Problem 81 - College Trigonometry

Question

Recall from Section \(3.4\) that given a complex number \(z=a+b i\) its complex conjugate, denoted \(\bar{z}\), is given by \(\bar{z}=a-b i\) (a) Prove that \(|\bar{z}|=|z|\). (b) Prove that \(|z|=\sqrt{z \bar{z}}\) (c) Show that \(\operatorname{Re}(z)=\frac{z+\bar{z}}{2}\) and \(\operatorname{Im}(z)=\frac{z-\bar{z}}{2 i}\) (d) Show that if \(\theta \in \arg (z)\) then \(-\theta \in \arg (\bar{z})\). Interpret this result geometrically. (e) Is it always true that \(\operatorname{Arg}(\bar{z})=-\operatorname{Arg}(z) ?\)

Step-by-Step Solution

Answer

(a) \(|\bar{z}|=|z|\); (b) \(|z|=\sqrt{z\bar{z}}\); (c) \(\operatorname{Re}(z)=\frac{z+\bar{z}}{2}\), \(\operatorname{Im}(z)=\frac{z-\bar{z}}{2i}\); (d) \(-\theta \in \arg (\bar{z})\); (e) Not always.

Step 1: Prove that \(|\bar{z}|=|z|\)

Given a complex number \(z = a + bi\), its conjugate is \(\bar{z} = a - bi\). The magnitude of \(z\) is \(|z| = \sqrt{a^2 + b^2}\) and the magnitude of \(\bar{z}\) is \(|\bar{z}| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2}\). Thus, \(|\bar{z}| = |z|\).

Step 2: Prove that \(|z|=\sqrt{z \bar{z}}\)

We have \(z = a + bi\) and \(\bar{z} = a - bi\). The product \(z \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\) since \(i^2 = -1\). Therefore, \(\sqrt{z \bar{z}} = \sqrt{a^2 + b^2} = |z|\).

Step 3: Show \(\operatorname{Re}(z)=\frac{z+\bar{z}}{2}\) and \(\operatorname{Im}(z)=\frac{z-\bar{z}}{2i}\)

For \(z = a + bi\) and \(\bar{z} = a - bi\), the real part is \(\operatorname{Re}(z) = a\). Calculating \(\frac{z+\bar{z}}{2} = \frac{(a+bi) + (a-bi)}{2} = \frac{2a}{2} = a\). For the imaginary part, \(\operatorname{Im}(z) = b\), and \(\frac{z - \bar{z}}{2i} = \frac{(a+bi)-(a-bi)}{2i} = \frac{2bi}{2i} = b\).

Step 4: Show \(-\theta \in \arg (\bar{z})\) if \(\theta \in \arg (z)\)

If \(z = re^{i\theta}\), where \(r = |z|\) and \(\theta = \arg(z)\), then \(\bar{z} = re^{-i\theta}\). Thus \(\arg(\bar{z}) = -\theta\), interpreting geometrically means the conjugate reflects across the real axis.

Step 5: Determine if \(\operatorname{Arg}(\bar{z})=-\operatorname{Arg}(z)\) is always true

The principal argument \(\operatorname{Arg}(z)\) usually takes values in \((-\pi, \pi]\). For \(\bar{z}\), the principal argument is \(-\theta\), which is in \((-\pi, \pi]\), thus \(\operatorname{Arg}(\bar{z})\) can equal \(-\operatorname{Arg}(z)\), but due to angles wrapping around, it is not always \(-\theta\) exactly.

Key Concepts

Complex Conjugate

To get the complex conjugate of a complex number, flip the sign of the imaginary part. For example, for a complex number \( z = a + bi \), its conjugate is \( \bar{z} = a - bi \). This switch in sign is useful in various mathematical operations.

• It is used to simplify the division of complex numbers.
• It plays a critical role in finding the modulus (or absolute value) of complex numbers.
• Conjugates help by 'focusing' the point exactly opposite across the real axis.

This reflection property is not only algebraic but also has a visual geometric interpretation, useful to understand complex planes.

Modulus of a Complex Number

The modulus of a complex number can be thought of as its "distance" from the origin in the complex plane. For a complex number \( z = a + bi \), its modulus is denoted as \( |z| \) and calculated by the formula:\[ |z| = \sqrt{a^2 + b^2} \]

• This is the geometric length of the vector represented by the complex number.
• The modulus is always a non-negative number.
• It embodies the Pythagorean theorem in the plane of complex numbers, relating length directly to its components.

The relationship \( |\bar{z}| = |z| \) holds true, as reversing the sign of the imaginary component does not change the distance from the origin.

Real and Imaginary Parts of Complex Numbers

Each complex number \( z = a + bi \), contains a real part \(\operatorname{Re}(z)\) and an imaginary part \(\operatorname{Im}(z)\). These can be extracted using:

• The real part: \( \operatorname{Re}(z) = \frac{z + \bar{z}}{2} \)
• The imaginary part: \( \operatorname{Im}(z) = \frac{z - \bar{z}}{2i} \)

These formulas help to distinguish and separate the contributions of different parts of a complex number.

• The real part works as a "horizontal" shift on the complex plane.
• The imaginary represents the "vertical" movement.

Separation ensures clearer interpretation of these numbers when they represent physical phenomena like electrical currents or waves.

Argument of a Complex Number

The argument of a complex number is the angle \( \theta \) that the complex number's 'position vector' makes with the positive real axis. Notation used for this is \( \arg(z) \) for any complex number \( z \). If \( z = re^{i\theta} \), then \( \theta \) is the angle made by the line connecting the origin with the point \( (a, b) \).

• It typically ranges from \( -\pi \) to \( \pi \).
• This angle helps position any complex number within a unit circle.
• The principal argument is the specific angle used within this standard range.

An interesting property is that for a conjugate \( \bar{z} \), the argument is the negative: \( \arg(\bar{z}) = -\arg(z) \). Geometrically, this represents a reflection over the real axis, which is important when considering rotations and symmetries in the plane.