Problem 24 - Advanced Engineering Mathematics

Question

\(|\operatorname{Re}(z)|=|x|\) is the same as \(\sqrt{x^{2}}\) and \(|z|=\sqrt{x^{2}+y^{2}} .\) since \(y^{2} \geq 0\) the inequality \(\sqrt{x^{2}} \leq \sqrt{x^{2}+y^{2}}\) is true for all complex numbers.

Step-by-Step Solution

Answer

The inequality \(|x| \leq |z|\) holds because \(\sqrt{x^2 + y^2} \geq \sqrt{x^2}\) for any complex number.

Step 1: Understanding Re(z) and Modulus

The problem is about understanding the relationship between the real part of a complex number and its modulus. The real part of a complex number \( z = x + yi \) is \( x \), which is denoted as \( \operatorname{Re}(z) = x \). The modulus of \( z \) is given by \( |z| = \sqrt{x^2 + y^2} \). We want to verify that \(|\operatorname{Re}(z)| = |x|\) and \( |x| \leq |z| \).

Step 2: Analyze |Re(z)| and |x|

By definition, the absolute value of the real part of \( z \) is \( |\operatorname{Re}(z)| = |x| \). From the description given, we have \( |x| = \sqrt{x^2} \), which is the definition of modulus for any real number \( x \). Both sides yield the positive root, thus confirming equality.

Step 3: Modulus |z| and Its Components

The modulus of \( z \) is expressed as \( |z| = \sqrt{x^2 + y^2} \). This represents the Euclidean distance from the complex number \( z \) to the origin in the complex plane. Since \( y^2 \geq 0 \), \( \sqrt{x^2 + y^2} \) is always greater than or equal to \( \sqrt{x^2} \).

Step 4: Compare |x| and |z|

We know \( |x| = \sqrt{x^2} \). Since \( y^2 \geq 0 \), it implies \( \sqrt{x^2} \leq \sqrt{x^2 + y^2} \). This inequality holds for all complex numbers \( z = x + yi \) since any non-negative \( y^2 \) contributes positively to \( |z| \). Thus \( |x| \leq |z| \) is true.

Key Concepts

Real Part of Complex Number

In the world of complex numbers, every number has both a real part and an imaginary part. For any complex number written as \( z = x + yi \), the real part is the portion represented by \( x \). When we look at \( \operatorname{Re}(z) \), this precisely refers to the real part of the complex number, which is simply \( x \).

Here's a way to think about it:

• Complex numbers are like points on a plane, and \( x \) provides the 'horizontal' coordinate.
• The real part significantly influences the complex number's position on this horizontal axis.

Understanding the real part \( \operatorname{Re}(z) = x \) helps us visualize where the complex number sits relative to the real number line. Practically speaking, when analyzing equations involving complex numbers, isolating the real part allows for solving and verifying mathematical properties related to this component.

Modulus of Complex Number

The modulus, also known as the magnitude, of a complex number \( z = x + yi \), is its distance from the origin in the complex plane. Mathematically, this is expressed as \( |z| = \sqrt{x^2 + y^2} \).

Think of the modulus as a way to measure how 'far' the number is from zero:

• It's like using a ruler to determine the length of a vector drawn from the origin to the point \( (x, y) \).
• The modulus plays a crucial role when comparing the sizes of complex numbers.

The calculation method is akin to using the Pythagorean theorem, where the real part \( x \) and the imaginary part \( y \) are the legs of the triangle, and the modulus \( |z| \) is the hypotenuse. This geometric interpretation is intuitive and visually helps in understanding unequal comparisons.

Inequalities in Complex Numbers

When we speak about inequalities with complex numbers, we're often comparing their magnitudes. An essential concept is understanding that the modulus of a real part is always less than or equal to the modulus of the entire complex number.

For \( z = x + yi \), this inequality is shown as \( |x| \leq |z| \).

• This inequality claims that the absolute value of the real part alone cannot exceed the total distance \( z \) is from the origin.
• If \( y = 0 \), then \( |z| \) would indeed equal \( |x| \), making them equal.

These inequalities illuminate relationships between different components of a complex number. They assure that additional components, like \( y \, i \), can only add to the distance, not lessen it. This becomes a solid foundation when handling complex number algebra, particularly in verifying solutions and ensuring calculations are consistent with theoretical bounds.