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

["Understanding the Complex Number Simplification: ( z = \frac{-1 - i + 1 - i}{2} = -i )", "Complex numbers are fundamental in mathematics, engineering, and physics, allowing us to solve problems involving magnitudes and directions. One common task is simplifying complex expressions, which can clarify their real and imaginary components.", "Let’s analyze the expression:", "[\nz = \frac{-1 - i + 1 - i}{2}\n]", "Step 1: Combine like terms in the numerator", "Rearranging and grouping the real and imaginary parts:", "[\n-1 + 1 - i - i = ( -1 + 1 ) + ( -i - i ) = 0 - 2i = -2i\n]", "Step 2: Divide by the denominator", "Now substitute back into the equation:", "[\nz = \frac{-2i}{2} = -i\n]", "Thus, the simplified form of the expression is:", "[\nz = -i\n]", "Why This Simplification Matters", "- Concise representation: Expressing complex numbers in standard form ( a + bi ) (where ( a ) is the real part and ( b ) the imaginary coefficient) enhances clarity.\n- Visual and computational ease: The result (-i) clearly shows that (z) lies purely on the imaginary axis in the complex plane.\n- Foundational for further concepts: This form is crucial when solving equations, analyzing series, or working with oscillations and waves in applied sciences.", "Key Takeaway", "Simplifying ( z = \frac{-1 - i + 1 - i}{2} ) step-by-step reveals:", "[\nz = -i\n]", "This powerful result not only resolves the expression cleanly but also reinforces essential skills in handling complex numbers—essential tools in modern mathematics and science.", "---", "Keywords: complex number simplification, ( z = \frac{-1 - i + 1 - i}{2} ), complex numbers solved, ( -i ) representation, imaginary number basics", "Meta Description: Learn how simplifying ( z = \frac{-1 - i + 1 - i}{2} ) yields ( -i ). A clear guide to manipulating complex expressions and understanding their standard form."]