The real number system allows for straightforward comparisons using inequalities like “greater than” or “less than.” But Can We Compare Two Complex Numbers in the same way? The answer is no. This article explores why comparing complex numbers using inequalities is not possible.
Understanding Ordered Systems
A number system is considered “ordered” if it has a subset, often called “P,” that meets specific criteria:
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Exclusivity: For any non-zero number in the system, either the number itself or its negative counterpart belongs to P, but not both. This ensures a clear distinction between positive and negative values.
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Closure under Addition and Multiplication: If two numbers belong to P, their sum and product must also belong to P. This maintains consistency within the positive subset.
In the real number system, the set of positive numbers fulfills these conditions. If the difference between two real numbers (x – y) is positive, we say x is greater than y (x > y).
Why Complex Numbers Can’t Be Ordered
The complex number system, unlike the real number system, lacks a subset that satisfies the requirements of an ordered system. Let’s consider the imaginary unit “i” to understand why.
If we assume “i” belongs to the positive subset P, then according to the second rule of ordered systems, i i = -1 should also be in P. Consequently, (-1) i = -i would also belong to P. However, this contradicts the first rule, which states that only one of i or -i can be in P.
The same contradiction arises if we assume -i belongs to P. This leads to the conclusion that no subset P can exist within the complex number system that satisfies the conditions of an ordered system.
Comparing Magnitudes: An Alternative
While we cannot directly compare complex numbers using inequalities, we can compare their magnitudes or absolute values. The magnitude of a complex number represents its distance from the origin on the complex plane. This comparison is based on real numbers and therefore allows for ordering.
Conclusion
The fundamental difference in structure between real and complex number systems prevents direct comparison using inequalities. The concept of order, as defined for real numbers, doesn’t apply to complex numbers. However, comparing magnitudes offers a way to assess the relative size of complex numbers based on their distance from the origin. This distinction is crucial in understanding the unique properties and applications of complex numbers in various fields of mathematics and beyond.
