Problem 71 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The sum \(z + \overline{z}\) is a real number, specifically \(2a\).

Step 1: Define the Complex Number

The complex number \(z\) is given as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Identify the Complex Conjugate

The complex conjugate of \(z\) is denoted as \(\overline{z}\) and is equal to \(a - bi\).

Step 3: Compute the Sum of \(z\) and \(\overline{z}\)

Add the complex number \(z\) and its conjugate \(\overline{z}\): \(z + \overline{z} = (a + bi) + (a - bi)\).

Step 4: Simplify the Expression

Simplify the expression: \((a + bi) + (a - bi) = a + bi + a - bi\). The imaginary parts \(+bi\) and \(-bi\) cancel each other out, resulting in \(2a\).

Step 5: Conclusion

The result is \(2a\), which is a real number because \(a\) is a real number. Thus, \(z + \overline{z}\) is a real number.

Key Concepts

Complex Conjugate

When working with complex numbers, it's essential to understand the concept of a complex conjugate. The complex conjugate of a complex number essentially mirrors the imaginary part across the real axis. If we have a complex number represented as \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \).
This concept is crucial because the conjugate has the effect of eliminating the imaginary part when added to the original complex number. It helps simplify the computation in various mathematical contexts, such as solving polynomial equations or simplifying expressions.
In our exercise, it's straightforward to see how the conjugate plays a crucial role:

• The expression for adding \( z \) and \( \overline{z} \) is \( z + \overline{z} = (a + bi) + (a - bi) \).
• The \(+bi\) and \(-bi\) cancel each other due to their opposing signs.
• This leaves us with a real number, \(2a\).

Understanding the complex conjugate is not only about opposites in sign; it’s about recognizing its utility in rendering complex expressions manageable.

Real Numbers

Real numbers are at the foundation of all complex numbers. They include both the positive and negative numbers, as well as zero, without any fractional parts of an imaginary unit \(i\). The importance of real numbers cannot be overstated since every complex number can be broken down into a real component and an imaginary component.
In our exercise, the result of the sum \( z+\overline{z} \) is \(2a\), which, as highlighted, is purely a real number. This computation clearly shows that:

• The real number \(a\) from the expression \(a + bi\) plays a central role when we work out the sum with its conjugate.
• When dealing with roots, plotting graphs, or other mathematical applications, isolating real numbers helps make sense of the broader picture.
• They provide the baseline upon which complex numbers are structured, particularly in scenarios where imaginary components vanish.

Each interaction with complex numbers brings us back to their real parts, mirrored by the result of their sum with their conjugate.

Imaginary Parts

The imaginary part of a complex number is what gives it complexity, quite literally. In any complex number, such as \(z = a + bi\), the term \(bi\) represents the imaginary component. Here, \(b\) is a real number coefficient, and \(i\) is the imaginary unit, satisfying the equation \(i^2 = -1\).
Working with imaginary parts can be abstract because they cannot be located on the traditional number line. Instead, they're visualized on the imaginary axis perpendicular to the real number line in a complex plane. This helps in understanding geometric interpretations of complex numbers.
In this particular exercise:

• Adding the complex conjugate cancels out the imaginary components: \( (a + bi) + (a - bi) = 2a \).
• This cancellation clearly demonstrates that the imaginary parts often influence the behavior of a complex number until neutralized by its conjugate.
• Thus, while imaginary parts introduce complexities, they can be elegantly managed through operations like conjugation.

The significance of imaginary parts in complex numbers echoes through calculations, providing depth in fields ranging from electrical engineering to quantum mechanics.