Problem 93
Question
Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ \bar{z} $$
Step-by-Step Solution
Verified Answer
\(\bar{z} = 1 - 2i\)
1Step 1: Understanding Complex Conjugates
To find the complex conjugate of a complex number, you replace its imaginary part with its negative. For example, if the complex number is given as \(a + bi\), its complex conjugate is \(a - bi\).
2Step 2: Applying the Concept to \(z\)
Given the complex number \(z = 1 + 2i\), the real part is \(1\) and the imaginary part is \(2i\). Therefore, the complex conjugate \(\bar{z}\) is obtained by changing the sign of the imaginary part.
3Step 3: Computing \(\bar{z}\)
Replace the imaginary part of \(z = 1 + 2i\) with its negative to get \(\bar{z} = 1 - 2i\).
Key Concepts
Complex ConjugateReal PartImaginary Part
Complex Conjugate
When dealing with complex numbers, one important operation is finding the complex conjugate. This operation involves changing the sign of the imaginary part of a complex number while keeping the real part unchanged. When you have a complex number of the form \( a + bi \), its conjugate is \( a - bi \).
Understanding complex conjugates is crucial because they are used in various applications, such as simplifying expressions and calculating magnitudes. For instance, conjugates are often used when dividing complex numbers to ensure the denominator becomes a real number.
To illustrate, in our original exercise, the complex number \( z = 1 + 2i \) changes to its conjugate \( \bar{z} = 1 - 2i \). By flipping the imaginary part's sign, we achieve the conjugate of the complex number. This simple operation is fundamental in handling and manipulating complex numbers.
Understanding complex conjugates is crucial because they are used in various applications, such as simplifying expressions and calculating magnitudes. For instance, conjugates are often used when dividing complex numbers to ensure the denominator becomes a real number.
To illustrate, in our original exercise, the complex number \( z = 1 + 2i \) changes to its conjugate \( \bar{z} = 1 - 2i \). By flipping the imaginary part's sign, we achieve the conjugate of the complex number. This simple operation is fundamental in handling and manipulating complex numbers.
Real Part
One of the essential components of a complex number is the real part. A complex number is generally expressed in the form \( a + bi \), where \( a \) represents the real part. It is simply the non-imaginary portion of the complex number.
Identifying and understanding the real part is crucial for operations involving complex numbers. Since the real part resembles a traditional real number, it allows complex numbers to situate uniquely within the complex plane.
Identifying and understanding the real part is crucial for operations involving complex numbers. Since the real part resembles a traditional real number, it allows complex numbers to situate uniquely within the complex plane.
- In our example, for \( z = 1 + 2i \), the real part is \( 1 \).
- This part remains unchanged when finding the complex conjugate.
Imaginary Part
The imaginary part of a complex number is the component that contains \( i \), the imaginary unit, which satisfies \( i^2 = -1 \). A complex number \( a + bi \) features \( bi \) as its imaginary part. Managing the imaginary part correctly is crucial for performing operations on complex numbers.
The imaginary part is where much of the unique property of complex numbers comes from. Changing the sign of this part is the key action when determining the complex conjugate.
The imaginary part is where much of the unique property of complex numbers comes from. Changing the sign of this part is the key action when determining the complex conjugate.
- In \( z = 1 + 2i \), the imaginary part is \( 2i \).
- For its conjugate \( \bar{z} \), the imaginary part becomes \( -2i \).
Other exercises in this chapter
Problem 93
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View solution Problem 94
Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ z+u $$
View solution