Finding Complex Number Points: A Step-by-Step Guide

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Finding Complex Number Points: A Step-by-Step Guide

Hey guys! Let's dive into the world of complex numbers and figure out how to determine the set of points in the plane that satisfy certain conditions related to their affixes (that's just a fancy word for their position in the complex plane). We'll break down the problem step by step, making sure it's super clear and easy to follow. Get ready to explore the exciting realm of complex numbers! This guide will cover how to find the set of points in the plane based on different conditions related to the real and imaginary parts of a complex number, z. This is a fundamental concept in complex analysis, and understanding it will give you a solid foundation for further exploration. We will analyze the different cases step by step, making sure you grasp the concepts. Let’s get started and unravel the mysteries of complex numbers together, covering everything you need to know to master these types of problems. We'll explore various conditions, including constraints on the real part, imaginary part, and combined restrictions. This knowledge will be crucial for understanding complex analysis. We will break down each case, ensuring a clear and comprehensive understanding. By the end, you'll be able to confidently solve problems involving the location of complex numbers in the complex plane.

a) Re(z) = 1

Okay, let's start with the first condition: Re(z) = 1. What does this even mean? Well, if we have a complex number z, we can write it in the form z = x + yi, where x is the real part (Re(z)) and y is the imaginary part (Im(z)). In this case, the condition Re(z) = 1 simply tells us that the real part of our complex number z must be equal to 1. So, x = 1. This means we're looking for all complex numbers where the real part is fixed at 1, regardless of the imaginary part. Imagine the complex plane: the real axis is the horizontal axis, and the imaginary axis is the vertical axis. The set of all points that satisfy this condition forms a vertical line that passes through the point where the real part is 1. This line is parallel to the imaginary axis (the y-axis). Therefore, the set of points consists of all complex numbers of the form z = 1 + yi, where y can be any real number. Think of it like a line running straight up and down on the complex plane, crossing the real axis at 1. The solution is essentially all points where the x-coordinate is 1, creating a vertical line. This is the set of all complex numbers with a real part equal to 1. We are essentially saying that for any value of y, x is fixed at 1, hence the vertical line. This line extends infinitely in both directions along the imaginary axis. We have successfully determined the set of points for Re(z) = 1. Now, let's move on to the next part!

b) Im(z) = √2

Now, let's tackle the second condition: Im(z) = √2. This time, we are given that the imaginary part of z is equal to the square root of 2. So, in our z = x + yi form, we know that y = √2. This means that the imaginary part is fixed at √2, while the real part x can vary freely. On the complex plane, this corresponds to a horizontal line that passes through the point where the imaginary part is √2. This line is parallel to the real axis (the x-axis). The set of points that satisfies this condition includes all complex numbers of the form z = x + √2i, where x can be any real number. Visualize a straight line moving across the complex plane at a constant height equal to √2 on the imaginary axis. All these points have an imaginary component of √2. The real part of the complex number can be any number. We've defined the set of points. The set of points satisfying this condition forms a horizontal line. This line extends infinitely in both directions along the real axis. This line goes straight across at the height of the square root of 2 on the imaginary axis. So we're essentially saying the y-coordinate is always the square root of 2. This defines a horizontal line. The set includes all complex numbers with an imaginary part equal to √2. Keep up the good work; you’re doing great!

c) -1 ≤ Re(z) ≤ 2

Alright, let's get to the third condition, which is a bit more interesting: -1 ≤ Re(z) ≤ 2. This means that the real part of z, which is x, must be between -1 and 2, including -1 and 2. So, we're looking at a range of values for x. The imaginary part y can take any real value. On the complex plane, this will form a vertical strip between the vertical lines x = -1 and x = 2. The set includes all complex numbers of the form z = x + yi, where -1 ≤ x ≤ 2 and y is any real number. Imagine two vertical lines: one at x = -1 and another at x = 2. The set of points consists of all the points that lie between these two lines, including the lines themselves. The width of this strip is determined by the range of the real part. The set of points satisfying this condition forms a vertical strip. Every point within this strip has a real part that falls between -1 and 2. We're essentially saying that the real part, or x, is restricted to being between -1 and 2. Thus, the solution is a vertical strip. The imaginary part, y, can vary freely. Therefore, any point within that region is part of the solution. We've successfully identified the set of points for this condition.

d) -2Discussion category :

I apologize, but there seems to be an incomplete part in your prompt. The last condition you mentioned, is missing its details. I can provide the appropriate solution if you give the rest of the problem. Please provide the condition so I can provide the right answer. The final step involves defining the set of points. With your continued support, together we will determine the solution to the problem.

General Approach and Tips

Here are some helpful tips to remember when solving these types of problems:

  • Visualize the Complex Plane: Always try to sketch the complex plane and the geometric representation of the solution. This will help you understand the problem better.
  • Separate Real and Imaginary Parts: Remember that a complex number z can be written as x + yi, where x is the real part and y is the imaginary part.
  • Understand Inequalities: Be careful with inequalities. Less than or equal to (≤) means the boundary is included, while less than (<) means the boundary is not included.
  • Practice, Practice, Practice: The more problems you solve, the more comfortable you'll become with complex numbers.

Conclusion

Great job, everyone! We've successfully gone through several examples of how to determine sets of points in the complex plane based on different conditions for the real and imaginary parts of a complex number. Always remember to break down the problems, visualize them, and use the basic form of a complex number z = x + yi. Keep practicing, and you'll become a pro at this. If you have any more questions, feel free to ask. Keep up the amazing work, and keep exploring the fascinating world of mathematics! The key is to carefully consider the given conditions and translate them into a geometric representation. Keep up the excellent work! You’ve got this!