Understanding the Cardinality of Real and Complex Numbers

In set theory, the concept of cardinality is used to describe the size or the number of elements in a set. In this article, we will explore the cardinality of real and complex numbers and understand why both sets are uncountably infinite.

Real Numbers

The set of real numbers, denoted as (mathbb{R}), consists of all rational and irrational numbers. This set is uncountably infinite, which means it is impossible to list all real numbers in a sequence. The cardinality of the real numbers is denoted as (mathfrak{c}), the cardinality of the continuum. This concept is significant because despite being infinite, the set of real numbers is larger in size than the set of natural numbers.

Complex Numbers

The set of complex numbers, denoted as (mathbb{C}), includes all numbers of the form (a bi), where (a) and (b) are real numbers, and (i) is the imaginary unit. Each complex number is a pair of real numbers. Therefore, the cardinality of the set of complex numbers is the same as the cardinality of the real numbers, which is also (mathfrak{c}).

Key Facts Explained

Let's elucidate some fundamental facts to understand why both the sets of real numbers and complex numbers are uncountably infinite:

Fact 1: Infinite Subset Implies Infinite Set

If a set (A) contains an infinite subset (B), then (A) is also infinite. This is a basic principle that helps us understand the nature of infinite sets. For example, if we consider the set of all complex numbers (mathbb{C}), it contains an infinite subset of real numbers (mathbb{R}). Consequently, (mathbb{C}) must be infinite.

Fact 2: Subset Relations Between Number Sets

The set of natural numbers (denoted as (mathbb{N})) is a subset of the real numbers ((mathbb{R})), and the real numbers ((mathbb{R})) are a subset of the complex numbers ((mathbb{C})). This hierarchical relationship further emphasizes the infinite nature of these sets:

Fact 3: Infinite Natural Numbers

The set of natural numbers (mathbb{N}) is explicitly defined as an infinite set. This fact sets the stage for understanding the infinite nature of both the real and complex number sets. Since (mathbb{N} subset mathbb{R} subset mathbb{C}), it follows that both (mathbb{R}) and (mathbb{C}) are infinite sets.

From these three facts, we can conclude that both the set of real numbers and the set of complex numbers are uncountably infinite, each with the cardinality (mathfrak{c}). This means that both sets cannot be put into a one-to-one correspondence with the set of natural numbers, highlighting the vastness of these infinite sets.

Conclusion

In summary, the sets of real numbers and complex numbers share the same cardinality and are each uncountably infinite. The concept of cardinality in set theory elegantly captures the size of these infinite sets, providing a deeper understanding of the nature of real and complex numbers.