If \( a = -1 \), \( b = 1 \), \( (-1 + i)^2 = 1 - 2i + i^2 = -2i \) ✅ - Verified Servers

["Understanding ((-1 + i)^2): Why It Equals (-2i) – A Detailed Explanation", "When exploring complex numbers, one of the most fundamental calculations is squaring ( -1 + i ). For math students and educators, understanding this expression not only reinforces algebra and complex arithmetic but also opens doors to deeper insights in engineering, physics, and digital signal processing.", "### The Calculation Step-by-Step", "Start with the complex number:", "[
\n(-1 + i)^2
\n]", "Using the binomial expansion formula ( (a + b)^2 = a^2 + 2ab + b^2 ), substitute ( a = -1 ), ( b = i ):", "[
\n(-1 + i)^2 = (-1)^2 + 2(-1)(i) + (i)^2
\n]", "Now compute each term:", "- ( (-1)^2 = 1 )
\n- ( 2(-1)(i) = -2i )
\n- ( i^2 = -1 ) (by the fundamental property of complex numbers)", "Putting it all together:", "[
\n1 - 2i + (-1) = (1 - 1) - 2i = -2i
\n]", "So,", "[
\n\boxed{(-1 + i)^2 = -2i}
\n]", "### Why This Matters: The Significance of Complex Squaring", "This simple expression is more than an algebra exercise—it’s a building block for working with imaginary numbers in equations, phasor analysis, and quantum mechanics. Specifically:", "- The result(-2i) highlights how real and imaginary parts interact under multiplication.
\n- Knowing such identities eases manipulation in formulas involving roots, exponentials, and Fourier transforms.
\n- It confirms the structure of complex arithmetic: combinations of real and imaginary components behave predictably, yet uniquely.", "### Visualizing the Result: The Complex Plane", "Graphically, (-1 + i) lies in the second quadrant of the complex plane. Squaring it transforms this point: the result, (-2i), lies purely along the negative imaginary axis—reflecting how squaring affects magnitude and direction in polar form.", "### Conclusion", "When ( a = -1 ), ( b = 1 ), the squaring of the complex number ( -1 + i ) elegantly yields ( -2i )—a concise yet powerful example of complex arithmetic. This identity is invaluable both as a teaching moment and a practical tool in advanced mathematics and scientific applications.", "✅ Final Result:
\n[
\n(-1 + i)^2 = -2i
\n\quad \ ext{when} \quad a = -1,\ b = 1
\n]", "---", "Keywords: complex numbers, squaring complex numbers, ( (-1 + i)^2 ), imaginary unit ( i ), algebraic identities, math education, complex arithmetic, complex plane, ( i^2 = -1 )", "Meta Description:
\nLearn why ( (-1 + i)^2 = -2i ) using step-by-step reasoning. Explore the importance of complex squaring in math and science with clear examples and visual insight."]