Problem 36 - Advanced Engineering Mathematics

Question

In Problems 33-36, find all complex numbers for which the given statement is true. $$ z^{2}=(\bar{z})^{2} $$

Step-by-Step Solution

Answer

Complex numbers satisfying the equation are purely imaginary or real.

Step 1: Understanding the equation

The given equation is \( z^2 = (\bar{z})^2 \), where \( z \) is a complex number and \( \bar{z} \) is its complex conjugate. This equation states that squaring \( z \) is equal to squaring its conjugate.

Step 2: Break down the complex number

A complex number \( z \) can be expressed as \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit (\( i^2 = -1 \)). Therefore, its conjugate \( \bar{z} \) is \( x - yi \).

Step 3: Expanding both sides of the equation

Calculate \( z^2 = (x + yi)^2 = x^2 + 2xyi - y^2 \). Similarly, \( (\bar{z})^2 = (x - yi)^2 = x^2 - 2xyi - y^2 \).

Step 4: Equating real and imaginary parts

Since \( z^2 = (\bar{z})^2 \), equate the real and imaginary parts: For the real parts, \( x^2 - y^2 = x^2 - y^2 \) holds true automatically. For the imaginary parts, \( 2xy = -2xy \), thus \( 4xy = 0 \).

Step 5: Solving for x and y

The equation \( 4xy = 0 \) implies that either \( x = 0 \) or \( y = 0 \). If \( x = 0 \), \( z = yi \) and \( \bar{z} = -yi \). If \( y = 0 \), \( z = x \) (a real number), and \( \bar{z} = x \).

Step 6: Finding the solution set

Considering the solutions, \( z = yi \) or \( z = x \). Therefore, all solutions are purely imaginary numbers (\( x = 0 \)) or real numbers (\( y = 0 \)).

Key Concepts

Complex Conjugate

When discussing complex numbers, one of the fundamental concepts is the complex conjugate. A complex number \( z \) is typically written in the form \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit, equivalent to \( \sqrt{-1} \). The complex conjugate of \( z \), denoted as \( \bar{z} \), is expressed as \( \bar{z} = x - yi \).

• To find the complex conjugate, you simply change the sign of the imaginary part.
• Conjugates are crucial in operations like division of complex numbers, as they help to rationalize the denominator.

The relationship between a complex number and its conjugate is useful in simplifying expressions and solving equations involving complex numbers.

Real Numbers

Real numbers form the basis of complex numbers; they include all the rational and irrational numbers. In the context of complex numbers, real numbers are considered when the imaginary part is zero.

• A complex number is purely real if it can be expressed as \( z = x + 0i = x \).
• For instance, number 3 is a real number and can be viewed as \( 3 + 0i \).

Real numbers play an integral role in complex numbers since they define the horizontal axis in the complex plane.
Understanding real numbers is vital, as they are one of the simplest forms of complex numbers, represented by \( x + yi \) where \( y = 0 \).

Imaginary Unit

Complex numbers incorporate a special element called the imaginary unit, symbolized by \( i \). The imaginary unit itself is defined as \( i^2 = -1 \). This unique property allows it to be used to express numbers that have no real part.

• An imaginary number takes the form \( yi \), where \( y \) is a real number.
• For example, \( 4i \) is an imaginary number, indicating four times the imaginary unit.

The imaginary unit enables the construction of complex numbers, \( x + yi \), allowing for the expression of numbers beyond the real number system.
It is essential for representing phases in electrical engineering and many other fields.

Squaring Complex Numbers

Squaring complex numbers requires the application of algebraic rules to both real and imaginary components of the number. For a complex number \( z = x + yi \):

• To square it, compute \( z^2 = (x + yi)^2 = x^2 + 2xyi - y^2 \).
• The real part of \( z^2 \) is \( x^2 - y^2 \), while the imaginary part is \( 2xy \).

When squaring the complex conjugate \( \bar{z} = x - yi \), we get:

• \( (\bar{z})^2 = (x - yi)^2 = x^2 - 2xyi - y^2 \).
• The real part remains the same, \( x^2 - y^2 \), while the imaginary part becomes \(-2xy \).

Squaring complex numbers is helpful in solving equations, as shown in the exercise where setting \( z^2 = (\bar{z})^2 \) leads us to determine that the solutions are either purely real or purely imaginary numbers.