Question

Man verifiziere fir \(z=x+\mathrm{i} y \in \mathrm{C}\) die Ungleichungen und $$ \frac{|x|+|y|}{\sqrt{2}} \leq|z|=\sqrt{x^{2}+y^{2}} \leq|x|+|y| $$ $$ \max \\{|x|,|y|\\} \leq|z| \leq \sqrt{2} \max \\{|x|,|y|\\} $$

Short answer

The given inequalities hold for any complex number \(z = x + yi\).

Step-by-step solution

Understand Complex Number Modulus

The modulus (magnitude) of a complex number \(z = x + yi\) is given by \(|z| = \sqrt{x^2 + y^2}\). This represents the Euclidean distance from the origin to the point \((x, y)\) in the complex plane.

Prove \(\frac{|x| + |y|}{\sqrt{2}} \leq |z| \leq |x| + |y|\)

To show \(\frac{|x| + |y|}{\sqrt{2}} \leq |z|\):- Recognize that \(|x|^2 + |y|^2 \geq 2|x||y|\) by the AM-GM inequality.- Hence, \(x^2 + y^2 \geq 2|x||y|\), so \(|z| \geq \frac{|x| + |y|}{\sqrt{2}}\).To show \(|z| \leq |x| + |y|\):- From the triangle inequality in an absolute value context, \(\sqrt{x^2 + y^2} \leq \sqrt{x^2} + \sqrt{y^2} = |x| + |y|\).

Prove \(\max\{|x|, |y|\} \leq |z| \leq \sqrt{2} \max\{|x|, |y|\}\)

To show \(\max\{|x|, |y|\} \leq |z|\):- Without loss of generality, assume \(|x| \geq |y|\), then \(|z| = \sqrt{x^2 + y^2} \geq \sqrt{x^2} = |x|\), hence the lower bound.To show \(|z| \leq \sqrt{2} \max\{|x|, |y|\}\):- If \(|x| = \max\{|x|, |y|\}\), then \(|z| = \sqrt{x^2 + y^2} \leq \sqrt{x^2 + x^2} = |x| \sqrt{2}\). Similarly if \(|y| = \max\{|x|, |y|\}\), giving the upper bound.

Key concepts

Complex Numbers

Complex numbers are numbers of the form \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. These numbers are crucial in mathematics as they extend the one-dimensional number line into the two-dimensional complex plane. You can think of them as ordered pairs \((x, y)\), which allows us to visualize them graphically. Each complex number corresponds to a point in the complex plane, with the real number \( x \) representing the horizontal axis and the imaginary part \( y \) the vertical axis.
Understanding complex numbers becomes particularly important as they allow for solutions to polynomial equations that real numbers alone cannot satisfy. For example, there is no real number that will satisfy the equation \( x^2 + 1 = 0 \), but using complex numbers, we find the solutions are \( i \) and \( -i \), where \( i \) is the imaginary unit and satisfies \( i^2 = -1 \).
Manipulating complex numbers involves performing operations like addition, subtraction, multiplication, and division, which are used in various engineering, physics, and mathematics applications.

Euclidean Distance

The Euclidean distance in the context of complex numbers is the modulus or magnitude of the number. For \( z = x + yi \), the modulus is \(|z| = \sqrt{x^2 + y^2}\). This represents the straight-line distance between the point \((x, y)\) in the complex plane and the origin \((0,0)\).
Calculating this distance helps us understand the size or length of the vector in the plane represented by the complex number. For instance, if you have the complex number \( z = 3 + 4i \), its Euclidean distance or modulus from the origin is \( \sqrt{3^2 + 4^2} = 5 \). Using Euclidean distance, we can compare complex numbers and determine their proximity to the origin or to each other, much like measuring distances in geometry.
The concept of distance in this context is quite intuitive and complements the geometrical representation of complex numbers, showing how algebraic expressions correspond to geometric properties.

Inequalities in Mathematics

Inequalities in mathematics are comparisons between two values or expressions, showing that one is larger or smaller than the other. Inequalities are essential for defining ranges and constraints within mathematical problems. In terms of complex numbers, inequalities often involve comparing the magnitudes or moduli of these numbers.
For instance, the inequality \( |z| \leq |x| + |y| \) is a way of comparing distances in the complex plane. It states that the modulus of a complex number \( z \) is always less than or equal to the sum of the absolute values of its components \( x \) and \( y \). This can be particularly useful for proving other mathematical properties or solving equations that involve complex numbers.
Inequalities are fundamental in many areas of mathematics, including calculus, algebra, and analysis, as they provide essential tools for understanding the behavior and properties of mathematical systems and structures.

AM-GM Inequality

The AM-GM inequality is a key principle in mathematics that relates arithmetic means and geometric means. It states that for any non-negative numbers \( a \) and \( b \), the arithmetic mean is always greater than or equal to the geometric mean:

• Arithmetic mean (AM) : \( \frac{a + b}{2} \)
• Geometric mean (GM) : \( \sqrt{ab} \)

Thus, \( \frac{a + b}{2} \geq \sqrt{ab} \), with equality if and only if \( a = b \).
This inequality is frequently used in proofs and problem-solving as it provides a means of evaluating expressions involving sums and products. In the context of complex numbers, it is used to derive bounds for the modulus of a complex number, applicable in proving inequalities like \( |z| \geq \frac{|x| + |y|}{\sqrt{2}} \).
Understanding and applying the AM-GM inequality can simplify complex mathematical arguments and is a vital concept in mathematical olympiads, inequality theory, and optimization problems.

Triangle Inequality

The triangle inequality is a fundamental principle in mathematics, particularly in relation to complex numbers and metric spaces. It states that for any points \(A\), \(B\), and \(C\), the sum of the lengths of any two sides of a triangle is always greater than or equal to the length of the third side. For complex numbers, this is expressed as \( |z_1 + z_2| \leq |z_1| + |z_2| \) where \(z_1\) and \(z_2\) are complex numbers.
This concept is crucial for understanding the properties of distances in spaces and holds under any metric satisfying the definition of distance. In the context of the original problem, it allows us to conclude that \( |z| \leq |x| + |y| \), ensuring that the magnitude of a complex number is always less than or equal to the sum of the magnitudes of its components.
The triangle inequality not only applies to complex numbers but is also a critical property in vector spaces, normed spaces, and real analysis, forming the foundation for many mathematical theorems and concepts.