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📘intermediate algebra review

key term - Argument (Phase)

Citation:

Definition

The argument, or phase, of a complex number refers to the angle or direction of the number on the complex plane. It represents the orientation of the complex number relative to the positive real axis. The argument is a crucial concept in understanding and working with complex numbers, particularly in the context of their geometric representation and applications in various mathematical and engineering domains.

5 Must Know Facts For Your Next Test

  1. The argument of a complex number is measured in radians, with the positive real axis corresponding to an argument of 0 radians and the positive imaginary axis corresponding to an argument of $\frac{\pi}{2}$ radians.
  2. The argument of a complex number can be calculated using the inverse tangent function, $\tan^{-1}\left(\frac{\Im(z)}{\Re(z)}\right)$, where $\Im(z)$ is the imaginary part and $\Re(z)$ is the real part of the complex number $z$.
  3. The argument of a complex number is unique up to multiples of $2\pi$ radians, meaning that the argument can be shifted by any integer multiple of $2\pi$ without changing the underlying complex number.
  4. The argument of a product of complex numbers is the sum of the arguments of the individual complex numbers, and the argument of a quotient of complex numbers is the difference of the arguments of the numerator and denominator.
  5. The argument of a complex number is an important concept in various applications, such as electrical engineering (e.g., impedance, power factor), signal processing (e.g., Fourier analysis), and quantum mechanics (e.g., wave functions).

Review Questions

  • Explain the geometric interpretation of the argument of a complex number on the complex plane.
    • The argument of a complex number $z = a + bi$ represents the angle between the positive real axis and the line segment connecting the origin to the point $(a, b)$ on the complex plane. This angle is measured counterclockwise from the positive real axis and is expressed in radians. The argument provides information about the orientation or direction of the complex number on the complex plane, which is an important concept in understanding and working with complex numbers.
  • Describe how the argument of a complex number is related to its polar form representation.
    • The polar form of a complex number $z = a + bi$ is given by $z = r(\cos \theta + i \sin \theta)$, where $r$ is the modulus (or magnitude) of the complex number and $\theta$ is the argument (or phase) of the complex number. The argument $\theta$ represents the angle that the complex number makes with the positive real axis on the complex plane. This polar form representation is useful for performing operations on complex numbers, such as multiplication, division, and exponentiation, as the argument and modulus can be manipulated separately.
  • Explain how the argument of a complex number is transformed when performing various operations, such as multiplication, division, and exponentiation.
    • The argument of a complex number exhibits the following properties when performing various operations: 1. Multiplication: The argument of the product of two complex numbers is the sum of the arguments of the individual complex numbers. 2. Division: The argument of the quotient of two complex numbers is the difference of the arguments of the numerator and denominator. 3. Exponentiation: The argument of the complex number raised to a power is the product of the power and the argument of the original complex number. These properties are important in understanding the behavior of complex numbers and their applications in various fields, such as electrical engineering, signal processing, and quantum mechanics.