\[ z = \frac{-1 - i -1 + i}{2} = \frac{-2}{2} = -1 \]

["Understanding the Complex Number Calculation: ( z = \frac{-1 - i -1 + i}{2} = -1 )", "When working with complex numbers, simplifying expressions step-by-step ensures accurate results. One such expression is:", "[\nz = \frac{-1 - i - 1 + i}{2} = \frac{-2}{2} = -1\n]", "### Breaking Down the Expression", "This equation combines real and imaginary parts in a numerator:", "- Real parts: (-1 - 1 = -2)\n- Imaginary parts: (-i + i = 0)", "Thus, the numerator simplifies cleanly to (-2), leaving:", "[\nz = \frac{-2}{2}\n]", "### Step-by-Step Simplification", "1. Group real and imaginary components:\n ((-1 - 1) + (-i + i) = -2 + 0i)", "2. Divide both parts by the denominator (2):\n (\frac{-2}{2} + \frac{0i}{2} = -1 + 0i = -1)", "This confirms that the complex number simplifies to a pure real value.", "### Why This Calculation Matters", "Understanding such expressions is essential in fields like electrical engineering, signal processing, and quantum mechanics, where complex numbers represent oscillating phenomena or wave functions. While (z) is real here, similar methods apply when comparing complex solutions or simplifying matrix operations involving complex entries.", "### Final Result", "[\nz = \frac{-1 - i - 1 + i}{2} = \frac{-2}{2} = -1\n]", "This computation demonstrates how combining and simplifying complex terms step-by-step leads to precise, real-valued results. Mastering these fundamentals strengthens problem-solving skills when working with more advanced algebraic or analytical tasks involving complex quantities.", "---", "Keywords: complex numbers, imaginary unit i, simplification, z calculation, complex arithmetic, algebra tutorial, complex expression, real vs imaginary parts, math fundamentals"]