Problem 2 - Algebra and Geometry

Question

Show that \(z^{2}=2 i\) if and only if \(z=\pm(1+i)\).

Step-by-Step Solution

Answer

\(z^2 = 2i\) if and only if \(z = \pm(1+i)\).

Step 1: Understand the Equation

We need to show that the equation \(z^2 = 2i\) is true if and only if \(z = \pm(1+i)\). This is an assertion that needs proof in both directions: (1) proving that if \(z^2 = 2i\) then \(z = \pm(1+i)\), and (2) proving that if \(z = \pm(1+i)\) then \(z^2 = 2i\).

Step 2: Set Up General Complex Number for z

To start, let's express \(z\) as a complex number: \(z = x + yi\), where \(x\) and \(y\) are real numbers. We then have \(z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi\). We will set this equal to \(2i\).

Step 3: Compare Real and Imaginary Parts

From the equation \(z^2 = 2i\), it implies that the real part is zero and the imaginary part is 2. Therefore, \(x^2 - y^2 = 0\) and \(2xy = 2\).

Step 4: Solve the Real Part Equation

Solve \(x^2 - y^2 = 0\) which means \(x^2 = y^2\). This implies \(y = \pm x\).

Step 5: Solve the Imaginary Part Equation

From \(2xy = 2\), we get \(xy = 1\). Now, substitute \(y = \pm x\) into this equation, leading to two cases: - When \(y = x\), \(x^2 = 1\), so \(x = \pm 1\).- When \(y = -x\), \(-x^2 = 1\) can't be satisfied for real \(x\).

Step 6: Derive Solutions for z

From the valid case \(x = \pm 1\) and \(y = x\), we find that \(z\) can be \(1+i\) or \(-1-i\). Checking by substituting back, both satisfy \(z^2 = 2i\). This completes the proof that if \(z^2 = 2i\), then \(z = \pm(1+i)\).

Step 7: Verify the Reverse Direction

Next, if \(z = (1+i)\) or \(z = -(1+i)\), calculate \(z^2\): - For \(z = 1+i\), \((1+i)^2 = 1 + 2i - 1 = 2i\).- For \(z = -(1+i)\), \((-1-i)^2 = 1 + 2i - 1 = 2i\).Thus both expressions equal \(2i\), showing the reverse direction is also true.

Key Concepts

Complex Equations

Complex equations involve unknowns that take the form of complex numbers. Each complex number can be written as \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part, multiplied by the imaginary unit \(i\), which satisfies \(i^2 = -1\).
In the given exercise, we are asked to solve the complex equation \(z^2 = 2i\).
To solve such an equation, we express \(z\) in its general form \(z = x + yi\) and substitute it into the equation.
This results in expressing the right side of the equation in terms of real and imaginary parts.
Then, by matching these parts with the corresponding parts of the number on the other side, we form two separate equations, which are easier to solve.

Real and Imaginary Parts

When dealing with complex numbers, it helps to separate them into real and imaginary components.
This method is useful for solving equations like \(z^2 = 2i\).
By expanding \((x + yi)^2\), we get \(x^2 - y^2\) as the real part and \(2xy\) as the imaginary part.
For the equation to hold true, the real part should be equal to 0, and the imaginary part should be equal to 2, as per \(2i\).

• The equation for the real part is \(x^2 - y^2 = 0\).
• The equation for the imaginary part is \(2xy = 2\).

Once these equations are developed, the process can continue by solving them independently to find values of \(x\) and \(y\).
This separation is a pivotal step in many complex number equations.

Proof Techniques

Proof techniques for equations involving complex numbers often include showing that both directions of a conditional are true.
This is also known as a bidirectional proof or "if and only if" proof.
In this exercise, we need to prove that if \(z^2 = 2i\), then \(z = \pm(1+i)\), and vice versa.

• **Forward Direction**: Start by assuming \(z^2 = 2i\) and show that this leads to \(z = \pm(1+i)\).
• **Backward Direction**: Assume \(z = \pm(1+i)\) and verify that \(z^2 = 2i\).

Checking both directions ensures the sufficiency and necessity of the statements.
Such proofs build a comprehensive understanding by validating assumptions in both ways.

Solving Equations

Solving complex equations like \(z^2 = 2i\) requires a combination of algebraic manipulation and logical reasoning.
Initially, represent the unknown \(z\) as \(x + yi\). From here, identify and solve equations for real and imaginary parts separately.

• **Real Part**: Solve \(x^2 - y^2 = 0\), implying \(y = \pm x\).
• **Imaginary Part**: Solve \(2xy = 2\), leading to \(xy = 1\).

Substitute \(y = x\) into \(xy = 1\), leading to \(x^2 = 1\), so \(x = \pm 1\).
With these values, \(z\) is either \(1+i\) or \(-1-i\).
Finally, verify these solutions by substituting back into the original equation to ensure they satisfy \(z^2 = 2i\).
This holistic approach ensures comprehensive equation solving.