Problem 18 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Plot \((-5, 6)\) and \((-5, -6)\) on the complex plane.

Step 1: Identify the Complex Number Components

The given complex number is \( z = -5 + 6i \). Here, \(-5\) is the real part, and \(6\) is the imaginary part of the complex number.

Step 2: Determine the Complex Conjugate

The complex conjugate of a complex number changes the sign of the imaginary part. Therefore, the complex conjugate of \( z = -5 + 6i \) is \( z^* = -5 - 6i \).

Step 3: Plotting the Complex Number on the Complex Plane

On the complex plane, the horizontal axis represents the real part, and the vertical axis represents the imaginary part. Plot the point corresponding to \( z = -5 + 6i \). Move 5 units to the left along the real axis (since it's negative) and 6 units up along the imaginary axis.

Step 4: Plotting the Complex Conjugate on the Complex Plane

Similarly, plot the point for \( z^* = -5 - 6i \). Start from the origin, move 5 units to the left along the real axis and 6 units down along the imaginary axis.

Step 5: Verify the Plot

You should see two points on the complex plane. One point should be located at \((-5, 6)\) and the other at \((-5, -6)\). These two points are symmetric with respect to the real axis.

Key Concepts

complex plane

The complex plane is a two-dimensional plane where each point represents a complex number. It is similar to the Cartesian coordinate system, but here:

• The horizontal axis is called the real axis.
• The vertical axis is called the imaginary axis.

In this plane, a complex number is represented as a point or a vector. For example, the complex number \(-5 + 6i\) is depicted by locating \(-5\) on the real axis (x-axis) and moving 6 units up along the imaginary axis (y-axis).
The power of the complex plane is in visualizing operations with complex numbers. Adding complex numbers involves a simple vector addition, and multiplying results in a rotation and change in magnitude. Such a graphical representation helps to understand complex operations and relationships more profoundly.

complex conjugate

A complex conjugate of a complex number is formed by changing the sign of its imaginary part. If a complex number is given by \( z = a + bi \), its complex conjugate is \( z^* = a - bi \).
This transformation has several useful properties, notably in simplifying the division of complex numbers and calculating magnitudes.

• When you multiply a complex number by its conjugate, you eliminate the imaginary component, producing a real number: \( z imes z^* = (a + bi)(a - bi) = a^2 + b^2 \).
• The conjugates are symmetrical about the real axis in the complex plane. For instance, \( z = -5 + 6i \) and \( z^* = -5 - 6i \) are symmetric, lying at \((-5, 6)\) and \((-5, -6)\) respectively.

Understanding complex conjugates is essential for complex arithmetic and certain engineering applications.

real and imaginary parts

Complex numbers are composed of two main components: the real part and the imaginary part. These components allow complex numbers to extend two-dimensional numeric representations.

• The real part of a complex number \( z = a + bi \) is \(a\). It corresponds to horizontal movement on the complex plane.
• The imaginary part is \(b\). It involves movement along the vertical axis and is typically prefixed with \(i\), which represents \(\sqrt{-1}\).

By recognizing these parts, you can perform operations such as addition, subtraction, and multiplication of complex numbers with clarity.
Notably, the real and imaginary parts enable applications across fields like physics and engineering where numbers with direction and magnitude (i.e., vectors) are crucial. Identifying \(z = -5 + 6i\) as having a real part of \(-5\) and an imaginary part of \(6\) also assists in plotting and understanding the properties of complex numbers.