the set $ U$ consists of all positive rationals whose square is greater than $ 2$. Our intuition about comparing magnitudes says that the number $ \sqrt{2}$ should be greater than all members of $ L$ and less than all members of $ U$.

Let's now observe the same thing on the number line. When we are dividing set of rationals into two subsets, we are effectively dividing the points of the number line corresponding to rationals into two parts. Now if the number line consisted only of points corresponding to rational numbers then we would expect that a division of the number line be achieved by a unique point such that all the points to the left of it lie in one part and all the points to the right of the line lie in the other part. The dividing point itself could lie in any of two the parts. However as geometrical arguments show the line has many points which don't correspond to any rational number, but the idea we have mentioned about the points of line being divided in two parts holds true as far as our intuition goes.

Thus we expect that if all the points of the number line are divided into two parts such that points of one part lie to the left of the points of other part then there must be a single unique point which makes this division such that all points to the left of it lie in one part and all points to the right of it lie in other part. This intuitive property of a straight line is the idea behind the continuity of the line that it has no gaps in it. We would like to have our number system expanded in a similar way to have no gaps. Thus we would want our set of number to have the following property:

If all the numbers be divided into two sets $ L, U$ such that every member of $ L$ is less than any member of $ U$ then there must be a number $ \alpha$ such that all numbers less than $ \alpha$ belong to $ L$ and all the numbers greater than $ \alpha$ belong to $ U$, the number $ \alpha$ may itself lie in one of the sets $ L, U$.

Now it is easy to see that the set of rational numbers does not have this property. And to establish this we again consider the division of rationals into $ L, U$ based on above discussion about $ \sqrt{2}$. Thus let \begin{align} L &= \{x \mid x \in \mathbb{Q}, x \leq 0\} \cup \{x \mid x \in \mathbb{Q}, x > 0, x^{2} < 2\}\notag\\ U &= \mathbb{Q} - L = \{x \mid x \in \mathbb{Q}, x > 0, x^{2} > 2\}\notag\ \end{align} But as we can see there is no rational number which makes this division into $ L$ and $ U$. For if such a rational number existed it would have to be in either $ L$ or $ U$ and in case it belonged to $ L$ it would have to be the greatest member of $ L$ (as numbers greater than it are to lie in $ U$) and if it belonged to $ U$ it would have to be the least member of $ U$. In any case we are going to show that neither $ L$ has a greatest member nor $ U$ has a least member. This is intuitively obvious if we look at the definition of $ L$ and $ U$, but a more rigorous proof is needed here.

Clearly $ 1 \in L$ and $ 2 \in U$ and we can find as many rationals between $ 1$ and $ 2$ as we please. In fact given any positive rational number $ c$ we can interpolate so many numbers between $ 1$ and $ 2$ that the gap between each of the two consecutive numbers is less than $ c$. To do this we only need to interpolate more than $ n = [1/c] + 1$ rationals between $ 1$ and $ 2$. Out of these interpolated numbers there is a last which belongs to $ L$ and the next one belongs to $ U$. So given any positive rational $ c$ we have been able to find a rational $ a \in L$ and a rational $ b \in U$ such that $ b - a < c$. Also from the construction above it is obvious that both $ a, b$ can be taken to be less than $ 2$. It follows that $ b^{2} - a^{2} = (b - a)(b + a) < c \cdot 4 = 4c$ and since $ c$ was arbitrary we could replace it by $ c/4$ to arrive at $ b^{2} - a^{2} < c$ or $ (b^{2} - 2) + (2 - a^{2}) < c$. Now both $ b^{2} - 2$ and $ 2 - a^{2}$ are positive and hence each of them needs to be less than $ c$.

We have thus established that given any positive rational $ c$ we can find a rational $ a \in L$ and a rational $ b \in U$ such that $ 2 - a^{2} < c$ and $ b^{2} - 2 < c$. It is now clear that $ L$ can't have a greatest member. For if it had greatest member $ l$ then we could chose $ c = 2 - l^{2}$ and find $ a \in L$ such that $ 2 - a^{2} < c = 2 - l^{2}$ or $ l^{2} < a^{2}$ or $ a > l$ which contradicts that $ l$ is the greatest member. Similar we can establish that $ U$ has no least member.

We have thus been able to find a division of the rationals into two parts for which there corresponds no rational number which makes this division possible and it is this fact which shows the inadequacy of rational number system in representing all the points of the number line. However it may be possible that for some kinds of division there is rational number which makes this division. For example if \begin{align} L &= \{x \mid x \in \mathbb{Q}, x \leq 0\} \cup \{x \mid x \in \mathbb{Q}, x > 0, x^{2} < 1\}\notag\\ U &= \mathbb{Q} - L = \{x \mid x \in \mathbb{Q}, x > 0, x^{2} \geq 1\}\notag \end{align} then we have a rational $ 1$ which makes this division and it is the least member of $ U$. If you see carefully this example and the example related to $ \sqrt{2}$ above you might notice the difference in definition of $ U$. We used $ x^{2} \geq 1$ here and not $ x^{2} > 1$ because if we do $ x^{2} > 1$ then we are excluding the rational number $ 1$ from the division process and that is not allowed.

Now back to the problem of making sense of the number $ \sqrt{2}$. After all the above discussion it makes sense to define $ \sqrt{2}$ as the number which makes the division of $ \mathbb{Q}$ into $ L, U$ as follows: \begin{align} L &= \{x \mid x \in \mathbb{Q}, x \leq 0\} \cup \{x \mid x \in \mathbb{Q}, x > 0, x^{2} < 2\}\notag\\ U &= \mathbb{Q} - L = \{x \mid x \in \mathbb{Q}, x > 0, x^{2} > 2\}\notag \end{align} However the only numbers we have at our disposal are rational numbers and we have seen that no rational is making this above division so we are still having a conundrum. The way out is to define the symbol $ \sqrt{2}$ as a shorthand for these pair of sets of $ L, U$ defined above in the same way that we can redefine the rational number $ 1$ as the pair of following sets: \begin{align} L &= \{x \mid x \in \mathbb{Q}, x \leq 0\} \cup \{x \mid x \in \mathbb{Q}, x > 0, x^{2} < 1\}\notag\\ U &= \mathbb{Q} - L = \{x \mid x \in \mathbb{Q}, x > 0, x^{2} \geq 1\}\notag \end{align} We formalize the whole discussion in a more general context in what follows.

Sections and Real Numbers

The peculiar way of dividing rationals into two sets was first described by Richard Dedekind in his paper Stetigkeit und Irrationale Zahlen (Continuity and Irrational Numbers) and since then it has become a standard practice. We define a section of rational numbers as a pair of non-empty sets $ L, U$ such that

• $ L \cup U = \mathbb{Q}, L \cap U = \Phi$
• $ a \in L, b \in U \Rightarrow a < b$
• $ L$ has no greatest member (this is for convenience/convention)

The set $ L$ is called the lower class of the section and set $ U$ is called the upper class of the section. The section itself is symbolically denoted by $ \langle L, U\rangle$. We now define a real number as a section of rational numbers. Now we have two possibilities:

1. The real number $ \langle L, U\rangle$ may be such that $ U$ has a least member $ u$. In this case we say that the real number corresponds to the rational number $ u$ and we can use the same symbol $ u$ to denote the real number also. This way we can identify all the rational numbers as special kinds of real numbers.
2. The real number $ \langle L, U\rangle$ may be such that $ U$ has no least member. In this case we say that the real number corresponds to an irrational number or is an irrational number.

We now see that we have been able to extend the system of numbers to include rationals and irrationals alike to form bigger system of real numbers. Our task will be completed if we can show that it is possible to define all the common algebraical operations and order relations on real numbers and that these processes satisfy the usual rules which hold good for the system of rational numbers. We will be brief in these matters however and focus on those properties of real number which distinguish it from the rational numbers.

Order Relations on Real Numbers

Let $ \alpha = \langle L_{1}, U_{1}\rangle, \beta = \langle L_{2}, U_{2}\rangle$ be two real numbers. If

• $ L_{1} = L_{2}$ (or equivalently $ U_{1} = U_{2}$) we say that the real numbers $ \alpha$ and $ \beta$ are equal and we write $ \alpha = \beta$.
• $ L_{1} \subset L_{2}$ (or equivalently $ U_{2} \subset U_{1}$) we say that $ \alpha$ is less than $ \beta$ and write $ \alpha < \beta$.
• $ L_{2} \subset L_{1}$ (or equivalently $ U_{1} \subset U_{2}$) we say that $ \alpha$ is greater than $ \beta$ and write $ \alpha > \beta$.

Under these definition it is easy to see that if $ \alpha = \langle L, U\rangle$ is an irrational number then $ \alpha$ is greater than all members of $ L$ and less than all members of $ U$. Thus we have been able to define $ \sqrt{2}$ using the means of a section of rational numbers in such a way that it is greater than all members of its lower class and less than all members of the upper class and therefore in a sense is the point which makes this division.

The real number whose lower class consists of negative rationals and whose upper class consists of zero and positive rationals is called the real number zero and is denoted by the conventional symbol $ 0$. Real numbers greater than $ 0$ are termed positive real numbers and those less than $ 0$ are termed negative real numbers.

All the usual rules of the inequalities which hold good for rationals also hold good for the reals (for example $ \alpha < \beta, \beta < \gamma \Rightarrow \alpha < \gamma$) under these definitions as can be checked easily.

Algebraical Operations on Real Numbers

To define addition of two real number $ \alpha = \langle L_{1}, U_{1}\rangle, \beta = \langle L_{2}, U_{2}\rangle$ we define a new section as follows: $$L = \{ x \mid x = a + b, a \in L_{1}, b \in L_{2}\},\,\,\,\, U = \mathbb{Q} - L$$ It is important to note here that the above division constitutes a section of rationals (i.e. it satisfied the properties stated in definition of a section) and this section defines the sum $ \alpha + \beta$ of the given real numbers $ \alpha$ and $ \beta$.

To define subtraction is is necessary to define the negative of a real number. First let $ \alpha = \langle L, U\rangle$ be an irrational number and then we define the negative of $ \alpha$ denoted by $ -\alpha$ by the following section: $$ L^{\prime} = \{ -x \mid x \in U\},\,\,\, U^{\prime} = \{ -x \mid x \in L\}$$ If $ \alpha$ is a rational number $ r$ then we define $ -\alpha$ as the section made by the rational number $ -r$. Now we can define $ \alpha - \beta$ as $ \alpha + (-\beta)$.

All the usual rules of addition and subtraction hold for the case of real numbers also. We need to mention an important result here which the reader will have no difficulty establishing: The sum of a rational and an irrational number is an irrational number. The sum of two irrationals may however turn out to be rational (i.e. adding a number and its negative gives rational zero).

To work with multiplication we first need to confine ourselves to positive real numbers only. Then let us assume that $ \alpha, \beta$ are two positive real numbers such that $ \alpha = \langle L_{1}, U_{1}\rangle$ and $ \beta = \langle L_{2}, U_{2}\rangle$. We define a new section of rationals as follows: $$ L = \{ x \mid x \in \mathbb{Q}, x \leq 0\} \cup \{x \mid x = ab, 0 < a \in L_{1}, 0 < b \in L_{2}\},\,\,\, U = \mathbb{Q} - L$$ The above section defines the product of $ \alpha, \beta$ and is denoted by $ \alpha\cdot\beta$ or $ \alpha\beta$. In order to handle the cases when numbers involved are negative or $ 0$ we define the following assuming $ \alpha, \beta$ positive: $$ 0 \cdot 0 = 0, 0 \cdot \alpha = \alpha \cdot 0 = 0, (-\alpha)\cdot\beta = -(\alpha\beta) = \alpha\cdot(-\beta)$$ Like in case of addition we need to understand that product of an irrational number and a non-zero rational number is irrational.

To define division it is necessary to define reciprocal of a real number. If $ \alpha$ is a rational number $ r$ and is non-zero then we define reciprocal of $ \alpha$ as the section generated by the rational number $ 1/r$. If $ \alpha = \langle L, U\rangle$ is irrational we need to first assume that it is positive and then we can define the reciprocal $ 1/\alpha$ of $ \alpha$ by the following section: $$L^{\prime} = \{x \mid x \in \mathbb{Q}, x \leq 0\} \cup \{x \mid x = 1/a, a \in U\},\,\,\, U^{\prime} = \mathbb{Q} - L^{\prime}$$ The reciprocal of a negative irrational number can be then defined by $ 1/(-\alpha) = -(1/\alpha)$. Reciprocal of zero is not defined and division by a non-zero real number can now be defined in an obvious manner by $ \alpha/\beta = \alpha\cdot(1/\beta)$.

Using these definitions it is possible to prove that all the common laws of algebra which are satisfied by the rationals are also satisfied by the real numbers. But we don't need to do this in elaborate detail. What we really to need to observe here is some common features about the definition of real numbers and their operations. We note that real numbers are essentially defined by infinite sets of rationals and so is the case with the operations defined on them. The irrational numbers can not be represented using a finite system of previously defined numbers whatsoever. For example if you need to add two irrationals you have to in effect add an infinity of rationals from two classes. Any finite combination of usual operations (+, -, *, /) on rational numbers will always lead to rational numbers only. Therefore to have an understanding of the real number system it is necessary to appreciate the processes of a totally different kind which involve discussions about an infinity of numbers of some kind. By the phrase "an infinity of" we mean "more than any specified positive integer".

With these remarks about the system of real numbers we will now discuss another important property of the real number system called denseness.

Denseness of Real Numbers

By denseness of the number system we loosely mean that between any two numbers of the system there lies another number of the same system. Clearly this holds for the system of rational numbers. The same property holds true for the system of real numbers too. But we would want to establish more.

Between any two given distinct real numbers there lies a rational number and an irrational number.

And by repeating the same argument there lie an infinity of rational and irrationals between any two given distinct real numbers. To establish this result we take two real numbers $ \alpha = \langle L_{1}, U_{1}\rangle$ and $ \beta = \langle L_{2}, U_{2}\rangle$ such that $ \alpha < \beta$. This clearly means that $ L_{1} \subset L_{2}$ and hence all the numbers which lie in $ L_{2}$ but not in $ L_{1}$ clearly lie between $ \alpha$ and $ \beta$. Thus we are able to find an infinity of rationals between the two given numbers.

It is bit tricky and indirect to find an irrational number between the two given numbers. Our strategy would be use the fact that $ \sqrt{2}$ defined as a section above is an irrational number and we will somehow find a rational number $ r$ such that $ \alpha < r + \sqrt{2} < \beta$. The number $ r + \sqrt{2}$ would clearly then be an irrational number lying between $ \alpha$ and $ \beta$. This is just translating the point corresponding to $ \sqrt{2}$ by a rational distance so that it falls somewhere between $ \alpha$ and $ \beta$. For clarity let us write $ \sqrt{2} = \langle L, U\rangle$ where $$ L = \{x \mid x \in \mathbb{Q}, x \leq 0\} \cup \{x \mid x \in \mathbb{Q}, x > 0, x^{2} < 2\},\,\,\,\, U = \mathbb{Q} - L$$ If $ \alpha < \sqrt{2} < \beta$ then we are done and we don't have to establish anything further. So let us assume that $ \sqrt{2} \leq \alpha$ (the case $ \beta \leq \sqrt{2}$ can be handled similarly). Let us take two rationals $ a, b$ between $ \alpha$ and $ \beta$ such that $ a < b$. We can now find two rational $ c, d$ such that $ c \in L, d \in U$ and $ d - c < b - a$. Also it is possible to take $ d < a$. Now consider the rational number $ r = b - d$ which is clearly positive as $ b - d > b - a > 0$. Also let's focus on the irrational number $ \gamma = \sqrt{2} + r$. Since $ c < \sqrt{2} < d$ it follows that $ c + r < \sqrt{2} + r = \gamma < d + r \Rightarrow c + r <\gamma < b$. Now $ c + r = c + b - d > a$ so that $ a < \gamma < b$. Since we have $ a, b$ in between $ \alpha$ and $ \beta$ it follows that we have $ \alpha < \gamma < \beta$. We thus have found an irrational number lying between $ \alpha$ and $ \beta$.

So we have established the denseness property of real numbers and now it is clearly possible to understand that there is no least positive real number.

In order to keep the length of the post to a reasonable size I need to end it here. In the next post we will discuss the properties of real number system which truly differentiate it from the system of rational numbers.

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