Real Numbers

What Are Real Numbers?

When we think about numbers, we often envision a seemingly endless sequence of digits.

However, there’s a crucial distinction between familiar numbers like fractions and integers and those that belong to a broader universe: the real numbers, denoted by \( \mathbb{R} \).

This set, which includes both rational numbers \( \mathbb{Q} \) and irrational numbers \( \mathbb{I} \), serves as a bridge between the discrete and the continuous.

How Are Real Numbers Different from Rational Numbers?

The set of rational numbers, \( \mathbb{Q} \), is infinite and ordered, but it is not complete.

For instance, there are points on the number line, such as \( \sqrt{2}, \sqrt{3}, \text{and } \pi \), that cannot be represented by any rational number.

In other words, if we plot the rational numbers \( \mathbb{Q} \) on the number line, we leave “gaps” that no rational number can fill.

A classic example is the diagonal of a square with a side length of 1. Using the Pythagorean Theorem, we find that the diagonal measures \( \sqrt{2} \). This number cannot be expressed as a fraction because its decimal expansion is infinite and non-repeating. Therefore, \( \sqrt{2} \) is an irrational real number, filling one of the “gaps” left by rational numbers.

The set \( \mathbb{R} \) goes beyond rational numbers, filling these gaps on the number line.

A defining feature of \( \mathbb{R} \) is its completeness: every point on the number line corresponds to a real number, and every real number corresponds to a point on the line.

This makes \( \mathbb{R} \) a set that fully “fills” the continuous line.

In this way, \( \mathbb{R} \) is more than just a collection of numbers—it’s a tool that allows us to explore the continuous realm, from infinitely large (positive or negative infinity) to infinitesimally small (infinitesimals).

This enables us to analyze complex phenomena and achieve a deeper understanding of the natural world.

Properties of the Real Numbers Set

Let’s take a closer look at the key properties of the set of real numbers \( \mathbb{R} \).

• Unbounded
  The set \( \mathbb{R} \) is infinite, but it is “larger” than the set of natural numbers \( \mathbb{N} \) or rational numbers \( \mathbb{Q} \) because real numbers have an uncountable cardinality. It is unbounded both below ( \( -\infty \) ) and above ( \( +\infty \) ).

  For example, take any real number \( n \in \mathbb{R} \). You can always find another real number \( M \in \mathbb{R} \) such that \( M > n \). This means \( \mathbb{R} \) has no greatest element and is unbounded above. Similarly, for any \( n \), there exists a real number \( m \in \mathbb{R} \) such that \( m < n \). This demonstrates that \( \mathbb{R} \) is also unbounded below.

• Ordered
  The set \( \mathbb{R} \) is ordered, as any two real numbers \( a \) and \( b \) can be compared to determine whether \( a < b \), \( a = b \), or \( a > b \). This order enables us to represent real numbers on a number line, where each number occupies a unique position.
• Dense
  The set of real numbers is dense because, between any two real numbers \( a \) and \( b \) (with \( a < b \)), there is always another real number \( c \) such that \( a < c < b \). While this property also applies to rational numbers, \( \mathbb{R} \) includes both rational and irrational numbers.
• Complete
  The set of real numbers \( \mathbb{R} \) is complete because every point on the number line corresponds to a real number, and vice versa. This completeness distinguishes \( \mathbb{R} \) from the set of rational numbers \( \mathbb{Q} \).
• Continuous
  From a topological perspective, \( \mathbb{R} \) is continuous because it has no “gaps” on the number line. In other words, \( \mathbb{R} \) forms a continuum.

Now, let’s explore the algebraic properties of real numbers in detail.

The set \( \mathbb{R} \) is equipped with operations such as addition and multiplication, and it satisfies the following properties:

• Closure
  The sum or product of any two real numbers is also a real number, ensuring that \( \mathbb{R} \) remains self-contained under these operations.
• Associativity
  This property guarantees that the grouping of terms does not affect the result of addition or multiplication. $$ (a + b) + c = a + (b + c) $$ $$ (a \cdot b) \cdot c = a \cdot (b \cdot c) $$
• Commutativity
  This principle highlights the symmetry of addition and multiplication. $$ a + b = b + a $$ $$ a \cdot b = b \cdot a $$
• Identity Elements
  Two special numbers, \( 0 \) and \( 1 \), serve as the additive and multiplicative identities, respectively. $$ a + 0 = a $$ $$ a \cdot 1 = a $$
• Inverses
  For every \( a \neq 0 \), there exists a real number \( \frac{1}{a} \) such that \( a \cdot \frac{1}{a} = 1 \), allowing multiplication to be undone.
• Distributivity
  This property connects multiplication and addition, enabling the simplification of algebraic expressions. $$ a \cdot (b + c) = a \cdot b + a \cdot c $$

In conclusion, the set of real numbers is a mathematical framework that goes beyond rational numbers, allowing for precise modeling of continuous phenomena.

Despite its vastness and completeness, the set of real numbers is not the ultimate numerical system. There is an even broader and more powerful extension: the set of complex numbers \( \mathbb{C} \), which expands the mathematical universe with new dimensions for calculation and representation.