Problem 20 - Algebra and Trigonometry

Question

Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$ z_{1}=-1+i, \quad z_{2}=2-3 i $$

Step-by-Step Solution

Answer

Plot \(z_1 = (-1, 1), z_2 = (2, -3), z_1+z_2 = (1, -2)\), and \(z_1 z_2 = (1, 5)\) on the complex plane.

Step 1: Identify Location of z1 and z2

We have two complex numbers, \(z_1 = -1 + i\) and \(z_2 = 2 - 3i\). In the complex plane, \(z_1\) corresponds to the point \((-1, 1)\) and \(z_2\) to the point \((2, -3)\).

Step 2: Plot z1 and z2 on the Complex Plane

On the complex plane, plot the point \((-1, 1)\) for \(z_1\) and \((2, -3)\) for \(z_2\). The x-axis represents the real part, and the y-axis represents the imaginary part.

Step 3: Calculate z1 + z2

Add the complex numbers: \(z_1 + z_2 = (-1 + i) + (2 - 3i) = 1 - 2i\). The result corresponds to the point \((1, -2)\).

Step 4: Plot z1 + z2 on the Complex Plane

Plot the point \((1, -2)\) for \(z_1 + z_2\) on the complex plane. This represents the sum of \(z_1\) and \(z_2\).

Step 5: Calculate z1*z2

Use the distributive property to multiply the complex numbers: \(z_1 \cdot z_2 = (-1 + i)(2 - 3i)\). First, distribute the terms: \((-1 \cdot 2) + (-1 \cdot (-3i)) + (i \cdot 2) + (i \cdot (-3i))\). Simplify to obtain \(-2 + 3i + 2i - 3i^2\). Since \(i^2 = -1\), this becomes \(-2 + 5i + 3 = 1 + 5i\).

Step 6: Plot z1*z2 on the Complex Plane

Plot the point \((1, 5)\) for \(z_1 \cdot z_2\) on the complex plane, which represents the product of \(z_1\) and \(z_2\).

Key Concepts

Complex Plane

The complex plane is a crucial concept when understanding complex numbers. It is a two-dimensional plane where each point represents a complex number. The horizontal axis is known as the real axis, and it corresponds to the real part of a complex number. Meanwhile, the vertical axis is called the imaginary axis, representing the imaginary part of the complex number.

For instance, the complex number \( z_1 = -1 + i \) would be plotted at the point \((-1, 1)\) because -1 is the real part and \( i \) (or 1 times \( i \)) is the imaginary part. Likewise, \( z_2 = 2 - 3i \) translates to the point \((2, -3)\).

Visualizing complex numbers in this way, on the complex plane, helps in understanding operations like addition and multiplication as movements within this space. The complex plane provides a visual context that is immensely helpful for grasping the relationships between complex numbers.

Addition of Complex Numbers

When we add complex numbers, we consider the real parts and imaginary parts separately. This operation resembles vector addition in two-dimensional geometry if you think of real and imaginary components as coordinates.

For example, adding \( z_1 = -1 + i \) and \( z_2 = 2 - 3i \):

• Add the real parts: \(-1 + 2 = 1\)
• Add the imaginary parts: \(1i - 3i = -2i\)

This gives us the new complex number, \( z_1 + z_2 = 1 - 2i \). On the complex plane, this addition corresponds to moving from point \( (-1, 1) \) to \( (1, -2) \).

Complex addition can be visually checked by plotting each number and then their sum, ensuring they form a parallelogram with the sum point being the diagonal vertex, a relation similar to vector addition rules.

Multiplication of Complex Numbers

Multiplying complex numbers requires the use of distribution, just like multiplying two binomials. Remember that \( i^2 = -1 \) is a key component in simplifying these products.

Let's multiply \( z_1 = -1 + i \) and \( z_2 = 2 - 3i \). We apply distributive property here:

• \((-1 \cdot 2) = -2\)
• \((-1 \cdot -3i) = 3i\)
• \((i \cdot 2) = 2i\)
• \((i \cdot -3i) = -3i^2 = 3\) (because \(-i^2 = 1\))

Combining these results, we have:
\(-2 + 3i + 2i + 3 = 1 + 5i\).

This final number \(1 + 5i\) is then plotted on the complex plane at point \((1, 5)\). This operation can be visualized as a rotation and scaling on the plane, unlike the straightforward path seen in addition.