Problem 80 - Algebra and Trigonometry

Question

Recall that the symbol \(\overline{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z=\overline{z}\) if and only if \(=\) is real.

Step-by-Step Solution

Answer

z is real if and only if \( z = \overline{z} \).

Step 1: Introduce the Complex Numbers

Let the complex numbers be defined as follows: \( z = a + bi \) and \( \overline{z} = a - bi \). The complex conjugate \( \overline{z} \) is formed by changing the sign of the imaginary part.

Step 2: Condition for Conjugate Equivalence

According to the given condition, we are to verify if \( z = \overline{z} \) implies that \( z \) is real. Let's examine this equality: \( a + bi = a - bi \).

Step 3: Evaluate Equality

For the equation \( a + bi = a - bi \) to hold true, both the real and imaginary parts must be equal. Thus, we require \( a = a \) and \( bi = -bi \).

Step 4: Solve Imaginary Part

The equality \( bi = -bi \) can only hold if \( b = 0 \). This indicates that the imaginary part of \( z \) must be zero, meaning \( z \) is purely real.

Step 5: Conclude the Realness

Since the imaginary component must be zero for \( z = \overline{z} \) to be valid, it follows that \( z \) must be a real number, i.e., \( z = a + 0i = a \).

Key Concepts

Complex Conjugate

A **complex conjugate** is a fundamental concept when dealing with complex numbers. For any complex number \(z = a + bi\), the complex conjugate is represented as \(\overline{z} = a - bi\). The concept involves merely flipping the sign of the imaginary part of the complex number. This means if the imaginary part in \(z\) is \(bi\), in the conjugate \(\overline{z}\) it becomes \(-bi\).
\(\)

• **Example**: If \( z = 3 + 4i \), then the complex conjugate \( \overline{z} = 3 - 4i \).
• The real part remains unchanged, while the imaginary part changes its sign.

Understanding complex conjugates is crucial since they are often used in operations such as division of complex numbers or solving polynomial equations. It's also useful in simplifying calculations like finding the modulus of a complex number, where we multiply the number by its conjugate.

Purely Real Numbers

A complex number is called **purely real** when its imaginary part is zero. This means that despite being expressed in the form \(a + bi\), the number only shows up as \(a\) because \(b = 0\). Therefore, it does not have an imaginary component at all.
\(\)

• If \(z = a + 0i\), then \(z\) is purely real.
• **Example**: \(5 + 0i\) is purely real, which can simply be written as \(5\).

In the context of the given problem, when a complex number equals its conjugate, it results in a purely real number because \(bi = -bi\) implies that \(b = 0\).
Recognition of purely real numbers is essential as it aids in distinguishing between complex numbers and helps avoid calculation errors where imaginary numbers might unintentionally be included.

Imaginary Part Zero

The intriguing scenario where the **imaginary part is zero** of a complex number is significant in understanding the condition for a complex number to be purely real. For the equality \(a + bi = a - bi\) to hold, the implication is that the imaginary components must cancel out, hence \(bi = -bi\).
\(\)

• This happens if and only if \(b = 0\).
• When this condition is met, the complex number reduces to \(a + 0i = a\), instilling it firmly as a purely real number.
• **Example**: \(7 + 0i = 7\) is a number with its imaginary part equal to zero.

Understanding why the imaginary part is zero is crucial in solving many algebraic problems involving complex numbers. It delineates the boundaries where complex numbers act like real numbers, simplifying various mathematical processes. Recognizing when this occurs can help in efficiently handling complex equations and their solutions.