Problem 22 - Algebra and Trigonometry

Question

Sketch the set in the complex plane. $$ \\{z=a+b i | a>1, b>1\\} $$

Step-by-Step Solution

Answer

The set is all points in the first quadrant above the line \(y=1\) and to the right of the line \(x=1\).

Step 1: Understand the Expression

We have a set of complex numbers written as \(z = a + bi\) where \(a\) and \(b\) are real numbers, \(a > 1\), and \(b > 1\). The complex number \(z\) is in the form \(a + bi\) which means \(a\) is the real part and \(b\) is the imaginary part of \(z\).

Step 2: Determine the Region in the Complex Plane

The condition \(a > 1\) tells us that all points must lie to the right of the vertical line \(x = 1\). The condition \(b > 1\) tells us that all points must lie above the horizontal line \(y = 1\). Thus, the set includes all points in the complex plane that are in the region to the right of \(x = 1\) and above \(y = 1\).

Step 3: Sketch the Region

On the complex plane, label the horizontal axis as the real axis (representing \(a\)) and the vertical axis as the imaginary axis (representing \(b\)). Draw a dashed vertical line at \(x = 1\) and a dashed horizontal line at \(y = 1\). The set is the area in the first quadrant to the right of this vertical line and above the horizontal line, forming an open region.

Key Concepts

Complex Numbers

Understanding complex numbers is essential for dealing with the world of the complex plane. Complex numbers are expressed in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The real part \(a\) and the imaginary part \(bi\) together form these unique numbers.
Complex numbers allow us to solve equations that have no real solutions. They extend the concept of one-dimensional number lines into a two-dimensional plane.

• The real part represents the horizontal direction, often labeled as the x-axis.
• The imaginary part represents the vertical direction, known as the y-axis.

Using complex numbers, we can explore many real-world phenomena like electrical engineering problems or even certain fluid dynamics issues by considering both the magnitude and the angular direction of values in this plane.

Real Part

The real part of a complex number plays a crucial role in defining its position on the complex plane. It denotes the horizontal location of the point represented by the complex number on this plane. For a complex number \(z = a + bi\):

• \(a\) is the real part.

In practical terms, if the real part \(a\) of a complex number is greater than 1, it implies that the point lies to the right of the line \(x = 1\).
Understanding the real part helps us to locate where a complex number falls along the x-axis in the complex plane.

Imaginary Part

The imaginary part of a complex number is equally important and determines the vertical position of a point on the complex plane. For a given complex number \(z = a + bi\):

• \(b\) is the imaginary part, which when multiplied by \(i\) gives the imaginary component of the number.

This part of the complex number affects how far up or down a number is plotted on the y-axis.
If \(b > 1\), this means the imaginary part of the complex numbers in our set makes them lie above the line \(y = 1\). This clearly helps in knowing which part of the plane the number will represent.

Inequalities in Complex Plane

Inequalities help us understand the regions of the complex plane that a set of complex numbers can occupy. For the expression \(z = a + bi\) with \(a > 1\) and \(b > 1\), these inequalities define a specific region in the plane.

• \(a > 1\) translates to points being right of the line \(x = 1\).
• \(b > 1\) translates to points being above the line \(y = 1\).

When sketching these regions, it's essential to note that inequalities are open, which means the "border" (lines \(x = 1\) and \(y = 1\) themselves) is not included in the set.
This forms an open area in the first quadrant that represents all complex numbers that have both their real and imaginary parts greater than 1.