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Chapter 13: Problem 11

\(\bar{z}\) is the conjugate of \(z\) : (a) \(|\bar{z}|=|z|\) (b) \(\arg z=\arg \bar{z}\) (c) \(z \bar{z}\) is real (d) \(z / z\) is real (e) \(\bar{z}\) is the mirror image of \(z\) in the \(y\)-axis.

Short Answer

Expert verified
(a) True, (b) False, (c) True, (d) True, (e) False

Step by step solution

01

Title - Define the Conjugate of a Complex Number

Given a complex number \(z = a + bi\), its conjugate is defined as \(\bar{z} = a - bi\). This helps us understand the rest of the properties.
02

- Verify \(|\bar{z}| = |z|\)

The magnitude of a complex number \(z = a + bi\) is \(|z| = \sqrt{a^2 + b^2}\). For the conjugate \(\bar{z} = a - bi\), the magnitude is \(|\bar{z}| = \sqrt{a^2 + b^2}\), which is identical to \(|z|\). Thus, \(|\bar{z}| = |z|\) is true.
03

- Verify \(\arg z = \arg \bar{z}\)

The argument of \(z = a + bi \) is \(\theta = \arctan\left(\frac{b}{a}\right)\), and for \(\bar{z} = a - bi\), the argument is \(\theta = \arctan\left(\frac{-b}{a}\right)\). Since arguments are taken modulo \(2\pi\), \(\arg \bar{z} = -\arg z\). Therefore, \(\arg z = \arg \bar{z}\) is not generally true.
04

- Verify \(z \bar{z}\) is Real

Multiplying \(z = a + bi\) by its conjugate \(\bar{z} = a - bi\): \[ z \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \] Since \(a^2 + b^2\) is a real number, \(z \bar{z}\) is real.
05

- Verify \(\frac{z}{z}\) is Real

Simplifying \(\frac{z}{z}\): \[ \frac{z}{z} = 1 \] Since 1 is a real number, \(\frac{z}{z}\) is real.
06

- Verify \(\bar{z}\) is the Mirror Image of \(z\) in the Y-Axis

The conjugation \(\bar{z} = a - bi\) reflects \(z\) over the real axis, not the \(y\)-axis. Thus, it is not a mirror image in the \(y\)-axis.

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Key Concepts

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

Conjugate of a Complex Number
A complex number is written in the form of \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The conjugate of a complex number \(z\) is denoted as \(\bar{z}\). The conjugate of \(z\) is found by changing the sign of the imaginary part. For the complex number \(z = a + bi\), the conjugate \(\bar{z}\) is defined as \(\bar{z} = a - bi\).
  • The conjugate effectively mirrors the complex number across the real axis.
  • This property helps in various algebraic operations, such as simplifying expressions and solving equations involving complex numbers.
Conjugation is a fundamental concept that will repeatedly appear in your study of complex numbers.
Magnitude of Complex Numbers
The magnitude (or modulus) of a complex number \(z = a + bi\) is a measure of its distance from the origin in the complex plane. It is denoted as \(|z|\) and calculated using the formula:
\[ |z| = \sqrt{a^2 + b^2} \]
  • This formula follows from the Pythagorean theorem.
  • The magnitude is always a non-negative value.
The magnitude of a complex number remains the same even after conjugation. For example, if \(z = a + bi\), then \(|z| = |a + bi| = \sqrt{a^2 + b^2}\) and \(|\bar{z}| = |a - bi| = \sqrt{a^2 + b^2}\). Therefore, \(|z| = |\bar{z}|\). This property is useful in operations involving the magnitude of complex numbers.
Argument of Complex Numbers
The argument of a complex number \(z = a + bi\), denoted as \(\arg(z)\), is the angle \(\theta\) formed with the positive real axis in the complex plane. It is given by:
\[ \arg(z) = \arctan\left(\frac{b}{a}\right) \]
  • Note that \(\arctan\left(\frac{b}{a}\right)\) gives an angle in the correct quadrant based on the signs of \(a\) and \(b\).
  • Arguments are typically measured in radians and taken modulo \(2\pi\).
For the complex number and its conjugate:
  • \(\arg(z) = \arctan\left(\frac{b}{a}\right)\)
  • \(\arg(\bar{z}) = \arctan\left(\frac{-b}{a}\right)\)
These values give opposite angles, implying \(\arg(\bar{z}) = -\arg(z)\). It's important to note that \(\arg(z)\) and \( \arg(\bar{z})\) are generally not equal but are related through symmetry.
Real and Imaginary Parts of Complex Numbers
A complex number \(z = a + bi\) consists of two parts:
  • The real part, denoted \(\text{Re}(z)\), is \(a\).
  • The imaginary part, denoted \(\text{Im}(z)\), is \(b\).
These parts can be visualized on the complex plane, where the real part corresponds to the horizontal (x) axis, and the imaginary part corresponds to the vertical (y) axis.
  • For \(z = a + bi\), \(\text{Re}(z) = a\) and \(\text{Im}(z) = b\).
  • For \(\bar{z} = a - bi\), \(\text{Re}(\bar{z}) = a\) and \(\text{Im}(\bar{z}) = -b\).
Understanding the separation of real and imaginary parts is crucial for solving complex algebraic equations and performing complex arithmetic. It's also foundational for other advanced topics in complex analysis.

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Most popular questions from this chapter

If \(a=3-\mathrm{i}\) and \(b=1+2 \mathrm{i}\), find the moduli of: (a) \(2 a+3 b\) (b) \(\frac{a}{2 b}\)

The equation \(x^{2}+3 x+1=0\) has: (a) no roots (b) one real and one complex root (c) two imaginary roots (d) two real roots (e) two complex roots.

\(-\frac{5 \pi}{6}\) is an argument of: (a) \(\cos \frac{5 \pi}{6}-i \sin \frac{5 \pi}{6}\) (b) \(\cos \frac{11 \pi}{6}+i \sin \frac{11 \pi}{6}\) (c) \(-\sqrt{3}-\mathrm{i}\) (d) \(\sqrt{3-i}\) (e) \(-\sqrt{3}+\mathrm{i}\)

When \(\sqrt{3}-\mathrm{i}\) is divided by \(-1-\mathrm{i}\) the modulus and argument of the quotient are respectively: (a) \(2 \sqrt{2}, \frac{7 \pi}{12}\) (b) \(\sqrt{2},-\frac{11 \pi}{12}\) (c) \(\sqrt{2}, \frac{7 \pi}{12}\) (d) \(2 \sqrt{2},-\frac{11 \pi}{12}\) (e) \(\sqrt{2}, \frac{11 \pi}{10}\).

5) Express the complex number \(\frac{5+12 \mathrm{i}}{3+4 \mathrm{i}}\) in the form \(a+\mathrm{i} b\) and in the form \(r(\cos \theta+\mathrm{i} \sin \theta)\) giving the values of \(a, b, r, \cos \theta, \sin \theta\).
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