Problem 57 - 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 \( \overline{z+w} = \bar{z} + \bar{w} \) is verified as true.

Step 1: Understand the Definition

The exercise asks us to verify the property \( \overline{z+w} = \bar{z} + \bar{w} \). Here, \( z \) and \( w \) are complex numbers, and the bar symbol represents the complex conjugate of a number. Recall that for a complex number \( z = a + bi \), its conjugate is \( \bar{z} = a - bi \).

Step 2: Express Complex Numbers

Let's express the complex numbers \( z \) and \( w \) in the standard form. Let \( z = a + bi \) and \( w = c + di \), where \( a, b, c, \) and \( d \) are real numbers.

Step 3: Find the Conjugate of the Sum

Calculate \( z + w \) first: \( z + w = (a + bi) + (c + di) = (a+c) + (b+d)i \). Now find the conjugate of this sum: \( \overline{z+w} = \overline{(a+c) + (b+d)i} = (a+c) - (b+d)i \).

Step 4: Find the Conjugate of Each Number

Calculate the conjugate of each complex number separately. \( \bar{z} = a - bi \) and \( \bar{w} = c - di \). Then, find their sum: \( \bar{z} + \bar{w} = (a - bi) + (c - di) = (a+c) - (b+d)i \).

Step 5: Compare the Two Results

From Steps 3 and 4, we have \( \overline{z+w} = (a+c) - (b+d)i \) and \( \bar{z} + \bar{w} = (a+c) - (b+d)i \). Since the expressions are the same, the property \( \overline{z+w} = \bar{z} + \bar{w} \) is verified.

Key Concepts

Complex Numbers

Complex numbers are fundamental in many areas of mathematics, engineering, and physics. They allow us to solve equations that have no real solutions. A complex number is expressed in the standard form as \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).

• The part \( a \) is known as the real part of the complex number.
• The part \( b \) is called the imaginary part.

The beautiful aspect of complex numbers is their ability to be manipulated just like real numbers, but they include an extra dimension that accounts for the imaginary part. This offers a richer framework for addressing various mathematical problems. When working with complex numbers, understanding their conjugates is crucial, especially in simplification processes and division.

Algebraic Properties

Like real numbers, complex numbers follow specific algebraic properties. These properties ensure you can perform basic arithmetic operations such as addition, subtraction, multiplication, and division with ease.

• **Addition and Subtraction**: To add or subtract complex numbers, simply add or subtract the corresponding real and imaginary parts. Given two complex numbers, \( z = a + bi \) and \( w = c + di \), their sum is \( (a+c) + (b+d)i \).
• **Multiplication**: To multiply, you apply the distributive property, just like with polynomials, remembering that \( i^2 = -1 \). The product of \( z \) and \( w \) is \( (ac - bd) + (ad + bc)i \).
• **Conjugation**: The conjugate of a sum extends to \( \overline{z+w} = \bar{z} + \bar{w} \), where "overline" indicates the conjugation operator. This property is extremely useful for simplifications.

Verification of Properties

Verification of algebraic properties involves proving that certain principles hold for all applicable cases or specific scenarios. In the case of complex numbers, the property \( \overline{z+w} = \bar{z} + \bar{w} \) is a pivotal rule worth validating.
To verify this property, we start by representing complex numbers \( z = a + bi \) and \( w = c + di \). First, compute their sum: \( z + w = (a + c) + (b + d)i \). Then, find the conjugate of this sum: \( \overline{z+w} = (a+c) - (b+d)i \).
Next, compute the conjugate of each individual number: \( \bar{z} = a - bi \) and \( \bar{w} = c - di \), and add them: \( \bar{z} + \bar{w} = (a+c) - (b+d)i \). Compare both results: they are identical, confirming the property.
This verification illustrates that the algebraic property holds for any given complex numbers, enhancing our understanding of complex arithmetic and calculus.