Problem 14 - Algebra and Trigonometry

Question

\(13-14\) : Sketch the complex number \(z\) and its complex conjugate \(z\) on the same complex plane. $$ z=-5+6 i $$

Step-by-Step Solution

Answer

Sketch \((-5, 6)\) for \(z\) and \((-5, -6)\) for its conjugate.

Step 1: Identify the Complex Number

The given complex number is \( z = -5 + 6i \). This number can be expressed as having a real part of \(-5\) and an imaginary part of \(6\).

Step 2: Determine the Complex Conjugate

The complex conjugate of a complex number \( z = a + bi \) is \( \overline{z} = a - bi \). Thus, the complex conjugate of \( z = -5 + 6i \) is \( \overline{z} = -5 - 6i \).

Step 3: Sketch \(z\) on the Complex Plane

On the complex plane, the real axis is horizontal and the imaginary axis is vertical. To plot \( z = -5 + 6i \), start at the origin \((0,0)\), move 5 units to the left (along the real axis) to \(-5\), and then move 6 units up (along the imaginary axis) to the point \((-5, 6)\).

Step 4: Sketch \(\overline{z}\) on the Complex Plane

To plot \( \overline{z} = -5 - 6i \), start at the origin \((0,0)\), move 5 units to the left to \(-5\), and then move 6 units down to the point \((-5, -6)\).

Step 5: Analyze the Points

Notice that \( z = -5 + 6i \) and \( \overline{z} = -5 - 6i \) are symmetric with respect to the real axis on the complex plane.

Key Concepts

Complex Plane

The complex plane is an incredibly useful tool for visualizing complex numbers, which are composed of both a real part and an imaginary part. Imagine it as a two-dimensional graph, similar to the Cartesian plane, but with specific axes representing different components of the complex number. In this plane:

• The Real Axis, running horizontally, is where you plot the real part of a complex number.
• The Imaginary Axis, running vertically, corresponds to the imaginary part of the number.

Each complex number can be depicted as a point or a vector from the origin to this point. For example, the complex number \(z = -5 + 6i\) can be plotted by moving \(-5\) units on the real axis (left of origin) and \(6\) units along the imaginary axis (upwards), ending up at the point \(-5, 6\). This geometric representation is not only intuitive but also provides insights on relationships such as symmetry and magnitude.

Complex Conjugate

The concept of the complex conjugate involves creating a mirror image of a complex number across the real axis on the complex plane. Given a complex number \(z = a + bi\), its complex conjugate is represented as \(\overline{z} = a - bi\). This simply involves changing the sign of the imaginary part.

• For \(z = -5 + 6i\), the complex conjugate would be \(\overline{z} = -5 - 6i\).

This operation has practical applications in mathematics, particularly in simplifying division of complex numbers and in solving polynomial equations. Moreover, when you plot both a complex number and its conjugate, they appear symmetrically with respect to the real axis. In our example, both \((-5, 6)\) and \((-5, -6)\) highlight this symmetry visually on the complex plane, demonstrating how the real part remains unchanged while the imaginary part is reflected.

Graphing Complex Numbers

Graphing complex numbers involves plotting them on the complex plane, where each number corresponds to a specific point. The process is straightforward and involves identifying both the real part and the imaginary part.

• Start at the origin of the complex plane, corresponding to the point \((0,0)\).
• Move horizontally along the real axis according to the real part of the number. A negative value dictates moving left.
• From this new position, move vertically along the imaginary axis according to the imaginary part. Move up for positive values and down for negative.

For example, to graph \(z = -5 + 6i\), start from the origin, move \(-5\) units horizontally (left), and \(6\) units vertically (up) to plot the point \((-5, 6)\). To graph its conjugate \(\overline{z} = -5 - 6i\), you start the same way but move \(6\) units downwards, locating the point \((-5, -6)\). This visualization assists in comparing complex numbers and understanding their algebraic relationships.