Understanding Positive and Negative Numbers in Complex Numbers

The concept of positive and negative numbers is a fundamental one in mathematics, particularly in the realm of ordered fields. However, when we venture into the complex number system, mathbb{C}, the traditional notions of positivity and negativity become less straightforward. This article aims to explore and clarify these concepts, providing a deeper understanding of why and how these terms are applied in the context of complex numbers.

The Definition and Prevalence of Ordered Fields

Firstly, to discuss positive and negative numbers, one has to be working in an ordered field. An ordered field is a field that is also a totally ordered set in which the order is compatible with the algebraic operations. In such a field, it is not possible to write zero as a sum of squares of non-zero elements. This property is crucial in defining positive and negative numbers. However, in the complex number system, mathbb{C}, we find that

0 1^2 cdot i^2

This means that mathbb{C} cannot be an ordered field, and the elements cannot be described as positive or negative in the same sense.

Complex Numbers: An Ordered Framework

Given that an ordered field framework doesn't apply directly, we must consider total ordering when dealing with complex numbers. Usually, in a totally ordered field, we say that z is positive if 0 leq z for some total ordering. This implies that exactly one of the following statements must hold:

However, extending this concept to complex numbers leads to issues. For example, if we insist that the product of two positive numbers is positive, and we set i > 0 or -i > 0, we encounter contradictions:

i cdot i -1 > 0

Additionally, i eq 0, so we have

i > 0 Rightarrow i > -i Rightarrow 0 > -i, which is a contradiction.

Imaginary Numbers and Their Geometric Interpretation

Imaginary numbers, such as i, represent vectors on the y-axis in the complex plane. To understand positivity and negativity in the context of imaginary numbers, it is essential to recognize that these concepts are used differently than in real numbers.

For purely imaginary numbers, we can visualize them on the y-axis. Imaginary numbers above the x-axis (the real numbers) are considered positive, while those below are considered negative.

This differentiation of terms can indeed be confusing, as the words "positive" and "negative" are used in different contexts: one refers to the X and X directions for real numbers, and the other to the Y and Y directions for imaginary numbers.

The Complex Plane and Polar Coordinates

A more intuitive way to understand complex numbers is through their representation in polar coordinates. In the complex plane, a complex number z can be expressed as:

z a bi r (cos theta i sin theta)

Here, r is the magnitude (or length), and theta is the angle (or argument) with respect to the positive real axis. The angle is measured in radians, with a full circle being 2pi.

When adding and multiplying complex numbers in polar coordinates:

To add two vectors, you simply slide the second one so its tail is at the head of the first vector, and the sum is from the origin to the head of the second vector.

To multiply two vectors, you multiply their lengths and add their angles.

The angle addition in multiplication has a deeper significance. This rule stems from Euler's famous equation, e^{ipi} -1, which relates an exponential to pi and imaginary numbers. This equation means that Ae^{itheta} is a vector with length A and angle theta with respect to the positive real axis.

Thus, multiplying two complex numbers:

Ae^{itheta} cdot Be^{iphi} AB e^{i(theta phi)}

This explains why the angles add when we multiply vectors: the angle of the product vector is the sum of the angles of the original vectors.

Conclusion

In conclusion, while the concepts of positive and negative numbers can be extended to imaginary numbers, the interpretation and application are context-dependent. The polar coordinate representation of complex numbers provides a more intuitive understanding of these operations, reinforcing the idea that these concepts are applied differently for real and imaginary parts.

Understanding the nuances of these operations is crucial for anyone working with complex numbers in fields such as engineering, physics, and advanced mathematics. The insights provided in this article should help demystify the confusing but fascinating world of complex numbers.