Problem 79 - Precalculus : Building Concepts and Connections

Question

Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that the real part of \(z\) is equal to \(\frac{z+\bar{z}}{2}\)

Step-by-Step Solution

Answer

The equality holds, as that's the definition of the real part of a complex number in terms of the complex number and its conjugate. The calculation shows that \(\frac{z + \bar{z}}{2}\) is indeed equal to \(a\), which is the real part of the complex number \(z = a + bi\).

Step 1: Definition of complex numbers and their conjugates

Start by writing out the given complex number \(z\) and its conjugate \(\bar{z}\). Remember, \(z = a + bi\) and \(\bar{z} = a - bi\), where \(a\) and \(b\) are the real and imaginary parts of the complex number, respectively.

Step 2: Addition of the complex number and its conjugate

Add the complex number \(z = a + bi\) to its conjugate \(\bar{z} = a - bi\). This is \((a + bi) + (a - bi)\). The result is \(2a\), as the imaginary parts cancel each other out.

Step 3: Calculation of the real part of \(z\)

Now, calculate the value of \(\frac{z + \bar{z}}{2}\). Since the sum of \(z\) and \(\bar{z}\) was found to be \(2a\), this is basically \(\frac{2a}{2}\). After simplification, this is equal to \(a\).

Key Concepts

Conjugate of a Complex Number

Understanding the conjugate of a complex number is crucial when dealing with complex arithmetic and algebra. A complex number is generally expressed in the form of \( z = a + bi \), where \( a \) is the real part and \( b \) the imaginary part, with \( i \) being the imaginary unit satisfying the property \( i^2 = -1 \).

The conjugate of complex number \( z \), denoted by \( \bar{z} \), is formed by changing the sign of the imaginary part, resulting in \( \bar{z} = a - bi \). The conjugation operation has several important properties:

• It preserves the magnitude of the original complex number, that is, \( |z| = |\bar{z}| \).
• When multiplying a complex number by its conjugate, the result is always a non-negative real number: \( z \times \bar{z} = (a + bi)(a - bi) = a^2 + b^2 \).
• Conjugates can help in division processes involving complex numbers by removing the imaginary part from the denominator.

These properties make conjugates a powerful tool for simplifying expressions and solving complex equations.

Real Part of a Complex Number

The real part of a complex number is, simply put, the 'real' component that does not involve the imaginary unit \( i \). For a complex number written as \( z = a + bi \), the real part is represented by \( a \).

To extract the real part from \( z \), one can utilize the conjugate of \( z \), \( \bar{z} = a - bi \), in a specific manner. By adding the original number and its conjugate, \( z + \bar{z} = (a + bi) + (a - bi) \), the imaginary parts are cancelled out, giving \( 2a \). This sum contains only the real part, doubled. Hence, taking half of this sum, \( \frac{z + \bar{z}}{2} = \frac{2a}{2} \), retrieves the original real part \( a \). This demonstrates how crucial and practical the concept of conjugation is, and its application in various complex number operations.

Imaginary Part of a Complex Number

Contrasting the real part, the imaginary part of a complex number is the component that involves the imaginary unit \( i \). If we have a complex number \( z = a + bi \), then the imaginary part is the term \( bi \), where \( b \) is a real number. Isolating the imaginary part requires a different approach compared to extracting the real part.

To determine the imaginary part, subtract the conjugate of the number from the original: \( z - \bar{z} = (a + bi) - (a - bi) \). This subtraction eliminates the real parts and doubles the imaginary part, giving us \( 2bi \). To get the coefficient of \( i \) alone, we take half of the difference, resulting in \( \frac{z - \bar{z}}{2i} = \frac{2bi}{2i} \), which simplifies to \( b \). This operation showcases another practical use of the conjugate in relation to the properties of complex numbers, facilitating the manipulation and understanding of these numbers in various mathematical contexts.