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Chapter 9: Problem 4

Find \(|z|\) and \(\bar{z}\), and verify that \(z \bar{z}=|z|^{2}\), if a. \(z=2+i\), b. \(z=3-4 i\).

Short Answer

Expert verified
For (a), \(|z| = \sqrt{5}\), \(\bar{z} = 2 - i\); for (b), \(|z| = 5\), \(\bar{z} = 3 + 4i\). Both verify \(z \bar{z} = |z|^2\).

Step by step solution

01

Identify the Components of z

For part (a), we have \(z = 2 + i\), where the real part is \(2\) and the imaginary part is \(1\). For part (b), \(z = 3 - 4i\), where the real part is \(3\) and the imaginary part is \(-4\).
02

Calculate |z| for Part (a)

For \(z = 2 + i\), the magnitude \(|z|\) is calculated as \(|z| = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}\).
03

Find \(\bar{z}\) for Part (a)

The conjugate of \(z = 2 + i\) is \(\bar{z} = 2 - i\).
04

Verify \(z\bar{z} = |z|^2\) for Part (a)

Compute \(z \bar{z} = (2 + i)(2 - i) = 2^2 - i^2 = 4 + 1 = 5\), and \(|z|^2 = (\sqrt{5})^2 = 5\). Therefore, \(z \bar{z} = |z|^2\) is verified for part (a).
05

Calculate |z| for Part (b)

For \(z = 3 - 4i\), the magnitude \(|z|\) is calculated as \(|z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
06

Find \(\bar{z}\) for Part (b)

The conjugate of \(z = 3 - 4i\) is \(\bar{z} = 3 + 4i\).
07

Verify \(z\bar{z} = |z|^2\) for Part (b)

Compute \(z \bar{z} = (3 - 4i)(3 + 4i) = 3^2 - (-4)^2 = 9 + 16 = 25\), and \(|z|^2 = 5^2 = 25\). Therefore, \(z \bar{z} = |z|^2\) is verified for part (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnitude of Complex Numbers
The magnitude of a complex number, also known as the modulus, is a measure of its size. It gives us a way to quantify how large or small a complex number is, just like absolute value for real numbers.
For a complex number written as \( z = a + bi \) where \( a \) is the real part and \( b \) is the imaginary part, the magnitude is calculated by taking the square root of the sum of the squares of the real and imaginary parts.
  • Formula: \(|z| = \sqrt{a^2 + b^2}\)
  • Example: For \( z = 2 + i \), the magnitude is \( |z| = \sqrt{2^2 + 1^2} = \sqrt{5} \).
  • For \( z = 3 - 4i \), the magnitude is \( |z| = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5 \).
The magnitude helps in understanding the distance of the complex number from the origin in the complex plane, which is a representation of complex numbers where the x-axis represents the real part and the y-axis represents the imaginary part.
Complex Conjugate
The complex conjugate of a complex number involves changing the sign of the imaginary part while keeping the real part unchanged. It is a crucial concept in complex number arithmetic.

If a complex number \( z \) is written as \( z = a + bi \), its conjugate, denoted as \( \bar{z} \), is \( \bar{z} = a - bi \).
  • Example: For \( z = 2 + i \), the conjugate is \( \bar{z} = 2 - i \).
  • For \( z = 3 - 4i \), the conjugate is \( \bar{z} = 3 + 4i \).
The process of finding the conjugate is simple: just flip the sign of the imaginary part. It is often used in division of complex numbers and in verifying properties such as \( z \bar{z} = |z|^2 \), which is a valuable check to confirm calculations.
Properties of Complex Numbers
Complex numbers have several interesting properties, making them unique in mathematics. One of the key properties relates to how they interact with their conjugates and magnitudes.
  • For any complex number \( z \), the relationship \( z \bar{z} = |z|^2 \) holds true.
  • This means when you multiply a complex number by its conjugate, the result is always a real number equal to the square of the magnitude of the original complex number.
  • Example: For \( z = 2 + i \), we found that \( z \bar{z} = (2+i)(2-i) = 5 = |z|^2 \).
  • For \( z = 3 - 4i \), \( z \bar{z} = 25 = |z|^2 \); hence, this property is verified.
This property is particularly useful in simplifying complex calculations and verifying results. It provides a way to turn a complex multiplication into a simpler, purely real number outcome, which can ease many computational problems involving complex numbers.

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Most popular questions from this chapter

Suppose that \(A\) is an \(n \times n\) matrix such that \(A^{*}=-A\). Show that a. A has eigenvalues of the form \(r i\), where \(r \in \mathbb{R}\), b. \(A\) is diagonalizable by a unitary matrix.

Find \(A^{2}\) and \(A^{4}\) if \(A=\left[\begin{array}{cc}1 & 1+i \\ -1+i & i\end{array}\right]\)

Describe the unitary diagonal matrices.

Find the sum \(z+w\) and the product \(z w\) if a. \(z=2+3 i, w=5-i\), b. \(z=1+2 i, w=2-i\).

Find a unitary matrix U and a diagonal matrix \(D\) such that \(D=U^{-1} A U\) for the given matrix \(A\).\(A=\left[\begin{array}{cc}9 & 3-i \\ 3+i & 0\end{array}\right]\)
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