Complex numbers, unlike real numbers, can’t be directly compared using greater than or less than symbols. This article explains why comparing complex numbers like 2 + 3i and 1 – 4i with inequalities is fundamentally different from comparing real numbers.
Understanding Ordered Systems
The key difference lies in the concept of an “ordered system.” An ordered system has a subset, often called ‘P’ (for positive numbers in the real number system), that follows two rules:
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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. In real numbers, either a number is positive or negative, but not both.
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Closure under Addition and Multiplication: If two numbers are in P, their sum and product must also be in P. For example, with positive real numbers, adding or multiplying them always results in a positive number.
Why Complex Numbers Can’t Be Ordered
The complex number system doesn’t have a subset that fulfills these rules. Let’s explore why, using the imaginary unit ‘i’ (where i² = -1):
Imagine if ‘i’ belonged to the hypothetical positive subset P. Following Rule 2, i i = -1 would also be in P. Then, (-1) i = -i would also have to be in P. But this violates Rule 1, as both ‘i’ and ‘-i’ can’t simultaneously exist in P.
The same contradiction arises if we assume -i belongs to P. Following the same logic, (-i) (-i) = -1 would be in P, and therefore (-1) (-i) = i would also be in P, leading to the same violation of Rule 1.
This fundamental contradiction demonstrates why a subset P satisfying the ordered system rules cannot exist within the complex number system. Consequently, comparing complex numbers using inequalities like ‘>’ or ‘<‘ is not meaningful.
The Magnitude of Complex Numbers: A Different Comparison
While direct comparison using inequalities is impossible, the magnitude (or modulus) of complex numbers offers a way to compare their distances from the origin on the complex plane. The magnitude of a complex number a + bi is calculated as √(a² + b²). This provides a real number value representing the distance, enabling comparisons of their magnitudes.
Conclusion
Comparing complex numbers using inequalities doesn’t work because the complex number system lacks the properties of an ordered system. The concept of positive and negative numbers, fundamental for comparison in real numbers, doesn’t directly translate to the complex plane. However, using the magnitude of complex numbers offers a viable alternative for comparing their distances from the origin. This distinction highlights a key difference between real and complex number systems.
