Problem 77 - 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. \(z+\overline{z}\) is a real number.

Step-by-Step Solution

Answer

\(z + \overline{z} = 2a\), which is a real number.

Step 1: Identify the complex conjugate

Given a complex number \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \). This means you change the sign of the imaginary part.

Step 2: Add the complex number to its conjugate

Write the sum of the complex number \( z \) and its complex conjugate \( \overline{z} \):\[z + \overline{z} = (a + bi) + (a - bi)\]

Step 3: Combine like terms

Combine the real parts and the imaginary parts in the expression:\[z + \overline{z} = a + bi + a - bi = 2a + 0i\]

Step 4: Conclude with real number result

Since the imaginary part is 0, the sum \( 2a + 0i \) simplifies to \( 2a \), which is a real number. Therefore, \( z + \overline{z} \) results in a real number.

Key Concepts

Complex Conjugate

In the world of complex numbers, the term "complex conjugate" refers to a pair of numbers that have the same real part but opposite imaginary parts. Given a complex number \( z = a + bi \), where \( a \) and \( b \) are real numbers, the complex conjugate is represented as \( \overline{z} = a - bi \).

What this means is you flip the sign of the imaginary component. This property is incredibly useful in various mathematical operations, as conjugating complex numbers can help simplify expressions and solve equations.

For example, when you add a complex number to its conjugate, the imaginary components cancel each other out, leaving only the real parts. This results in a real number, as seen in the formula \( z + \overline{z} = 2a \). Understanding complex conjugates is crucial as it forms the foundation for more advanced operations involving complex numbers.

Addition of Complex Numbers

Adding complex numbers is a straightforward process, especially if you break it down into its real and imaginary components. When you are dealing with two complex numbers, say \( z = a + bi \) and \( w = c + di \), addition involves combining the respective parts:

• Add the real parts together: \( a + c \)
• Add the imaginary parts: \( b + d \)

The result is a new complex number \( (a+c) + (b+d)i \), where the real part is \( a+c \) and the imaginary part is \( b+d \).

This method of addition maintains the properties of complex numbers and is essential in solving equations with them. For example, if you are given \( z \) and its conjugate \( \overline{z} = a - bi \), adding these results in \( (a + bi) + (a - bi) = 2a + 0i \), simplifying to a real number \( 2a \). The imaginary components effectively cancel out, demonstrating how powerful this property of addition is.

Imaginary Numbers

Imaginary numbers exist to extend the real number system and solve equations that regular numbers cannot. The basic unit of imaginary numbers is \( i \), defined by the property \( i^2 = -1 \). An imaginary number typically appears as part of a complex number, in the form \( bi \). Here, \( b \) is a real number, and \( i \) is the imaginary unit.

These numbers are crucial because they allow a broader range of answers in algebra and calculus. Unlike real numbers, imagining numbers exist on a separate plane of numbers through what's known as the "complex plane."

Using imaginary numbers can prove incredibly useful in complex conjugates and the addition of complex numbers. The imaginary parts help provide solutions to polynomial equations that have no real solutions. When you see terms like \( bi + di \), they represent the imaginary parts of complex numbers, which can add up together separately from the real parts to form a full complex number.