Problem 20 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=1-3 i $$

Step-by-Step Solution

Answer

Real part: 1, imaginary part: -3, modulus: \(\sqrt{10}\), argument: approximately -71.57 degrees.

Step 1: Identify the Real and Imaginary Parts

The complex number given is \( z = 1 - 3i \). The real part of \( z \), denoted as \( \operatorname{Re}(z) \), is 1. The imaginary part, denoted as \( \operatorname{Im}(z) \), is -3.

Step 2: Calculate the Modulus

The modulus \( |z| \) of a complex number \( z = a + bi \) is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \). For \( z = 1 - 3i \), \( a = 1 \) and \( b = -3 \). Plug in these values to get \( |z| = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \).

Step 3: Find the Argument

The argument \( \arg(z) \) can be found using the formula \( \tan^{-1}(\frac{b}{a}) \) where \( a \) is the real part and \( b \) is the imaginary part. For \( z = 1 - 3i \), compute \( \tan^{-1}(\frac{-3}{1}) \), which is approximately equal to \( -71.57^\circ \). This value represents the principal argument \( \operatorname{Arg}(z) \) which lies in the fourth quadrant.

Step 4: Form the Polar Representation

The polar form of a complex number is \( z = r(\cos(\theta) + i\sin(\theta)) \), where \( r = |z| \) and \( \theta = \arg(z) \). Here, \( r = \sqrt{10} \) and \( \theta \approx -71.57^\circ \) (or in radians \(-1.25\) approx), so the polar representation is \( \sqrt{10}(\cos(-1.25) + i\sin(-1.25)) \).

Key Concepts

Complex Numbers

Complex numbers are a fundamental part of mathematics that extend our number system beyond real numbers. They take the form of \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \). The real part of the complex number, \( a \), represents the component that lies along the real axis on the complex plane. Meanwhile, the imaginary part, \( b \), lies along the imaginary axis. Together, they give a complete description of the complex number in the complex plane.

• Real Component: Serves as the horizontal part of the complex number.
• Imaginary Component: Represents the vertical part, multiplied by \( i \).

Understanding complex numbers allows us to perform operations and solve equations that are not possible within the realm of real numbers alone. They are used extensively in fields such as engineering, physics, and signal processing.

Modulus of Complex Number

The modulus of a complex number is a measure of its 'size' or 'length' in the complex plane. Given a complex number \( z = a + bi \), the modulus is defined as \(|z| = \sqrt{a^2 + b^2}\). This yields a non-negative value representing the distance from the origin to the point \( (a, b) \) on the complex plane.

• Calculation: If \( z = 1 - 3i \), then \( a = 1 \) and \( b = -3 \), giving \(|z| = \sqrt{1^2 + (-3)^2} = \sqrt{10}\).
• Meaning: The modulus provides a way to gauge how far the number is from the origin, similar to the length of a vector.

The modulus is particularly useful when converting a complex number into its polar form, as it becomes the radial coordinate.

Argument of Complex Number

The argument of a complex number is the angle formed between the positive real axis and the line representing the complex number in the complex plane. To find this angle for a complex number \( z = a + bi \), we use the formula \( \arg(z) = \tan^{-1}(\frac{b}{a}) \). This angle is typically given in radians.

• Principal Argument: It lies within the interval \((-\pi, \pi] \) or equivalently \((-180^\circ, 180^\circ]\). For example, for \( z = 1 - 3i \), \( \arg(z) \approx -71.57^\circ \) or in radians \(-1.25\).
• Quadrant Consideration: Depending on the signs of \( a \) and \( b \), the argument may need adjustment to place it within the correct quadrant.

The argument provides a direction to the modulus, helping to fully describe the complex number in polar coordinates.

Trigonometric Representation of Complex Numbers

The trigonometric (or polar) representation of a complex number is a different way to express a complex number, reflecting both its magnitude and direction. It is given by:\[ z = r(\cos(\theta) + i\sin(\theta)) \]where \( r \) is the modulus \(|z|\) and \( \theta \) is the argument \( \arg(z) \).

• Modulus: Serves as the radius or distance from the origin in the polar form.
• Angle: \( \theta \), the argument, provides the direction.For example, for the complex number \( z = 1 - 3i \), the polar form becomes \( \sqrt{10}(\cos(-1.25) + i\sin(-1.25)) \).

This representation is particularly powerful in operations like multiplication and division of complex numbers as it simplifies them into operations on the moduli and arguments.