Purely Imaginary (z-1)/(z+1): When Does It Happen?
Hey everyone! Today, we're diving into a fascinating problem involving complex numbers. We're going to explore the condition under which the expression (z-1)/(z+1) becomes purely imaginary, given that z ≠-1. This is a classic problem that beautifully combines algebraic manipulation with geometric intuition in the complex plane. So, grab your thinking caps, and let's get started!
The Problem: Unveiling the Imaginary Nature of (z-1)/(z+1)
Our main focus is to prove that (z-1)/(z+1) is purely imaginary if and only if |z| = 1. In simpler terms, we want to show that the complex number (z-1)/(z+1) lies on the imaginary axis if and only if the magnitude (or modulus) of z is equal to 1. This magnitude, |z|, represents the distance of the complex number z from the origin in the complex plane.
To tackle this problem, we'll break it down into two parts:
- If |z| = 1, then (z-1)/(z+1) is purely imaginary: This means we'll start by assuming that the magnitude of z is 1 and then demonstrate that the resulting expression (z-1)/(z+1) has no real part.
- If (z-1)/(z+1) is purely imaginary, then |z| = 1: Here, we'll assume that (z-1)/(z+1) is purely imaginary and then prove that the magnitude of z must be 1.
By proving both directions, we establish the "if and only if" relationship, solidifying our understanding of the connection between the magnitude of z and the nature of (z-1)/(z+1).
Diving into the First Direction: Assuming |z| = 1
Let's kick things off by assuming that |z| = 1. This tells us that z lies on the unit circle in the complex plane. A powerful way to represent complex numbers on the unit circle is using the polar form. We can express z as:
z = cos(θ) + i sin(θ)
where θ is the argument (angle) of z. Now, let's substitute this into our expression (z-1)/(z+1):
(z-1)/(z+1) = (cos(θ) + i sin(θ) - 1) / (cos(θ) + i sin(θ) + 1)
To simplify this, we'll use a clever trick: multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. So, the conjugate of cos(θ) + i sin(θ) + 1 is cos(θ) - i sin(θ) + 1. Let's do the multiplication:
[(cos(θ) + i sin(θ) - 1) / (cos(θ) + i sin(θ) + 1)] * [(cos(θ) - i sin(θ) + 1) / (cos(θ) - i sin(θ) + 1)]
This looks a bit intimidating, but we'll carefully expand the numerator and denominator. After expanding and simplifying (which involves using trigonometric identities like sin²(θ) + cos²(θ) = 1), we arrive at:
[2i sin(θ)] / [2 + 2cos(θ)]
Further simplification gives us:
i [sin(θ) / (1 + cos(θ))]
Notice anything special about this result? It's purely imaginary! The expression is a real number [sin(θ) / (1 + cos(θ))] multiplied by the imaginary unit 'i'. This means the real part of (z-1)/(z+1) is zero, confirming that it lies on the imaginary axis.
Therefore, we've successfully shown that if |z| = 1, then (z-1)/(z+1) is purely imaginary.
Tackling the Converse: Assuming (z-1)/(z+1) is Purely Imaginary
Now, let's tackle the other direction. This time, we'll assume that (z-1)/(z+1) is purely imaginary and aim to prove that |z| = 1. Since (z-1)/(z+1) is purely imaginary, its real part must be zero. This is our key to unlocking the solution.
Let's represent the complex number z in its general form:
z = x + yi
where x and y are real numbers. Now, substitute this into our expression (z-1)/(z+1):
(z-1)/(z+1) = (x + yi - 1) / (x + yi + 1)
Again, we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (x + 1) + yi is (x + 1) - yi. Performing the multiplication:
[(x + yi - 1) / (x + yi + 1)] * [((x + 1) - yi) / ((x + 1) - yi)]
Expanding the numerator and denominator, we get a somewhat lengthy expression. However, our focus is on the real part of the result. After careful expansion and simplification, the real part of the resulting expression turns out to be:
(x² + y² - 1) / ((x + 1)² + y²)
Remember, we assumed that (z-1)/(z+1) is purely imaginary, which means its real part must be zero. Therefore, we can set the real part we just found equal to zero:
(x² + y² - 1) / ((x + 1)² + y²) = 0
For a fraction to be zero, its numerator must be zero (and the denominator must be non-zero). So, we have:
x² + y² - 1 = 0
Rearranging this equation, we get:
x² + y² = 1
Now, recall that the magnitude of a complex number z = x + yi is given by:
|z| = √(x² + y²)
Substituting x² + y² = 1 into the magnitude equation, we get:
|z| = √1 = 1
Therefore, we've successfully shown that if (z-1)/(z+1) is purely imaginary, then |z| = 1.
The Grand Conclusion: |z| = 1 is the Key!
We've now proven both directions of our statement: if |z| = 1, then (z-1)/(z+1) is purely imaginary, and if (z-1)/(z+1) is purely imaginary, then |z| = 1. This establishes the "if and only if" relationship.
In conclusion, the complex number (z-1)/(z+1) is purely imaginary if and only if the magnitude of z is equal to 1. This means that z must lie on the unit circle in the complex plane for (z-1)/(z+1) to lie on the imaginary axis.
This problem beautifully illustrates the interplay between algebra and geometry in the realm of complex numbers. By understanding the geometric interpretation of complex numbers and their magnitudes, we can solve problems like this with elegance and clarity.
I hope you enjoyed this exploration of complex numbers! Keep practicing, keep exploring, and you'll unlock even more of the fascinating world of mathematics.
