Question

Use \(z=a+b i\) and \(w=c+d i\) to show that \(\overline{z+w}=\bar{z}+\bar{w}\)

Short answer

\( \overline{z+w} = \bar{z} + \bar{w} \)

Step-by-step solution

Define the complex numbers

Given two complex numbers, define them as follows: Let \( z = a + bi \) and \( w = c + di \), where \( a, b, c, d \) are real numbers and \( i \) is the imaginary unit (\( i^2 = -1 \)).

Add the complex numbers

Add the two complex numbers \( z \) and \( w \). This results in: \[ z + w = (a + bi) + (c + di) = (a + c) + (b + d)i \]

Take the conjugate of the sum

Take the complex conjugate of \( z + w \). Recall that the conjugate of a complex number \( x + yi \) is \( x - yi \). Therefore, \[ \overline{z + w} = \overline{(a + c) + (b + d)i} = (a + c) - (b + d)i \]

Take the individual conjugates

Find the complex conjugates of \( z \) and \( w \) separately. These are: \[ \bar{z} = \overline{a + bi} = a - bi \] \[ \bar{w} = \overline{c + di} = c - di \]

Add the conjugates

Add the individual conjugates \( \bar{z} \) and \( \bar{w} \). This results in: \[ \bar{z} + \bar{w} = (a - bi) + (c - di) = (a + c) - (b + d)i \]

Compare the results

Compare the expressions obtained in Step 3 and Step 5. Both expressions are equal to \( (a + c) - (b + d)i \). Hence, \[ \overline{z + w} = \bar{z} + \bar{w} \]

Key concepts

complex conjugate

A complex conjugate is a very helpful concept in complex number operations. For any complex number in the form of \(z = x + yi\), its complex conjugate is defined as \(\bar{z} = x - yi\). The conjugate essentially flips the sign of the imaginary part, while keeping the real part constant.
The conjugate is useful because it preserves the magnitude of the complex number. For example, when multiplied by its conjugate, a complex number creates a real number: \[ z \bar{z} = (x + yi)(x - yi) = x^2 - y^2i^2 = x^2 + y^2 \]
One of the key properties of the complex conjugate is observed in the exercise: \[ \bar{z} + \bar{w} = \bar{(z + w)} \]
Understanding the complex conjugate not only helps in simplifying complex arithmetic but also has implications in fields like signal processing and quantum mechanics.

addition of complex numbers

The addition of complex numbers is straightforward. When adding two complex numbers taken as \(z = a + bi\) and \(w = c + di\), add the real parts and the imaginary parts separately. Here's a step-by-step breakdown:

• Step 1: Identify the real and imaginary parts. For \(z\), they are \(a\) and \(bi\), and for \(w\), they are \(c\) and \(di\).
• Step 2: Add the real parts: \(a + c\).
• Step 3: Add the imaginary parts: \(bi + di = (b + d)i\).

Thus, the sum of \(z \text{ and } w\) is \((a + c) + (b + d)i\). This concept lays the groundwork for understanding other operations with complex numbers, such as subtraction and even multiplication. It's essential to see how these basic operations work to dig deeper into more complex topics.

algebraic proof

Proving algebraic properties is crucial as it confirms the underlying rules you use in calculations. Let’s dissect the proof that \(\bar{z + w} = \bar{z} + \bar{w}\). This proof uses simple steps to verify the property:

• Step 1: Define the complex numbers as \z = a + bi\ and \w = c + di\.
• Step 2: Add the complex numbers: \z + w = (a + c) + (b + d)i\.
• Step 3: Take the conjugate of the sum: \(\bar{z + w} = (a + c) - (b + d)i\).
• Step 4: Find the individual conjugates: \(\bar{z} = a - bi\) and \(\bar{w} = c - di\).
• Step 5: Add the conjugates: \(\bar{z} + \bar{w} = (a - bi) + (c - di) = (a + c) - (b + d)i\).

Step 6: Compare: \( \bar{z + w} = \bar{z} + \bar{w} \). The proof follows logically. It confirms that the sum of the complex conjugates of two complex numbers is the same as the conjugate of their sum. This proven property is foundational to complex number theory and is used in various domains of mathematics and science.