Problem 17 - Algebra and Trigonometry

Question

Sketch the complex number \(z\) and its complex conjugate \(z\) on the same complex plane. $$ z=8+2 i $$

Step-by-Step Solution

Answer

Plot the points \((8, 2)\) and \((8, -2)\) on the complex plane.

Step 1: Identify the Real and Imaginary Parts

The complex number is given as \(z = 8 + 2i\). In this expression, \(8\) is the real part and \(2\) is the coefficient of the imaginary unit \(i\). Thus, the real part is \(8\) and the imaginary part is \(2\).

Step 2: Comprehend the Complex Conjugate

The complex conjugate of a complex number \(z = a + bi\) is \(\bar{z} = a - bi\). For our given number \(z = 8 + 2i\), the complex conjugate \(\bar{z}\) would be \(8 - 2i\).

Step 3: Plot the Complex Number On the Complex Plane

The complex plane consists of the real axis (horizontal) and the imaginary axis (vertical). The point corresponding to the complex number \(z = 8 + 2i\) can be plotted at the coordinates \((8, 2)\) since the real part is \(8\) and the imaginary part is \(2\).

Step 4: Plot the Complex Conjugate On the Complex Plane

Similarly, plot the complex conjugate \(\bar{z} = 8 - 2i\). This can be represented by the point \((8, -2)\) on the complex plane, with the real part \(8\) and the imaginary part \(-2\).

Step 5: Visualize the Points

Now on the complex plane, you should visualize two points: one at \((8, 2)\) representing \(z\), and another at \((8, -2)\) representing \(\bar{z}\). These points are symmetric with respect to the real axis.

Key Concepts

Real Part

In complex numbers, the real part is essentially the portion that doesn't involve the imaginary unit, denoted as \( i \). For a complex number \( z = a + bi \), the real part is the number \( a \). This component can be viewed like any real number on the familiar number line.
In the expression \( z = 8 + 2i \), the real part is simply \( 8 \).
The real part gives a sense of how far along the horizontal axis (real axis) the number is located when plotted on the complex plane.

• It does not involve any multiplication by \( i \).
• It is purely a real value, no imaginary component.

Imaginary Part

The imaginary part of a complex number includes the imaginary unit \( i \), which satisfies \( i^2 = -1 \). It's what makes a number complex as opposed to real. In the format \( z = a + bi \), the imaginary part is represented by \( b \), which is the coefficient of \( i \).
In our example \( z = 8 + 2i \), the imaginary part is \( 2 \).
Imaginary parts are graphed along the vertical axis (imaginary axis) on a complex plane.

• It's not identical to the value; it signifies the 'imaginary direction'.
• No imaginary multiplier (\( i \)) is needed in calculations for real numbers.

Complex Conjugate

A complex conjugate of a complex number is formed by changing the sign of the imaginary part. If you start with \( z = a + bi \), the complex conjugate, \( \bar{z} \), will be \( a - bi \).
For the complex number \( z = 8 + 2i \), the complex conjugate is \( \bar{z} = 8 - 2i \).
This swap creates a mirror image of the original complex number across the real axis when plotted on the complex plane.

• Complex conjugates are useful for simplifying division among complex numbers.
• They always have the same real part as the original number.

Complex Plane

The complex plane is like a two-dimensional graph where each complex number corresponds to a unique point. This involves a horizontal axis, called the real axis, and a vertical axis, known as the imaginary axis.
For our number \( z = 8 + 2i \), it appears on the plane at point \( (8, 2) \).
Its conjugate \( \bar{z} = 8 - 2i \) would appear at \( (8, -2) \).

• The real part of the complex number determines its position on the horizontal axis.
• The imaginary part sets its location on the vertical axis.

Complex numbers and their conjugates form symmetric points on the plane, reflecting across the real axis.