In this post, we discuss the fundamental properties of real numbers. In the post on types of numbers, we briefly discussed different kinds of numbers including real numbers. However, we need to look at real numbers from another perspective. It is because real numbers have unique properties that are important in real analysis. Recall that real analysis is a branch of mathematics that studies the behavior and properties of real numbers and real functions.

#### The field of real numbers

By definition, a set \(K\) equipped with two binary operations addition “\(+\)” and multiplication “\(\cdot\)” is a field if the following conditions are satisfied for all \(a\), \(b\), and \(c\) in \(K\):

- \(a+b \in K\).
- \((a+b) + c = a + (b+c)\).
- There is a special element \(0\) in \(K\) such that $$a+0 = 0 + a = a.$$
- There is an element \(-a\) in \(K\) with the following property: $$ a + (-a) = -a + a = 0.$$
- \(a+b = b + a\).
- \(a \cdot b \in K\).
- \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
- There is a special element \(1 \neq 0\) in \(K\) such that $$a \cdot 1 = 1 \cdot a = a.$$
- If \(a\neq 0\), then there is an element \(a^{-1}\) in \(K\) with the following property: $$ a \cdot a^{-1} = a^{-1} \cdot a = 1.$$
- \(a \cdot b = b \cdot a\).
- \(a \cdot (b+c) = a \cdot b + a \cdot c \).

The first essential property of real numbers is that the set of real numbers \(\mathbb{R}\) equipped with the usual addition and multiplication of real numbers is a field known as the field of real numbers.

#### The field of real numbers is an ordered field

A field \(K\) is an ordered field if there is a binary relation \(\leq\) such that for all \(a\), \(b\) and \(c\) in \(K\), we have the following properties:

- \(a \leq a\).
- \(a \leq b\) and \(b \leq c\) imply \(a \leq c\).
- \(a \leq b\) and \(b \leq a\) imply \(a = b\).
- Either \(a \leq b\) or \(b \leq a\).
- \(a \leq b\) implies \(a + c \leq b + c\).
- \(a \leq b\) and \(0 \leq c\) imply \(a \cdot c \leq b \cdot c\).

Note that by definition \(a < b \) means that \(a \leq b\) with \(a \neq b\). It is an easy exercise to see that the binary relation \(<\) has the following properties:

- \(a < b\) and \(b < c\) imply \(a < c\).
- Either \(a < b\) or \(b < a \) or \(a = b\).
- \(a < b\) implies \(a + c < b + c\).
- \(a < b\) and \(0 < c\) imply \(a \cdot c < b \cdot c\).

Though it is supposed that the field of real numbers is an ordered field, however, there are many fields containing the field of rational numbers \(\mathbb{Q}\) that are also ordered fields. In the following, we give such an example:

**Exercise**. Prove that $$K = \{a+b\sqrt{2}: a,b \in \mathbb{Q}\}$$ is an ordered field.

#### The concepts of infimum and supremum

In this section, we introduce the important concepts of infimum and supremum.

Let \(S\) be a subset of an ordered field \(K\). By definition,

- A lower bound of \(S\) is an element \(a \in K\) such that \(a \leq x\), for all \(x \in S\).
- A lower bound of \(S\) is an infimum of \(S\) if \(y \in K\) is a lower bound for \(S\), then \(y \leq a\). In other words, an infimum is the greatest lower bound.
- An upper bound of \(S\) is an element \(b \in K\) such that \(x \leq b\), for all \(x \in S\).
- An upper bound of \(S\) is a supremum of \(S\) if \(z \in K\) is an upper bound for \(S\), then \(b \leq z\). In other words, a supremum is the least upper bound.

A subset \(S\) of an ordered field \(K\) may not have an infimum or a supremum. However, if \(S\) has an infimum (a supremum), then it is unique. The infimum (the supermum) of a set \(S\), if it exists, is denoted by \(\inf(S)\) (\(\sup(S)\)).

We also need the following definitions:

- A subset \(S\) of an ordered field \(K\) is bounded above by an element \(z\) if \(z\) is larger than or equal to all elements of the subset \(S\); in other words, for all \(s \in S\), we have \(s \leq z\).
- A subset \(S\) of an ordered field \(K\) is bounded below from an element \(y\) if \(y\) is smaller than or equal to all elements of the subset \(S\); in other words, for all \(s \in S\), we have \(y \leq s\).

#### The field of real numbers has the least-upper-bound property

The least-upper-bound property states that any non-empty subset of real numbers that is bounded above has a least upper bound in real numbers.

Note that the field of rational numbers is an ordered field but it does not have the least-upper-bound property because the following set $$S = \{x \in \mathbb{Q} : x^2 < 2 \}$$ is bounded above but it does not have a supremum in rational numbers (prove it!).