Problem 7 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Skizzieren Sfe die folgenden komplexen Zahlen als Punkte der Ebene und berechnen sie ihre Beträget \(1-i \sqrt{3}, i+i^{2}+i^{3}+i^{4}+1^{5}, \frac{3+i \sqrt{7}}{4}\) und \(-2=\frac{3}{2} i\)

Step-by-Step Solution

Answer

The magnitudes are 2, 1, 1, and 2, respectively.

Step 1: Plot the Complex Number 1-i√3

The complex number is given by \(1 - i \sqrt{3}\). This can be represented on the complex plane where the real part is 1 and the imaginary part is \(-\sqrt{3}\). Plot this point at \((1, -\sqrt{3})\).

Step 2: Simplify and Plot the Complex Number i+i^2+i^3+i^4+i^5

The expression \(i + i^2 + i^3 + i^4 + i^5\) can be simplified using the cyclic nature of the imaginary unit: \(i = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), \(i^5 = i\). Sum these values to get \(i - 1 - i + 1 + i = i\). Plot the resulting point \((0, 1)\).

Step 3: Plot the Complex Number 3+i√7/4

Consider \(\frac{3 + i\sqrt{7}}{4}\). Simplify to get \(\frac{3}{4} + i\frac{\sqrt{7}}{4}\). The real part is \(\frac{3}{4}\) and the imaginary part is \(\frac{\sqrt{7}}{4}\). Plot the point \((\frac{3}{4}, \frac{\sqrt{7}}{4})\).

Step 4: Simplify and Plot the Complex Number -2=(3/2)i

The equation \(-2 = \frac{3}{2}i\) indicates that the imaginary part is actually \(-2\) for balance since the real part is zero. Correct to \(0 - 2i\). Plot the point \((0, -2)\).

Step 5: Calculate the Magnitude of Each Complex Number

The magnitude \(|z|\) of a complex number \(a + bi\) is \(\sqrt{a^2 + b^2}\). Compute for each number:- For \(1 - i\sqrt{3}\), \(|z| = \sqrt{1^2 + (-\sqrt{3})^2} = 2\).- For \(i\), \(|z| = \sqrt{0^2 + 1^2} = 1\).- For \(\frac{3}{4} + i\frac{\sqrt{7}}{4}\), \(|z| = \sqrt{\left(\frac{3}{4}\right)^2 + \left(\frac{\sqrt{7}}{4}\right)^2} = 1\).- For \(0 - 2i\), \(|z| = \sqrt{0^2 + (-2)^2} = 2\).

Key Concepts

Complex Plane

The complex plane is an incredibly useful tool when working with complex numbers. Essentially, it's a plane where each point represents a complex number. The horizontal axis (usually marked 'x') stands for the real part of the complex number, while the vertical axis (often labeled 'y') corresponds to the imaginary part. So any complex number expressed as \(a + bi\) can be easily plotted by positioning it at (a, b).

Here's why this is useful:

• Visualizing complex numbers helps us understand their properties better.
• Allows for geometric interpretation, which is helpful in many areas of mathematics and engineering.

The complex plane gives a geometric representation of operations with complex numbers, such as addition, subtraction, and scaling, and makes it easier to understand concepts like the magnitude of a complex number.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a curious and essential concept in complex numbers. By definition, \(i\) is a number whose square is equal to -1, i.e., \(i^2 = -1\). This might seem strange at first, but it enables mathematicians to extend the real number system into the complex realm, tackling equations that have no real solutions.

The powers of \(i\) have a cyclical nature, repeating every four iterations:

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

This cyclic pattern means any power of \(i\) can be simplified by knowing these four fundamental results. Understanding the imaginary unit's properties is crucial for performing algebraic manipulations in complex numbers.

Magnitude of Complex Numbers

Magnitude, sometimes referred to as the modulus or absolute value, is a measure of the size of a complex number, much like how we might think of the absolute value of a real number. For a complex number \(a + bi\), its magnitude is calculated using the formula: \[ |z| = \sqrt{a^2 + b^2} \]This formula is derived from treating the complex number as a line segment from the origin in the complex plane to the point (a, b). Hence, \(|z|\) represents the length of this line segment.

Some important notes:

• The magnitude is always a non-negative real number, as it describes a length.
• If the magnitude is 0, the complex number is the origin point itself.

Understanding magnitude is critical for calculating distances and understanding the behavior of complex numbers in applications such as signal processing.

Plotting Complex Numbers

Plotting complex numbers might seem daunting initially, but it's essentially just locating points on a plane. Every complex number \(a + bi\) becomes a point at (a, b) on the complex plane. Here's how the process usually works:

• Identify the real part 'a' and the imaginary part 'b'.
• Move 'a' units along the horizontal axis (real axis).
• Move 'b' units along the vertical axis (imaginary axis).

These steps help you plot any complex number efficiently.

Often plotting is used for visualizing complex addition, subtraction, and even multiplication. For example, the addition of two complex numbers results in a point that can be found by adding the corresponding coordinates on the plane, giving a geometric interpretation of these algebraic operations. This visualization supports the understanding of many higher-level math concepts and helps in problem-solving involving complex numbers.