Problem 18 - Precalculus Mathematics for Calculus

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 \((-5, 6)\) for \(z\) and \((-5, -6)\) for \(\overline{z}\) on the complex plane.

Step 1: Understand the Complex Numbers

The complex number given is \( z = -5 + 6i \). The real part is \(-5\), and the imaginary part is \(6\). The complex conjugate, denoted as \( \overline{z} \), is \( -5 - 6i \), which simply changes the sign of the imaginary part.

Step 2: Identify Coordinates for z

On the complex plane, the real part represents the horizontal axis, and the imaginary part represents the vertical axis. For \(z = -5 + 6i\), the point will have coordinates \((-5, 6)\). This means moving 5 units to the left (due to -5) and 6 units up from the origin.

Step 3: Identify Coordinates for Conjugate

The conjugate \( \overline{z} = -5 - 6i \) will have coordinates \((-5, -6)\). This means moving 5 units to the left and 6 units down from the origin on the complex plane.

Step 4: Plot the Complex Number and its Conjugate

Draw a two-dimensional plane where the horizontal axis is the real axis and the vertical axis is the imaginary axis. Plot the point \((-5, 6)\) for \(z\) and the point \((-5, -6)\) for \(\overline{z}\). These two points will be symmetric with respect to the real axis.

Key Concepts

Complex Plane

When dealing with complex numbers, the complex plane is fundamental. It is a two-dimensional plane where each point represents a complex number. This plane has two perpendicular axes:

• Real Axis: This is the horizontal axis. It represents the real part of complex numbers. For example, for the complex number \( -5 + 6i \), the real part \(-5\) is represented on this axis.
• Imaginary Axis: This is the vertical axis. It showcases the imaginary part of complex numbers. In the case of \( -5 + 6i \), the imaginary part \(6\) finds its place on this axis.

The complex plane allows for a visual representation of complex numbers, making it easier to understand their mathematical properties and operations. When plotting on the complex plane, the

• x-coordinate shows the real part.
• y-coordinate displays the imaginary part.

So, a point \((-5, 6)\) for the complex number \(-5 + 6i\) is marked by moving 5 units left for the real part and moving 6 units up for the imaginary part. Similarly, plotting its conjugate \(-5 - 6i\) involves moving 5 units left and 6 units down, reflecting it over the real axis.

Complex Conjugate

Complex conjugates are pairs of complex numbers that have the same real part but differ in the sign of their imaginary part. If you have a complex number \( z = a + bi \), its conjugate \( \overline{z} \) will be \( a - bi \).

• For example, for \( z = -5 + 6i \), the conjugate is \( \overline{z} = -5 - 6i \).
• Conjugates are essential in many mathematical operations, such as simplifying divisions of complex numbers.

When you plot a complex number and its conjugate on the complex plane, they will be reflections of each other across the real axis. This symmetry can be seen easily in our example: plot \( z = -5 + 6i \) at the point \((-5, 6)\), then its conjugate \( \overline{z} = -5 - 6i \) at \((-5, -6)\). This mirrored relationship helps in several applications, including calculating magnitudes and performing geometrical transformations.

Imaginary Part

The imaginary part of a complex number is significant for understanding complex numbers fully. A complex number is typically expressed as \( z = a + bi \), where \(i\) is the imaginary unit satisfying \( i^2 = -1 \). The imaginary part here is \( b \).

• In the number \( z = -5 + 6i \), the imaginary part is \(6\).
• This number indicates movement along the imaginary axis on the complex plane.

It's important because the imaginary part, in conjunction with the real part, determines the position of a complex number on the complex plane. The magnitude of the imaginary part also plays a crucial role in determining properties like the modulus or absolute value of the complex number, calculated as \( \sqrt{a^2 + b^2} \), combining both the real and imaginary components for distance from the origin.