Question

Berechnen Sie Real- und Imaginärteil der komplexen Zahl \(z=e^{i \frac{5 \pi}{6}}+\frac{2-i}{1+i}\).

Short answer

Real part: \( \frac{-\sqrt{3} + 1}{2} \), Imaginary part: \(-1\).

Step-by-step solution

Simplify the Exponential Term

The first term is in exponential form: \[ z_1 = e^{i \frac{5\pi}{6}} \]Using Euler's formula, we can express this as:\[ z_1 = \cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right) \]The angle \( \frac{5\pi}{6} \) corresponds to 150 degrees. From trigonometric tables, we find:\[ \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}, \quad \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \]Thus:\[ z_1 = -\frac{\sqrt{3}}{2} + i\frac{1}{2} \]

Simplify the Fractional Term

Now simplify \( z_2 = \frac{2-i}{1+i} \). Multiply numerator and denominator by the conjugate of the denominator:\[ z_2 = \frac{(2-i)(1-i)}{(1+i)(1-i)} \]Calculate the denominator:\[ (1+i)(1-i) = 1^2 + 1^2 = 2 \]Calculate the numerator:\[ (2-i)(1-i) = 2 \times 1 - 2 \times i - 1 \times i + i^2 = 2 - 3i - 1 = 1-3i \]Thus:\[ z_2 = \frac{1-3i}{2} = \frac{1}{2} - \frac{3}{2}i \]

Combine Both Terms

Now add the two simplified parts together to get the full complex number:\[ z = z_1 + z_2 = \left(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\right) + \left(\frac{1}{2} - \frac{3}{2}i\right) \]Combine real and imaginary components separately:Real part:\[ -\frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{-\sqrt{3} + 1}{2} \]Imaginary part:\[ \frac{1}{2} - \frac{3}{2} = -1 \]Thus, the complex number is:\[ z = \frac{-\sqrt{3} + 1}{2} - i \]

Identify Real and Imaginary Parts

From the expression for the complex number \( z \), identify the real and imaginary components:- Real part: \( \text{Re}(z) = \frac{-\sqrt{3} + 1}{2} \)- Imaginary part: \( \text{Im}(z) = -1 \)

Key concepts

Euler's Formula

Euler's formula is a fundamental bridge between trigonometry and complex numbers. It's expressed as \( e^{ix} = \cos(x) + i\sin(x) \). This formula shows how exponential functions with imaginary exponents relate to circular functions like sine and cosine.

This relationship is particularly useful when dealing with complex numbers in polar form as it allows us to combine rotational dynamics with exponential growth.

• The angle \( x \) in Euler's formula represents the rotation angle in the complex plane.
• When using Euler's formula, \( \cos(x) \) represents the real part and \( i\sin(x) \) represents the imaginary part.

For this exercise, Euler's formula helps us turn the exponential term \( e^{i \frac{5\pi}{6}} \) into the sum of \( \cos\left(\frac{5\pi}{6}\right) \) and \( i\sin\left(\frac{5\pi}{6}\right) \). This translates the problem into a simpler form which can then be easily manipulated.

Trigonometry

Trigonometry plays a crucial role in solving problems involving complex numbers—especially when these numbers are expressed in polar form. By understanding sine and cosine, we can easily manipulate the real and imaginary components of complex numbers.

In our exercise, we used trigonometric identities to find:

• \( \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \)
• \( \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \)

These values come from knowing the unit circle, which is essential for determining angle-related ratios.

Whether you’re converting back-and-forth between exponential form and rectangular form, or simply adding and subtracting complex numbers, understanding these trigonometric functions provides clarity and precision.

Imaginary Numbers

Imaginary numbers are numbers that involve the square root of (-1), denoted as \( i \). Imaginary numbers allow us to work with numbers that cannot be placed on the traditional number line.

In the example provided, the imaginary parts stem from both Euler's transformation and the division of complex numbers in the second term.

• The imaginary part from \( e^{i \frac{5\pi}{6}} \) results from \( i\sin\left(\frac{5\pi}{6}\right) \), giving us \( i\frac{1}{2} \).
• The imaginary part from \( \frac{2-i}{1+i} \) is calculated by expressing the fraction in a simpler form: \( \frac{-3}{2}i \).

By combining these calculations, the imaginary part of the entire complex number becomes \(-1\). This shows how imaginary numbers interact and combine with other numbers in complex expressions.

Real Numbers

Real numbers are found on the traditional number line and they are an indispensable part of complex numbers, displayed as the "x" component in the complex plane coordinate system.

In the given exercise, we discover real parts in two segments.

• From Euler's formula application, the real part is born out of the cosine function \( \cos\left(\frac{5\pi}{6}\right) \) resulting in \(-\frac{\sqrt{3}}{2}\).
• In the simplified fraction \( \frac{2-i}{1+i} \), we find innovation is necessary: after calculations, the fraction simplifies to give us an additional real part \( \frac{1}{2} \).

By collaborating these real parts, we arrived at a final real part of the overall complex number as \( \frac{-\sqrt{3}+1}{2} \). Real numbers act as constants in this blend, anchoring the complex numbers to our familiar real world.