Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-62: Verify the property. $$ \overline{z-w}=\bar{z}-\bar{w} $$

Step-by-Step Solution

Answer

The property \( \overline{z-w} = \overline{z} - \overline{w} \) is verified.

Step 1: Understand Complex Conjugate

The complex conjugate of a complex number \( z = a + bi \) is \( \overline{z} = a - bi \), where \( a \) and \( b \) are real numbers.

Step 2: Define Complex Numbers z and w

Assume \( z = a + bi \) and \( w = c + di \), where \( a, b, c, \) and \( d \) are real numbers.

Step 3: Subtract Complex Numbers

Calculate \( z - w = (a + bi) - (c + di) = (a - c) + (b - d)i \).

Step 4: Find the Conjugate of z - w

Using the formula from Step 1, the complex conjugate of \( z - w \) is \( \overline{z-w} = (a - c) - (b - d)i \).

Step 5: Find Individual Conjugates

Calculate the conjugates: \( \overline{z} = a - bi \) and \( \overline{w} = c - di \). Thus \( \overline{z} - \overline{w} = (a - bi) - (c - di) = (a - c) - (b - d)i \).

Step 6: Compare Results

Notice that \( \overline{z-w} = (a - c) - (b - d)i \) and \( \overline{z} - \overline{w} = (a - c) - (b - d)i \); they are the same.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part and can be expressed in the form \( z = a + bi \). Here, \( a \) is the real component and \( bi \) is the imaginary component, where \( i \) is the imaginary unit and satisfies \( i^2 = -1 \).
Complex numbers are useful in various fields such as engineering, physics, and mathematics, providing a way to work with two-dimensional quantities.
They're often represented on a plane known as the complex plane, where the x-axis represents real numbers and the y-axis represents imaginary numbers.

• The real part is the 'a' in \( a + bi \).
• The imaginary part is the 'bi' in \( a + bi \).
• Complex numbers can be added, subtracted, multiplied, and divided, following certain algebraic rules.

Real Numbers

Real numbers are numbers that can be found on the number line, including all the positive and negative integers, fractions, and irrational numbers. They're the building blocks of complex numbers.
When we talk about real numbers in the context of complex numbers, we refer to the part of a complex number that does not involve the imaginary unit \( i \).
Real numbers include:

• Integers like -3, 0, 4
• Fractions like \( \frac{1}{2} \)
• Irrational numbers like \( \sqrt{2} \) or \( \pi \)

Real numbers are crucial for defining the real part of a complex number, and understanding them is essential for manipulating complex expressions.

Complex Subtraction

Complex subtraction involves subtracting one complex number from another. To do this, you subtract both the real and imaginary parts separately.
Using two complex numbers \( z = a + bi \) and \( w = c + di \), the subtraction \( z - w \) becomes:

• Subtracting the real parts: \( a - c \)
• Subtracting the imaginary parts: \( b - d \)

Thus, the result is another complex number \( (a - c) + (b - d)i \).
This operation maintains the structure of complex numbers, as you end up with a new complex number where both components are derived from direct subtraction of the components of the numbers being subtracted. Understanding this practice is key for verifying properties like the one in the original exercise.