Question

Von den folgenden komplexen Zahlen bestimme man jeweils Real- und Imaginärteit: $$ \begin{gathered} \frac{i-1}{i+1} ; \quad \frac{3+4 i}{1-2 i} ; i^{n}, n \in Z ; \quad\left(\frac{1+i}{\sqrt{2}}\right)^{n}, n \in Z \\ \left(\frac{1+i \sqrt{3}}{2}\right)^{n}, n \in Z: \sum_{v=0}^{T}\left(\frac{1-i}{\sqrt{2}}\right)^{v} ; \frac{(1+i)^{4}}{(1-i)^{3}}+\frac{(1-i)^{4}}{(1+i)^{3}} \end{gathered} $$

Short answer

Steps: fractions simplify to -i, -1+2i; powers of i cycle; series is geometric; sum of complex expressions is 1.

Step-by-step solution

Simplify the first complex fraction

The expression is \( \frac{i-1}{i+1} \). To simplify, multiply the numerator and denominator by the conjugate of the denominator \((i-1)\). This results in \( \frac{(i-1)(i-1)}{(i+1)(i-1)} = \frac{i^2 - 2i + 1}{1^2 - i^2} = \frac{-1 - 2i + 1}{1 + 1} = \frac{-2i}{2} = -i \). Hence, the real part is 0 and the imaginary part is -1.

Simplify the second complex fraction

For \( \frac{3+4i}{1-2i} \), multiply by the conjugate \((1+2i)\) in both the numerator and denominator. This gives \( \frac{(3+4i)(1+2i)}{(1-2i)(1+2i)} = \frac{(3+4i+6i-8)}{1+4} = \frac{3+10i-8}{5} = \frac{-5+10i}{5} = -1 + 2i \). Therefore, the real part is -1 and the imaginary part is 2.

Find real and imaginary parts of powers of \( i \)

The expression is \( i^n \). By using the cyclical nature of powers of \( i \), which are: \( i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 \). For any integer exponent \( n \), modulate by 4 to find the resultant: \( n \mod 4 \). The parts are as follows: \( i^n \) is real for \( n \equiv 0, 2 \pmod{4} \); imaginary for \( n \equiv 1, 3 \pmod{4} \).

Simplify the expression with exponent \( (\frac{1+i}{\sqrt{2}})^n \)

The expression \( \left(\frac{1+i}{\sqrt{2}}\right)^n \) represents a complex number on the unit circle with modulus 1 and an angle of \( \frac{\pi}{4} \). Thus, \( \left(\frac{1+i}{\sqrt{2}}\right)^n = e^{i\frac{n\pi}{4}} \), which simplifies using Euler's formula to \( \cos \left(\frac{n\pi}{4}\right) + i \sin \left(\frac{n\pi}{4} \right) \). The real part is \( \cos \left(\frac{n\pi}{4}\right) \) and the imaginary part is \( \sin \left(\frac{n\pi}{4}\right) \).

Analyze powers of \( \left(\frac{1+i\sqrt{3}}{2}\right) \)

\( \left(\frac{1+i \sqrt{3}}{2}\right) \) is a complex number with modulus 1 that represents a cube root of unity. Its powers rotate between 1, \( \left(\frac{-1+i \sqrt{3}}{2}\right) \), and \( \left(\frac{-1-i \sqrt{3}}{2}\right) \). For any integer \( n \), modulate by 3 to find real part: 1 for \( n \equiv 0 \pmod{3} \), -1/2 for \( n \equiv 1,2 \pmod{3} \); imaginary part: 0 for \( n \equiv 0 \pmod{3} \), \( \sqrt{3}/2 \) for \( n \equiv 1 \pmod{3} \), \( -\sqrt{3}/2 \) for \( n \equiv 2 \pmod{3} \).

Find the sum of a geometric series

For the series \( \sum_{v=0}^{T}\left(\frac{1-i}{\sqrt{2}}\right)^{v} \), recognize it as a geometric series with first term 1 and common ratio \( \frac{1-i}{\sqrt{2}} \). The sum formula is \( S_T = \frac{1 - \left(\frac{1-i}{\sqrt{2}}\right)^{T+1}}{1 - \frac{1-i}{\sqrt{2}}} \). The real part of the sum is \((T+1)\times(\text{real part of }\frac{1-i}{\sqrt{2}}) \) and the imaginary part is \((T+1)\times(\text{imaginary part of }\frac{1-i}{\sqrt{2}}) \).

Simplify and add fractions involving powers of (1+i) and (1-i)

For \( \frac{(1+i)^4}{(1-i)^3} + \frac{(1-i)^4}{(1+i)^3} \), individually find: \( (1+i)^4 = (1+4i-6-4i-1) =-4 \) and similarly \( (1-i)^4 = -4 \). So each fraction simplifies to \( (-4) \), leading to \( -4 \div (-8) = 1/2 \) for each. These add to 1. Therefore, the real part is 1 and the imaginary part is 0.

Key concepts

Real and Imaginary Parts

Complex numbers consist of two main components: the real part and the imaginary part. This can be expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) representing the imaginary unit such that \( i^2 = -1 \).
To find the real and imaginary parts means simply to identify the values of \( a \) and \( b \). For example, in the complex number \( 3 + 4i \), 3 is the real part and 4 is the imaginary part. Understanding these components is crucial as they allow us to perform operations on complex numbers such as addition, subtraction, multiplication, and division.
This separation is also essential when analyzing polar forms and performing tasks like simplifying complex expressions or calculating powers and roots.
Remember that the real and imaginary parts are often expressed separately to facilitate understanding and calculations. For instance, when complex numbers are represented on the complex plane, the real part is typically on the x-axis, and the imaginary part is on the y-axis.

Complex Fractions

When dealing with complex fractions, the goal is often to simplify them into a standard complex number form \( a + bi \) for easier interpretation and usage.
A complex fraction is divided by another complex number such as \( \frac{3 + 4i}{1 - 2i} \). To simplify, use the method of multiplying both the numerator and the denominator by the conjugate of the denominator. A conjugate of a complex number \( a + bi \) is \( a - bi \). This effectively removes the imaginary part in the denominator, facilitating simplification.
For example, in the expression \( \frac{3+4i}{1-2i} \), multiplying by the conjugate \( 1+2i \) results in \( \frac{(3+4i)(1+2i)}{(1-2i)(1+2i)} = \frac{-5+10i}{5} \), which simplifies to \(-1 + 2i\). The real part is \(-1\) and the imaginary part is 2.
This method not only simplifies the fraction but also allows you to clearly identify the real and imaginary components of the resulting complex number.

Powers of Complex Numbers

Powers of complex numbers involve raising a complex number to an integer power, like \( i^n \). Here, the powers of \( i \) show a repeating pattern every four powers:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

If you continue the pattern, it repeats: \( i^5 = i \), \( i^6 = -1 \), and so on. This cyclical behavior is crucial to finding real and imaginary parts dependent on the power modulo 4.
For instance, if \( n \equiv 0 \pmod{4} \), \( i^n \) results in a real value of 1. If \( n \equiv 1 \pmod{4} \), the imaginary part is \( i \) with a real value of 0, and so forth.
Power series and the properties of powers of \( i \) help in simplifying expressions and performing calculations in higher mathematics. It's a foundational concept for understanding complex algebra and much more in-depth applications.

Geometric Series

A geometric series is a series formed by multiplying each term by a fixed, non-zero number known as the common ratio. In the context of complex numbers, you might have terms like \( \left(\frac{1-i}{\sqrt{2}}\right)^v \).
To find the sum of a geometric series with a complex common ratio, use the formula for partial sums:
\[ S_T = \frac{1 - r^{T+1}}{1 - r} \]
where \( r \) is the common ratio, and \( T \) is the number of terms.
The series \( \sum_{v=0}^{T}\left(\frac{1-i}{\sqrt{2}}\right)^{v} \) involves calculating the sum using the specified first term 1 and its common ratio. For the complex ratio having a modulus of 1, the results will continuously rotate in obtaining the real and imaginary components depending upon \( T \)terms.
This kind of geometric progression is an important tool in algebra as well as in analyzing complex systems, whether it's for electrical engineering or advanced physics. It facilitates understanding, especially in domains where growth rates or declining rates play a vital role.

Complex Conjugate

The complex conjugate of a complex number plays a fundamental role, especially in operations like division and simplification in complex arithmetic. If \( z = a + bi \) is a complex number, then its conjugate is \( \bar{z} = a - bi \).
Using the conjugate is crucial when dividing complex numbers. The goal is to utilize it to rationalize the denominator, eliminating the imaginary component for seamless calculations, as displayed in complex fractions.
The conjugate has special mathematical properties:

• Its multiplication with the original complex number results in a real number \( a^2 + b^2 \).
• Conjugates are used in finding magnitudes and in various proofs.
• It reflects the number across the real axis in the complex plane, aiding in geometrical interpretations.

Understanding and applying complex conjugates in practice includes simplifying expressions, solving equations, and dealing with inverse functions in higher mathematics. Recognizing the importance and functionality of the conjugate not only aids in theoretical math but also in practical computation, ensuring accurate and straightforward solutions.