The equation \( z^2 = 1 \) involves solving for \( z \) in the context of complex numbers. Let's break it down step by step.

### Solution in the Real Numbers:
If we restrict \( z \) to real numbers, solving \( z^2 = 1 \) is straightforward:
\[
z^2 = 1 \quad \Rightarrow \quad z = \pm 1
\]
So, \( z = 1 \) or \( z = -1 \).

### Solution in the Complex Numbers:
However, in the complex plane, we treat \( z \) as a complex number: \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. We can also express it using Euler's formula and polar form:
\[
z = re^{i\theta}
\]
where \( r \) is the magnitude (modulus) of \( z \) and \( \theta \) is the argument (angle).

The equation \( z^2 = 1 \) implies:
\[
r^2 e^{i 2\theta} = 1
\]
Since \( 1 \) can be written as \( e^{i 2k\pi} \) for any integer \( k \), this gives the following:
1. \( r^2 = 1 \), so \( r = 1 \) (since magnitudes are positive),
2. \( 2\theta = 2k\pi \), so \( \theta = k\pi \).

Thus, the two solutions are:
\[
z = e^{i0} = 1 \quad \text{and} \quad z = e^{i\pi} = -1
\]

So in both real and complex numbers, the solutions are \( z = 1 \) and \( z = -1 \).

Would you like further details on complex solutions or more questions on this topic?

### Related Questions:
1. How do you solve \( z^2 = -1 \) in the complex plane?
2. What is the geometric interpretation of the solutions of \( z^2 = 1 \) on the complex plane?
3. How does Euler's formula relate to solving quadratic equations in the complex plane?
4. What are the roots of unity, and how do they connect to solving equations like \( z^n = 1 \)?
5. Can you generalize the solution of \( z^2 = c \) for any complex number \( c \)?

**Tip:** Understanding complex numbers in polar form makes solving quadratic and higher-degree equations much easier!

Explore how to solve z^2 = 1 in both real and complex numbers using algebra and polar form. This detailed solution covers concepts like Euler's formula, the fundamental theorem of algebra, and the roots of unity. Ideal for Grades 11-12 and early college students.

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