Problem 15 - Algebra and Trigonometry

Question

\(15-16=\) Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$ z_{1}=2-i, \quad z_{2}=2+i $$

Step-by-Step Solution

Answer

The points are (2, -1), (2, 1), (4, 0) for addition, and (5, 0) for multiplication.

Step 1: Understand the Complex Numbers

We have two complex numbers: \( z_{1} = 2 - i \) and \( z_{2} = 2 + i \). These can be visualized on the complex plane where the real part is the x-coordinate and the imaginary part is the y-coordinate.

Step 2: Plot Complex Numbers on the Plane

Plot \( z_{1} = 2 - i \) at the point (2, -1) on the complex plane. Next, plot \( z_{2} = 2 + i \) at the point (2, 1). These points represent the locations of the complex numbers on the plane.

Step 3: Add the Complex Numbers

To find the sum \( z_{1} + z_{2} \), add the real parts and the imaginary parts separately: \( (2 + 2) + (-i + i) = 4 + 0 = 4 \). The point (4, 0) represents \( z_{1} + z_{2} \) on the complex plane.

Step 4: Multiply the Complex Numbers

To find the product \( z_{1} z_{2} \), use the distributive property: \((2 - i)(2 + i) = 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5 \). The point (5, 0) represents \( z_{1} z_{2} \) on the complex plane.

Step 5: Conclusion

We have plotted all necessary points: \( z_{1} = (2, -1) \), \( z_{2} = (2, 1) \), \( z_{1} + z_{2} = (4, 0) \), and \( z_{1} z_{2} = (5, 0) \) on the complex plane.

Key Concepts

Complex Plane

The complex plane is a powerful tool that helps us visually represent and understand complex numbers. To imagine it, think of a standard two-dimensional coordinate system. Here, the horizontal axis (x-axis) is the real axis, while the vertical axis (y-axis) is the imaginary axis.
For any complex number, you express it in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. This number can be plotted as a point whose real part corresponds to the x-coordinate and imaginary part corresponds to the y-coordinate.

• For \( z_1 = 2 - i \), plot at (2, -1).
• For \( z_2 = 2 + i \), plot at (2, 1).
• The point’s location provides insights into its modulus and argument, key attributes of complex numbers.

By using the complex plane, we easily perform operations like addition and multiplication, and visualize the results much like handling vectors in geometry.

Addition of Complex Numbers

Adding complex numbers is quite straightforward and can be likened to vector addition. For two complex numbers, \( z_1 = a + bi \) and \( z_2 = c + di \), you add them simply by summing the real parts and the imaginary parts separately.
For our example,

• Real parts: \( 2 + 2 = 4 \).
• Imaginary parts: \( -i + i = 0 \).
• So, \( z_1 + z_2 = 4 + 0 = 4 \).

Visually, this sum appears as a movement on the complex plane, analogous to adding the corresponding coordinates of two vectors.
On the plane, this takes the form where the endpoint of one number is connected to the head of the other, completing the triangle. This geometric representation reinforces the algebraic addition in a clear and simple way.

Multiplication of Complex Numbers

Multiplying complex numbers uses the distributive property, where each term in one complex number multiplies each term in the other. For two complex numbers, \( z_1 = a + bi \) and \( z_2 = c + di \), their product is given by:
\[ (a + bi)(c + di) = (ac - bd) + (ad + bc)i \]In our case with \( z_1 = 2 - i \) and \( z_2 = 2 + i \):

• Real part: \( 2 \times 2 - (-i) \times i = 4 + 1 = 5 \).
• Imaginary part: \( 2 \times i + (-i) \times 2 = 2i - 2i = 0 \).
• Thus, \( z_1 z_2 = 5 + 0 = 5 \).

The product, represented as a point on the real line (because the imaginary component is zero), shows how multiplication affects not only the magnitude of the numbers but can also alter their direction, just like rotating and stretching vectors on this geometric plane.