Why It's Wrong to Compare Complex Numbers Using '

Beyond the realm of basic real numbers, the world of complex numbers presents a unique set of challenges when it comes to ordering and comparison. A common misconception is that complex numbers can be ordered in the same way as real numbers, which is not the case. This article delves into the intricacies of comparing complex numbers and the distinctions between them and real numbers.

H1: Understanding Complex Numbers

Complex numbers, denoted by the form (a bi), where (a) and (b) are real numbers, and (i) is the imaginary unit, are fundamentally different from real numbers. Unlike their one-dimensional counterparts, complex numbers exist in a two-dimensional plane, often represented by an x-y axis. This characteristic distinguishes them from real numbers, which are one-dimensional and can be plotted on a simple number line.

H2: The Importance of Magnitude

One of the key distinctions between complex numbers and real numbers lies in how their values can be compared. In real numbers, we can easily say that (3 > 2) or (5

H3: What is Magnitude?

Magnitude, or absolute value, is a measure of the size of a complex number. The magnitude of a complex number (a bi) is given by the formula:

[|a bi| sqrt{a^2 b^2}]

Essentially, it represents the distance from the complex number to the origin (0,0) in the complex plane. This distance is always a real number, allowing us to compare the magnitudes of different complex numbers using standard real number operations.

H2: A Practical Example

Let's take the example of comparing (63i) and (42i). In the world of complex numbers, the direct comparison using () is nonsensical. For instance, is (63i [|63i| sqrt{63^2 0^2} sqrt{3969} 63] [|42i| sqrt{42^2 0^2} sqrt{1764} 42]

Since (63 > 42), we can say that the magnitude of (63i) is greater than the magnitude of (42i). This comparison is valid and meaningful because we are dealing with real numbers.

H2: Complex Numbers Can't Be Ordered like Real Numbers

Another illustration of the difficulty in comparing complex numbers is a graphical representation. Consider the following eight complex numbers plotted on the complex plane. Despite each number having a different representation, they all have the same magnitude:

Each of these numbers has a magnitude of 5 (the hypotenuse of a 3-4-5 triangle), but ordering them as we would real numbers is not possible:

[|3 4i| |4 3i| sqrt{3^2 4^2} sqrt{9 16} sqrt{25} 5]

As illustrated, (3 4i) and (4 3i) are not ordered like real numbers, even though their magnitudes are equal.

H2: The Formula and Application

Let’s apply this concept to a specific example. Consider the complex numbers (63i) and (42i) and use the magnitude formula to compare them:

[|63i| sqrt{63^2 0^2} sqrt{3969} 63] [|42i| sqrt{42^2 0^2} sqrt{1764} 42]

Clearly, (63 > 42), so the magnitude of (63i) is greater than the magnitude of (42i).

H2: Additional Complex Examples

For further illustration, consider the imaginary numbers (34i) and (43i). These complex numbers have different values but the same magnitude:

[|3 4i| sqrt{3^2 4^2} sqrt{9 16} sqrt{25} 5] [|4 3i| sqrt{4^2 3^2} sqrt{16 9} sqrt{25} 5]

Here again, we see that although (34i) and (43i) are not equal, we cannot compare them using () because their magnitudes are equal, both being 5.

H2: Conclusion

In conclusion, it is incorrect to compare complex numbers using the '' symbols, as this would imply an ordering that does not exist. Instead, we use the magnitude of complex numbers, a real number, to compare them. This distinction is crucial in understanding and working with complex numbers effectively. Understanding the true nature of complex numbers and their magnitudes is essential for accurate comparison and manipulation in mathematical and scientific applications.