Z=e^i(theta) then

1)z+1=

2)z-1=

Where (theta) is argument of z,answer the following questions 1 and 2 given above

To tackle the expressions involving the complex number \( z = e^{i\theta} \), we first need to understand what this representation means. The expression \( e^{i\theta} \) is derived from Euler's formula, which states that \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \). This means that \( z \) can be represented in the complex plane as a point on the unit circle, where the angle \( \theta \) is measured from the positive real axis.

Analyzing the Expressions

Now, let’s break down the two expressions you provided: \( z + 1 \) and \( z - 1 \).

1) Evaluating \( z + 1 \)

Substituting \( z \) into the first expression gives us:

Step 1: Substitute \( z \):
\( z + 1 = e^{i\theta} + 1 \)

Step 2: Rewrite using Euler's formula:
\( e^{i\theta} + 1 = \cos(\theta) + i\sin(\theta) + 1 = (\cos(\theta) + 1) + i\sin(\theta) \)

This expression represents a complex number where the real part is \( \cos(\theta) + 1 \) and the imaginary part is \( \sin(\theta) \). The geometric interpretation is that you are shifting the point \( z \) one unit to the right along the real axis.

2) Evaluating \( z - 1 \)

Now, let’s look at the second expression:

Step 1: Substitute \( z \):
\( z - 1 = e^{i\theta} - 1 \)

Step 2: Again, using Euler's formula:
\( e^{i\theta} - 1 = \cos(\theta) + i\sin(\theta) - 1 = (\cos(\theta) - 1) + i\sin(\theta) \)

In this case, the real part is \( \cos(\theta) - 1 \) and the imaginary part is \( \sin(\theta) \). Geometrically, this expression represents a point that is shifted one unit to the left along the real axis from the point \( z \).

Summary of Results

• For \( z + 1 \): The result is \( (\cos(\theta) + 1) + i\sin(\theta) \).
• For \( z - 1 \): The result is \( (\cos(\theta) - 1) + i\sin(\theta) \).

These expressions illustrate how the complex number \( z \) interacts with basic arithmetic operations, revealing both its real and imaginary components. Understanding these transformations is crucial in complex analysis and can help visualize complex numbers in the plane.