Properties of Argument in Complex Numbers

In the context of complex numbers, the argument refers to the angle that a complex number makes with the positive direction of the real axis on the complex plane. This angle is measured in radians and is often denoted by arg(z), where z is a complex number. The argument of a complex number is not unique; it is determined up to an integer multiple of 2π radians.

Definition

For a complex number z = x + yi, where x and y are real numbers, the argument is defined as the angle θ in polar coordinates such that:

z = r(cos(θ) + isin(θ))

where r = |z| is the modulus of z.

Principal Argument

The principal argument is the value of the argument that lies in the interval (-π, π]. It is denoted by Arg(z).

Properties of Argument

Here are some important properties of the argument of complex numbers:

1. Argument of a Positive Real Number: If z is a positive real number, then arg(z) = 0.

2. Argument of a Negative Real Number: If z is a negative real number, then arg(z) = π.

3. Argument of a Pure Imaginary Number: If z is a positive imaginary number, then arg(z) = π/2. If z is a negative imaginary number, then arg(z) = -π/2.

4. Argument of a Product: The argument of the product of two complex numbers is the sum of their arguments:

arg(z_1z_2) = arg(z_1) + arg(z_2)

1. Argument of a Quotient: The argument of the quotient of two complex numbers is the difference of their arguments:

arg(z_1/z_2) = arg(z_1) - arg(z_2)

1. Argument of a Power: The argument of a complex number raised to a power is the product of the power and the argument of the complex number:

arg(z^n) = n * arg(z)

1. Periodicity: The argument of a complex number is periodic with period 2π. This means that if θ is an argument of z, then θ + 2kπ is also an argument of z for any integer k.

2. Conjugate: The argument of the conjugate of a complex number is the negative of the argument of the complex number:

arg(\overline{z}) = -arg(z)

Table of Differences and Important Points

Property	Description	Example
Argument of a Positive Real Number	arg(z) = 0 for z > 0	arg(5) = 0
Argument of a Negative Real Number	arg(z) = π for z < 0	arg(-3) = π
Argument of a Pure Imaginary Number	arg(z) = π/2 for z = yi, y > 0; arg(z) = -π/2 for z = yi, y < 0	arg(4i) = π/2, arg(-4i) = -π/2
Argument of a Product	arg(z_1z_2) = arg(z_1) + arg(z_2)	arg((1+i)(1-i)) = arg(1+i) + arg(1-i) = π/4 + (-π/4) = 0
Argument of a Quotient	arg(z_1/z_2) = arg(z_1) - arg(z_2)	arg((1+i)/(1-i)) = arg(1+i) - arg(1-i) = π/4 - (-π/4) = π/2
Argument of a Power	arg(z^n) = n * arg(z)	arg((1+i)^2) = 2 * arg(1+i) = 2 * π/4 = π/2
Periodicity	arg(z) = θ + 2kπ for any integer k	arg(1+i) = π/4, arg(1+i) = π/4 + 2π
Conjugate	arg(\overline{z}) = -arg(z)	arg(\overline{1+i}) = -arg(1+i) = -π/4

Examples

Example 1: Argument of a Product

Let z_1 = 1 + i and z_2 = 2 - 2i. Find arg(z_1z_2).

Solution:

arg(z_1) = π/4 (since z_1 lies in the first quadrant)

arg(z_2) = -π/2 (since z_2 lies in the fourth quadrant)

Using the property of the argument of a product:

arg(z_1z_2) = arg(z_1) + arg(z_2) = π/4 + (-π/2) = -π/4

Example 2: Argument of a Quotient

Let z_1 = 3i and z_2 = -2. Find arg(z_1/z_2).

Solution:

arg(z_1) = π/2 (since z_1 is a positive imaginary number)

arg(z_2) = π (since z_2 is a negative real number)

Using the property of the argument of a quotient:

arg(z_1/z_2) = arg(z_1) - arg(z_2) = π/2 - π = -π/2

Example 3: Argument of a Power

Let z = 1 - i. Find arg(z^3).

Solution:

arg(z) = -π/4 (since z lies in the fourth quadrant)

Using the property of the argument of a power:

arg(z^3) = 3 * arg(z) = 3 * (-π/4) = -3π/4

These examples illustrate how the properties of the argument can be applied to calculate the argument of complex numbers in various operations. Understanding these properties is crucial for solving problems involving complex numbers in mathematics.