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Chapter 2: Problem 3

Express arg \((\bar{z})\) and \(\arg (-z)\) in terms of \(\arg (z)\).

Short Answer

Expert verified
Question: Express the argument of the complex conjugate of a complex number and the argument of the negative of a complex number in terms of the argument of the given complex number. Answer: 1. For the complex conjugate, the relationship between their arguments is given by: \(\arg (\bar{z}) = -\arg (z)\) 2. For the negative of a complex number, the relationship between their arguments is given by: \(\arg (-z) = \arg (z) + \pi,\) if \(\arg (z) < 0\) \(\arg (-z) = \arg (z) - \pi,\) if \(\arg (z) \ge 0\)

Step by step solution

01

Review the concept of the argument of a complex number

The argument of a complex number, denoted as \(\arg (z)\), represents the angle between the positive x-axis and the line connecting the complex number to the origin in the complex plane. Given a complex number \(z = a + bi\), the argument \(\arg (z)\) can be found using the following: \(\arg (z) = \tan^{-1}\left(\frac{b}{a}\right)\) where \(a\) is the real part and \(b\) is the imaginary part of the complex number.
02

Find the relationship between \(z\) and its conjugate \(\bar{z}\) in the complex plane

The complex conjugate of the complex number \(z = a + bi\) is denoted as \(\bar{z} = a - bi\). Geometrically, taking the conjugate of a complex number reflects it across the real axis. For the argument of the complex conjugate, we have: \(\arg (\bar{z}) = \tan^{-1}\left(\frac{-b}{a}\right)\) Since the reflection is across the real axis, the angle between the line connecting \(\bar{z}\) to the origin and the positive x-axis will have the negative value of the argument of \(z\). Therefore: \(\arg (\bar{z}) = -\arg (z)\)
03

Find the relationship between \(z\) and \(-z\) in the complex plane

The negative of a complex number is defined as \(-z = -a - bi\). Geometrically, taking the negative of a complex number rotates it by \(180^\circ\) or \(\pi\) radians about the origin. For the argument of the negative of the complex number, we have: \(\arg (-z) = \tan^{-1}\left(\frac{-b}{-a}\right)\) Since the rotation is by \(180^\circ\) or \(\pi\) radians, we need to find the sum of the argument of \(z\) and the angle of rotation. We should also consider the case when the sum exceeds \(180^\circ\) or \(\pi\) radians, in which case we need to subtract \(360^\circ\) or \(2\pi\) radians (to find the principle argument in the range \([-\pi, \pi)\)). Therefore: \(\arg (-z) = \arg (z) + \pi,\) if \(\arg (z) < 0\) \(\arg (-z) = \arg (z) - \pi,\) if \(\arg (z) \ge 0\) In conclusion, we have found the relationships between the arguments of a complex number \(z\), its conjugate \(\bar{z}\), and its negative \(-z\): 1. \(\arg (\bar{z}) = -\arg (z)\) 2. \(\arg (-z) = \arg (z) + \pi,\) if \(\arg (z) < 0\) 3. \(\arg (-z) = \arg (z) - \pi,\) if \(\arg (z) \ge 0\)

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Key Concepts

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

Argument of a Complex Number
When we deal with complex numbers, it's not just about the magnitude; the direction also matters. This is where the concept of the argument of a complex number comes into play. The argument refers to the angle a complex number makes with the positive x-axis on the complex plane. This angle is typically represented in radians.

For a complex number given by \(z = a + bi\), the argument is calculated using arctan, specifically \(\arg (z) = \tan^{-1}\left(\frac{b}{a}\right)\). This angle helps us understand the position of \(z\) in relation to the real axis. Knowing the argument allows us to write the complex number in polar form, which can be extremely helpful in multiplication or finding powers of complex numbers.
Complex Conjugate
The complex conjugate is a simple yet powerful concept in complex number theory. Given a complex number \(z = a + bi\), its complex conjugate is denoted as \(\bar{z} = a - bi\). Visually, if you plot both on the complex plane, the conjugate is the 'reflection' of the original number across the real axis.

This reflection changes the sign of the imaginary part, which, in turn, changes the sign of the argument. If \(z\) has an argument of \(\theta\), then \(\bar{z}\) will have an argument of \(-\theta\). It's vital to remember the effects of a conjugate in operations because it directly impacts trigonometric expressions and polar forms involving complex numbers.
Complex Plane Geometry
The geometry of the complex plane is fascinating. Each point on this plane represents a complex number where the horizontal axis is the real part and the vertical axis is the imaginary part. The complex plane allows us to apply geometric interpretations to complex number operations. For example, multiplication by \(i\) corresponds to a \(90^\circ\) rotation.

Similarly, when we take the conjugate or the negative of a complex number, we are performing geometric transformations - reflection across the real axis and rotation by \(180^\circ\), respectively. Understanding these geometric moves in the complex plane can often make complex number calculations more intuitive.
Principal Argument
The principal argument of a complex number is the 'main' argument that we refer to and it falls within the interval \([-\pi, \pi)\). When we perform operations like taking the negative of a complex number \(-z\), we might need to adjust the argument to keep it within this principal range.

If an operation results in an argument below \(-\pi\) or above \(\pi\), we subtract or add \(2\pi\) to bring it back to the principal argument. This concept ensures that every complex number has a unique argument in the prescribed interval, simplifying comparisons and other operations in the complex domain.

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Most popular questions from this chapter

Find polar representations for the following complex numbers: a) \(z_{1}=\cos a-i \sin a, \quad a \in[0,2 \pi)\) b) \(z_{2}=\sin a+i(1+\cos a), \quad a \in[0,2 \pi)\) c) \(z_{3}=\cos a+\sin a+i(\sin a-\cos a), \quad a \in[0,2 \pi)\) d) \(z_{4}=1-\cos a+i \sin a, \quad a \in[0,2 \pi)\)

Find the geometric images for the complex numbers \(z\) in each of the following cases: a) \(|z|=2\); b) \(|z+i| \geq 2\) c) \(|z-i| \leq 3\) d) \(\pi<\arg z<\frac{5 \pi}{4}\); e) \(\arg z \geq \frac{3 \pi}{2} ; \quad\) f) \(\arg z<\frac{\pi}{2}\) g) \(\arg (-z) \in\left(\frac{\pi^{4}}{6}, \frac{\pi}{3}\right)\); h) \(|z+1+i|<3\) and \(0<\arg z<\frac{\pi}{6}\).

Compute the following products using the polar representation of a complex number: a) \(\left(\frac{1}{2}-i \frac{\sqrt{3}}{2}\right)(-3+3 i)(2 \sqrt{3}+2 i)\) b) \((1+i)(-2-2 i) \cdot i\) c) \(-2 i \cdot(-4+4 \sqrt{3} i) \cdot(3+3 i)\) d) \(3 \cdot(1-i)(-5+5 i)\). Verify your results using the algebraic form.

Find the cartesian coordinates for the following points, given their polar coordinates: a) \(P_{1}\left(2, \frac{\pi}{3}\right)\); b) \(P_{2}\left(4,2 \pi-\arcsin \frac{3}{5}\right)\); c) \(P_{3}(2, \pi)\); d) \(P_{4}(3,-\pi)\) e) \(P_{5}\left(1, \frac{\pi}{2}\right)\); f) \(P_{6}\left(4, \frac{3 \pi}{2}\right)\).

Prove that de Moivre's formula holds for negative integer exponents.
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