Problem 61 - Precalculus: Functions and Graphs

Question

Verify the property. \(\bar{z}=z\) if and only if \(z\) is real.

Step-by-Step Solution

Answer

\(\bar{z}=z\) means \(z\) is real because the imaginary part must be zero.

Step 1: Define the Property

The property states that the complex conjugate of a complex number equals the complex number itself if and only if the number is real.

Step 2: Express the Complex Number

A complex number \( z \) can be expressed as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

Step 3: Write the Complex Conjugate

The complex conjugate of \( z \), denoted by \( \bar{z} \), is \( a - bi \).

Step 4: Set the Equality

Set the given equality \( \bar{z} = z \). This implies \( a - bi = a + bi \).

Step 5: Compare Real and Imaginary Parts

For the equality \( a - bi = a + bi \) to hold, the imaginary parts must be equal: \(-bi = bi\). This simplifies to \(-b = b\).

Step 6: Solve for the Imaginary Part

The equation \(-b = b\) implies that \( b = 0 \).

Step 7: Conclude on the Nature of \(z\)

Since \( b = 0 \), the complex number \( z = a + bi\) reduces to \( z = a \). Therefore, \( z \) is real.

Key Concepts

Complex Conjugate

The complex conjugate of a complex number is a core concept in mathematics, particularly useful in various fields like engineering and physics. If you have a complex number, say \( z = a + bi \), where \( a \) and \( b \) are real numbers, the complex conjugate of \( z \) is denoted as \( \bar{z} \) and is equal to \( a - bi \). This operation essentially "flips" the sign of the imaginary component while keeping the real part unchanged.

The significance of this operation lies in its properties:

• If you multiply a complex number by its complex conjugate, you get a non-negative real number: \( z\bar{z} = a^2 + b^2 \).
• This can help in simplifying division of complex numbers by eliminating the imaginary part in the denominator.
• It is also used in determining whether results are consistent when dealing with complex numbers subject to various operations, as was demonstrated in the original exercise.

Understanding complex conjugates helps you see patterns and relationships in complex numbers, letting you transform complex expressions into simpler, more usable forms.

Real Numbers

Real numbers are foundational in mathematics and are defined as any number that can be located on the number line. These include both positive and negative integers, fractions, and irrational numbers. In relation to complex numbers, if the imaginary part is zero, a complex number reduces to just its real part, making it a real number.

Consider the equation \( z = a + bi \). If \( b = 0 \), then \( z = a \), where \( a \) belongs to the set of real numbers. This is significant when understanding when a complex number equals its complex conjugate: it happens when the number is purely real with no imaginary part.

• This property is crucial because it indicates that if a complex conjugate \( \bar{z} = z \), \( z \) must be entirely real.
• Real numbers are fundamental in both everyday math and advanced calculations, underpinning countless mathematical concepts.

Recognizing when you're dealing with real numbers is key in simplifying and solving equations involving complex numbers.

Imaginary Part

In complex numbers, the imaginary part is the component that involves the imaginary unit \( i \), where \( i \) is defined as \( \sqrt{-1} \). This is what distinguishes complex numbers from real numbers. For a complex number \( z = a + bi \), \( b \) is the coefficient of the imaginary part.

Managing the imaginary part is crucial when analyzing complex numbers:

• When you have an expression of the form \( a + bi \), \( b \) must be zero for \( z \) to equal its complex conjugate.
• In the provided exercise, setting \( \bar{z} = z \) led to the realization that the imaginary part must cancel out, requiring \( b = 0 \).
• This cancellation results in a number that is purely real, as the imaginary component disappears.

Understanding the imaginary part helps in visualizing and working with complex numbers, allowing for conversion of complex relationships into real situations.