Proving Re(z) = Im(z) Given |z - 1| = |z - I| A Comprehensive Guide
Introduction
In the realm of complex numbers, the interplay between their geometric representation and algebraic properties often reveals fascinating insights. This article delves into one such instance, exploring the condition |z - 1| = |z - i| where z is a complex number. Our primary goal is to demonstrate that if this condition holds true, then the real part of z, denoted as Re(z), is equal to the imaginary part of z, denoted as Im(z). This exploration will not only solidify our understanding of complex number magnitudes but also enhance our ability to navigate the complex plane with greater confidence.
Unpacking the Magnitude of Complex Numbers
At the heart of our discussion lies the concept of the magnitude (or modulus) of a complex number. For any complex number z = a + bi, where a and b are real numbers and i is the imaginary unit (√-1), the magnitude of z, denoted as |z|, is given by the formula |z| = √(a² + b²). Geometrically, |z| represents the distance from the origin (0, 0) to the point (a, b) in the complex plane. This geometric interpretation is crucial as it provides a visual context for the algebraic manipulations we will undertake. When we encounter expressions like |z - 1| or |z - i|, we are essentially calculating the distance between the point representing z in the complex plane and the points representing the complex numbers 1 and i, respectively. Understanding this geometric significance is the key to unlocking the problem's solution. Now, let’s embark on a journey to understand the given condition in detail and derive the desired result, making sure to grasp every step with clarity and precision.
Elucidating the Condition |z - 1| = |z - i|
The given condition, |z - 1| = |z - i|, forms the cornerstone of our analysis. To truly grasp its implications, we must first dissect it into its fundamental components and interpret its geometric significance. Let us represent the complex number z in its standard form, z = x + yi, where x and y are real numbers representing the real and imaginary parts of z, respectively. With this representation in hand, we can rewrite the expressions within the magnitude symbols and proceed with our calculations.
Algebraic Transformation and Geometric Interpretation
Substituting z = x + yi into the given condition, we have |x + yi - 1| = |x + yi - i|. This can be further rearranged as |(x - 1) + yi| = |x + (y - 1)i|. Now, we apply the definition of the magnitude of a complex number to both sides of the equation. This yields √[(x - 1)² + y²] = √[ x² + (y - 1)²]. Squaring both sides to eliminate the square roots, we obtain the equation (x - 1)² + y² = x² + (y - 1)². This algebraic transformation is pivotal as it converts the complex number condition into a more manageable equation involving real variables x and y. Geometrically, this equation represents the set of all points z = x + yi in the complex plane that are equidistant from the points 1 + 0i and 0 + 1i. In other words, we are looking at the perpendicular bisector of the line segment joining the points (1, 0) and (0, 1) in the complex plane. Understanding this geometric interpretation provides us with an intuitive sense of the solution we are aiming to derive. With this groundwork laid, we can now proceed with the algebraic simplification of the equation to reveal the crucial relationship between x and y, ultimately proving that Re(z) = Im(z).
Proving Re(z) = Im(z)
Having established the algebraic form of our condition, the next step is to simplify the equation (x - 1)² + y² = x² + (y - 1)² and extract the relationship between x and y. This simplification process will directly lead us to the conclusion that Re(z) = Im(z). Let's meticulously perform the algebraic manipulations to unveil this key result.
Algebraic Simplification and the Crucial Relationship
Expanding both sides of the equation, we get x² - 2x + 1 + y² = x² + y² - 2y + 1. Now, we can observe that several terms cancel out: the x² terms, the y² terms, and the constant term 1. This simplification leaves us with the equation -2x = -2y. Dividing both sides by -2, we arrive at the remarkably simple yet powerful result: x = y. This equation is the cornerstone of our proof. It states that the real part of z (x) is equal to the imaginary part of z (y). Recalling that we defined z = x + yi, this result directly translates to Re(z) = Im(z), thus fulfilling the objective of our proof. The elegance of this result lies in its directness and the clear algebraic path that leads to it. The initial complex condition, when dissected and simplified, reveals a fundamental equality between the real and imaginary components of the complex number z. This equality has significant geometric implications, which we will delve into further, solidifying our understanding of the connection between algebra and geometry in the context of complex numbers. By meticulously simplifying the equation, we've successfully demonstrated that the condition |z - 1| = |z - i| indeed implies that the real and imaginary parts of z are equal.
Geometric Interpretation of Re(z) = Im(z)
Having algebraically proven that Re(z) = Im(z) when |z - 1| = |z - i|, it is highly insightful to explore the geometric implications of this result. The equality x = y, where x and y are the real and imaginary parts of z, respectively, has a clear geometric representation in the complex plane. Understanding this representation enriches our comprehension of the problem and reinforces the link between the algebraic and geometric aspects of complex numbers.
The Line of Symmetry in the Complex Plane
The equation x = y represents a straight line in the complex plane. Specifically, it is the line that bisects the first and third quadrants, passing through the origin at a 45-degree angle with respect to the real axis. Every point on this line has equal real and imaginary components. Therefore, if a complex number z satisfies the condition Re(z) = Im(z), then the point representing z in the complex plane must lie on this line. This geometric interpretation provides a visual confirmation of our algebraic result. Furthermore, it allows us to understand the initial condition |z - 1| = |z - i| from a geometric perspective. As we discussed earlier, this condition implies that z is equidistant from the points 1 and i in the complex plane. The set of all such points forms the perpendicular bisector of the line segment joining 1 and i. It is not coincidental that this perpendicular bisector is precisely the line x = y. The geometric significance of this line further solidifies our understanding of the problem. By visualizing the complex plane and the line x = y, we can intuitively grasp why the equality of the real and imaginary parts of z arises from the given condition. The geometric perspective adds another layer of depth to our analysis, making the solution more intuitive and memorable. This understanding is invaluable when tackling similar problems involving complex numbers and their geometric representations. Through a combination of algebraic manipulation and geometric interpretation, we have painted a comprehensive picture of the condition |z - 1| = |z - i| and its profound implications.
Conclusion
In this exploration, we have successfully demonstrated that if |z - 1| = |z - i|, then Re(z) = Im(z). We began by understanding the concept of the magnitude of complex numbers and the geometric significance of expressions like |z - 1| and |z - i|. By representing z as x + yi and applying the definition of magnitude, we transformed the given condition into an algebraic equation involving x and y. Through careful simplification, we arrived at the crucial result x = y, which directly implies that Re(z) = Im(z). Moreover, we delved into the geometric interpretation of this result, recognizing that x = y represents a line in the complex plane that bisects the first and third quadrants. This geometric perspective provided a visual confirmation of our algebraic findings and enhanced our understanding of the problem.
Synthesis of Algebraic and Geometric Insights
The beauty of this problem lies in the seamless interplay between algebra and geometry. The algebraic manipulations allowed us to transform the complex condition into a simple equation, while the geometric interpretation provided a visual context for understanding the solution. This synthesis of algebraic and geometric insights is a powerful tool in the realm of complex numbers. By mastering this interplay, we can approach similar problems with greater confidence and intuition. The key takeaway from this analysis is the ability to translate complex number conditions into algebraic equations and geometric representations. This skill is fundamental for solving a wide range of problems involving complex numbers. In conclusion, we have not only proven that Re(z) = Im(z) under the given condition but also gained a deeper appreciation for the rich interplay between algebra and geometry in the complex plane. This understanding will serve as a valuable asset in our continued exploration of mathematics and its diverse applications.
Final Thoughts
This article has meticulously walked through the process of proving that if |z - 1| = |z - i|, then Re(z) = Im(z). The journey involved understanding the magnitude of complex numbers, algebraic manipulation, and geometric interpretation. By connecting these different aspects of complex numbers, we've gained a comprehensive understanding of the problem and its solution. This holistic approach is essential for success in mathematics and its related fields. The ability to seamlessly transition between algebraic and geometric perspectives is a hallmark of mathematical proficiency. As we continue our mathematical journey, let us carry forward this understanding and apply it to new and challenging problems. The world of complex numbers is rich with fascinating concepts and applications, and by mastering the fundamentals, we unlock the door to deeper insights and discoveries.
