Problem 72 - 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 pure imaginary number

Step-by-Step Solution

Answer

\(z - \overline{z}\) equals \(2bi\), which is purely imaginary.

Step 1: Understanding the Complex Parts

Recall the definition of a complex number. A complex number is given by the expression \(z = a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Step 2: Finding the Complex Conjugate

The complex conjugate of a complex number \(z = a + bi\) is \(\overline{z} = a - bi\). This reverses the sign of the imaginary part.

Step 3: Calculating \(z - \overline{z}\)

Since \(z = a + bi\) and \(\overline{z} = a - bi\), we compute \(z - \overline{z} = (a + bi) - (a - bi)\).

Step 4: Simplifying the Expression

Simplify the expression: \((a + bi) - (a - bi) = a + bi - a + bi = 2bi\).

Step 5: Identifying the Type of Number

The result, \(2bi\), has no real part (the part before \(i\) is the real part, which in this case is zero), making it a pure imaginary number.

Key Concepts

Complex Conjugate

Imagine turning a number inside out by flipping its imaginary part. That's essentially what a complex conjugate does! For any complex number \( z = a + bi \), the complex conjugate is denoted as \( \overline{z} = a - bi \). Notice how the real part, \(a\), stays the same, while only the sign of the imaginary part, \(bi\), changes. This concept is pretty handy.

By subtracting a complex conjugate from its original number, we can eliminate certain parts or even transform it into a specific type, like a pure imaginary number. This property makes complex conjugates very useful in simplifying complex expressions and solving equations.

Pure Imaginary Number

A pure imaginary number is a special type of complex number that doesn't have a real part. In terms of format, it appears as \( bi \), where \( b \) is a real number and \( i \) is the imaginary unit (the square root of -1).

In our exercise, when we compute \( z - \overline{z} \), we end up with \( 2bi \). Note how there is no real part; the result is purely imaginary.

• The form \( 2bi \) signifies two times the imaginary part of the original number.
• Notice that the real components cancel each other out during subtraction, leaving us with only the imaginary portion.

Pure imaginary numbers are especially interesting because they show the power of symmetry in complex conjugates.

Definition of Complex Number

Let's break down what makes a complex number complex. A complex number is a blend of both real and imaginary components. It is typically expressed in the form \( z = a + bi \), where:

• \(a\) is termed the real part.
• \(b\) is termed the coefficient of the imaginary part.
• \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\).

The combination of these parts allows complex numbers to describe values that regular real numbers can't, especially in fields like engineering and physics.

Real and Imaginary Parts

Understanding the distinction between the real and imaginary parts of a complex number is key.

• The real part of a complex number, denoted as \(a\) in \(z = a + bi\), can be thought of like a regular real number along the x-axis on a graph.
• The imaginary part, represented by \(bi\), acts somewhat like a number along the y-axis, giving the number a direction off the traditional real number line.

Returning to the exercise, when computing \(z - \overline{z}\), the real parts cancel each other (as they are equal and opposite), highlighting just how purely imaginary numbers can result from such operations. Knowing how these parts interplay is fundamental in visualizing and solving problems involving complex numbers.