If \( a = 1 \), \( b = -1 \), then \( (a + bi)^2 = (1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i \) ✅
["Understanding ( (1 - i)^2 ) When ( a = 1 ) and ( b = -1 ): A Step-by-Step Breakdown", "When exploring complex numbers, one of the most essential calculations involves squaring expressions in the form ( a + bi ). Knowing how to expand expressions like ( (1 - i)^2 ) clearly helps build foundational knowledge for deeper studies in algebra, engineering, and physics.", "### What Are Complex Numbers?", "A complex number takes the form:\n[\nz = a + bi\n]\nwhere:\n- ( a ) is the real part,\n- ( b ) is the imaginary part,\n- ( i ) is the imaginary unit with ( i^2 = -1 ).", "In this article, we examine a specific case: when ( a = 1 ) and ( b = -1 ), meaning:\n[\nz = 1 - i\n]", "### Squaring ( 1 - i ): Step-by-Step Expansion", "We calculate ( (1 - i)^2 ) using the binomial expansion formula:\n[\n(z)^2 = (a + bi)^2 = a^2 + 2abi + (bi)^2\n]", "Substituting ( a = 1 ) and ( b = -1 ):\n[\n(1 - i)^2 = 1^2 + 2(1)(-1)i + (-i)^2\n]", "Now simplify each term:\n- ( 1^2 = 1 )\n- ( 2(1)(-1)i = -2i )\n- ( (-i)^2 = (-i)(-i) = i^2 = -1 ) (since ( i^2 = -1 ))", "Putting it together:\n[\n(1 - i)^2 = 1 - 2i - 1\n]", "Combine the real parts:\n[\n1 - 1 - 2i = -2i\n]", "So,\n[\n(1 - i)^2 = -2i\n]", "### Why This Matters", "This simple but powerful result shows how squaring complex numbers introduces both real and imaginary components—except when cancellation eliminates the real part, as seen here. Understanding such expansions prepares learners for more advanced manipulations in complex analysis, electrical engineering, signal processing, and quantum mechanics.", "### Summary", "- ( a = 1 ), ( b = -1 \Rightarrow z = 1 - i )\n- ( (1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i )\n- A clear demonstration of binomial expansion and imaginary unit properties", "Perfect squaring ( (a + bi)^2 ) when ( a = 1 ), ( b = -1 ) yields ( -2i ), reinforcing key sympolics in complex number arithmetic.", "---", "Whether you're studying for math competitions, programming algorithms involving complex analysis, or simply expanding your mathematical toolkit, mastering such computations is invaluable. Use this example as a milestone in your journey with complex numbers!"]
