$ (x + yi)(u + vi) = xu - yv + (xv + yu)i = 13 - 2i $.
["The Multiplication of Complex Numbers: Solving $ (x + yi)(u + vi) = 13 - 2i $ — A Step-by-Step Guide", "Complex numbers are a cornerstone of modern mathematics, engineering, and applied sciences. Understanding how to multiply complex numbers not only strengthens foundational knowledge but also enables advanced applications in signal processing, electrical engineering, and quantum mechanics. This article breaks down the multiplication of two complex numbers using the identity:\n$$(x + yi)(u + vi) = xu - yv + (xv + yu)i = 13 - 2i.$$", "---", "### Understanding Complex Numbers", "A complex number is expressed in the form $ z = x + yi $, where:\n- $ x $ and $ y $ are real numbers (the real and imaginary parts),\n- $ i $ is the imaginary unit, defined by $ i^2 = -1 $.", "Multiplying two complex numbers involves distributing each term, then using the identity $ i^2 = -1 $ to simplify.", "---", "### Multiplying $(x + yi)(u + vi)$", "Start by applying the distributive property:\n$$(x + yi)(u + vi) = x(u + vi) + yi(u + vi)$$\n$$= xu + xvi + yiu + yivi$$", "Now handle $ i^2 = -1 $ in the last term:\n$$yivi = yu(i^2) = yu(-1) = -yu.$$", "Rewriting the expression:\n$$xu + xvi + yiu - yu = (xu - yv) + (xv + yu)i.$$", "This matches the general form:\n$$\mathrm{(x + yi)(u + vi)} = (xu - yv) + (xv + yu)i.$$", "---", "### Matching to $13 - 2i$", "We are given:\n$$(x + yi)(u + vi) = 13 - 2i.$$", "By comparing real and imaginary parts, we obtain the system:\n$$\n\begin{align}\nxu - yv &= 13 \quad \ ext{(1)} \\nxv + yu &= -2 \quad \ ext{(2)}\n\end{align}\n$$", "This system can be used to solve for unknowns $x, y, u, v$, though multiple solutions exist depending on the values of two variables.", "---", "### Applications and Insights", "- Solving for Specific Values: Given three variables (e.g., $ x, y, u $), you can solve for $ v $.\n- Geometry of Complex Numbers: The multiplication corresponds geometrically to rotation and scaling in the complex plane — understandably, this product’s magnitude is $ \sqrt{13^2 + (-2)^2} = \sqrt{169 + 4} = \sqrt{173} $.\n- Use in Equations: Multiplying complex numbers is essential in solving high-degree polynomial equations, circuits, and control systems.", "---", "### Conclusion", "Multiplying complex numbers like $(x + yi)(u + vi)$ follows clear algebraic rules rooted in the real and imaginary units. By expanding $(x + yi)(u + vi)$ to $xu - yv + (xv + yu)i$, we obtain a real component and an imaginary component matching $13 - 2i$. Solving the resulting equations unlocks deeper insight into complex arithmetic, paving the way for solving real-world problems involving oscillatory systems, AC circuits, and quantum states.", "Understanding this process deepens mathematical fluency and opens doors to advanced topics in science and engineering.", "---", "Keywords: complex numbers multiplication, $(x + yi)(u + vi)$, multiplication = $13 - 2i$, $xu - yv + (xv + yu)i$, solving complex equations, imaginary unit $i$, applications of complex numbers", "Meta Description:\nLearn how to multiply two complex numbers $(x + yi)(u + vi)$ using the identity $xu - yv + (xv + yu)i = 13 - 2i$, including step-by-step expansion, variable matching, and real-world relevance."]
