Argument

Understanding the Argument of a Complex Number

In mathematics, particularly in complex analysis, the argument of a complex number is a measure of the angle by which the number is rotated around the origin in the complex plane. It is an important concept that helps in understanding the geometric interpretation of complex numbers.

Definition

The argument of a complex number $z = x + yi$, where $x$ and $y$ are real numbers and $i$ is the imaginary unit, is defined as the angle $\theta$ between the positive real axis and the line segment joining the origin to the point $(x, y)$ in the complex plane. This angle is measured in radians and is denoted as $\text{arg}(z)$.

The argument can be calculated using the arctangent function:

$$ \theta = \text{arg}(z) = \arctan\left(\frac{y}{x}\right) $$

However, since the arctangent function only gives values from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$, we need to adjust the angle based on the quadrant in which the complex number lies.

Quadrants and Argument

The complex plane is divided into four quadrants:

  • Quadrant I: $x > 0, y > 0$
  • Quadrant II: $x < 0, y > 0$
  • Quadrant III: $x < 0, y < 0$
  • Quadrant IV: $x > 0, y < 0$

The argument of a complex number depends on its quadrant:

Quadrant Argument ($\theta$)
I $\arctan\left(\frac{y}{x}\right)$
II $\pi + \arctan\left(\frac{y}{x}\right)$
III $-\pi + \arctan\left(\frac{y}{x}\right)$
IV $\arctan\left(\frac{y}{x}\right)$

Principal Value of the Argument

The principal value of the argument is the value of $\theta$ in the interval $(-\pi, \pi]$. It is denoted as $\text{Arg}(z)$.

Examples

Let's consider a few examples to understand the calculation of the argument of a complex number:

  1. For $z = 1 + i$, the complex number lies in Quadrant I. Therefore, $\text{arg}(z) = \arctan\left(\frac{1}{1}\right) = \frac{\pi}{4}$.

  2. For $z = -1 + i$, the complex number lies in Quadrant II. Therefore, $\text{arg}(z) = \pi + \arctan\left(\frac{1}{-1}\right) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.

  3. For $z = -1 - i$, the complex number lies in Quadrant III. Therefore, $\text{arg}(z) = -\pi + \arctan\left(\frac{-1}{-1}\right) = -\pi + \frac{\pi}{4} = -\frac{3\pi}{4}$.

  4. For $z = 1 - i$, the complex number lies in Quadrant IV. Therefore, $\text{arg}(z) = \arctan\left(\frac{-1}{1}\right) = -\frac{\pi}{4}$.

Properties of the Argument

  • The argument of a product of two complex numbers is the sum of their arguments: $\text{arg}(zw) = \text{arg}(z) + \text{arg}(w)$.
  • The argument of a quotient of two complex numbers is the difference of their arguments: $\text{arg}\left(\frac{z}{w}\right) = \text{arg}(z) - \text{arg}(w)$.
  • The argument of the conjugate of a complex number is the negative of the argument of the number: $\text{arg}(\overline{z}) = -\text{arg}(z)$.

Conclusion

The argument of a complex number is a fundamental concept in complex analysis. It provides a way to describe the orientation of a complex number in the complex plane and plays a crucial role in operations such as multiplication and division of complex numbers. Understanding the argument helps in visualizing complex functions and solving problems in various fields of science and engineering.