The real number system (ℝ) and the complex number system (ℂ) share many similarities. However, a key distinction lies in the concept of order. While real numbers can be easily compared using inequalities (greater than, less than), comparing complex numbers in the same way isn’t possible. This begs the question: Can You Compare Complex And Real Values in terms of order? The answer lies in understanding what makes a number system ordered.
Defining an Ordered Number System
An ordered number system possesses a subset, denoted as ‘P’, with specific characteristics:
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Exclusivity: For any non-zero number within the system, either the number itself or its negative counterpart belongs to subset P, but not both simultaneously. This ensures a clear distinction between positive and negative values.
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Closure under Addition and Multiplication: If two numbers belong to subset P, then both their sum and their product must also belong to P. This property maintains consistency within the ordered system.
In the real number system, the set P represents positive numbers. We define x as greater than y (x > y) if and only if the difference between them (x – y) is a positive number, hence belonging to P.
Why Complex Numbers Can’t Be Ordered
The complex number system lacks such a subset P, making it impossible to establish a consistent order. Let’s consider the imaginary unit ‘i’ (√-1).
If we assume ‘i’ belongs to P, then according to the second property of ordered systems, i multiplied by itself (i i = -1) should also belong to P. Consequently, -1 multiplied by i (-1 i = -i) should also be in P. This creates a contradiction, as both i and –i cannot simultaneously exist within P according to the first property.
The same contradiction arises if we assume -i belongs to P: (-i) (-i) = -1, leading to -1 (-i) = i belonging to P. Therefore, due to this inherent contradiction, the complex number system cannot have an ordered subset P, and thus, complex numbers cannot be compared using inequalities like real numbers. While we can compare magnitudes of complex numbers (distance from the origin in the complex plane), a definitive “greater than” or “less than” relationship cannot be established.
Conclusion: Comparing Complex and Real Values Based on Order
While both complex and real numbers form number systems, only the real number system possesses the properties of an ordered system. The inability to define a subset P with the required characteristics for complex numbers prevents the establishment of a consistent order. Consequently, direct comparisons of magnitude using inequalities are valid for real numbers but not for complex numbers. The difference stems from the fundamental nature of these systems, highlighting a crucial distinction in their mathematical properties.
