z_1 \cdot z_2 = (3 + 4i)(1 - 2i)

["Understanding the Product of Complex Numbers: z₁ · z₂ = (3 + 4i)(1 - 2i)", "When working with complex numbers, multiplication follows specific rules that combine both real and imaginary components. One fundamental calculation often explored is the product of two complex numbers:\nz₁ · z₂ = (3 + 4i)(1 - 2i)", "This expression not only demonstrates the mechanics of complex multiplication but also serves as a foundational example in fields such as engineering, electrical engineering, signal processing, and advanced mathematics.", "### What Are Complex Numbers?", "A complex number is expressed in the form a + bi, where:\n- a is the real part,\n- b is the imaginary part,\n- i is the imaginary unit, defined by the property that ( i^2 = -1 ).", "In our example,\nz₁ = 3 + 4i and z₂ = 1 – 2i", "To compute z₁ · z₂, we apply the distributive property (also known as the FOIL method):", "[\nz₁ \cdot z₂ = (3 + 4i)(1 - 2i)\n]", "### Step-by-Step Multiplication", "[\n= 3 \cdot 1 + 3 \cdot (-2i) + 4i \cdot 1 + 4i \cdot (-2i)\n]", "[\n= 3 - 6i + 4i - 8i^2\n]", "Recall that ( i^2 = -1 ), so:", "[\n= 3 - 6i + 4i - 8(-1)\n]", "[\n= 3 - 2i + 8 \quad \ ext{(since } -6i + 4i = -2i \ ext{ and } -8(-1) = +8\ ext{)}\n]", "[\n= 11 - 2i\n]", "### The Result: z₁ · z₂ = 11 – 2i", "Thus, multiplying ( z₁ = 3 + 4i ) by ( z₂ = 1 - 2i ) yields:", "[\n\boxed{(3 + 4i)(1 - 2i) = 11 - 2i}\n]", "This result combines a real number (11) and an imaginary number (–2i), illustrating how complex multiplication involves both components interacting through distribution and the fundamental rule ( i^2 = -1 ).", "### Why This Calculation Matters", "1. Algebraic Insight: Understanding complex multiplication reinforces key algebraic properties like distributivity and exponent rules.\n2. Practical Applications: Complex numbers appear in AC circuit analysis, quantum mechanics, computer graphics, and control systems. This computation is a building block for these advanced applications.\n3. Conceptual Clarity: Mastering such problems demystifies complex arithmetic and prepares learners for solving equations and manipulating expressions in the complex plane.", "### Summary", "- z₁ · z₂ = (3 + 4i)(1 – 2i) → Expands to a real + imaginary result.\n- Final Answer: ( \boxed{11 - 2i} )\n- Key Steps: Distributive law, substitution, simplification using ( i^2 = -1 ).\n- Real-World Use: Essential for engineering systems and theoretical mathematics.", "Understanding the product ( z₁ \cdot z₂ = (3 + 4i)(1 – 2i) = 11 - 2i ) is more than an arithmetic exercise—it’s a gateway to mastering complex numbers used ubiquitously in science and technology.", "---", "People Also Ask:\n- How do you multiply complex numbers?\n- What is the product of (3 + 4i)(1 – 2i)?\n- Why does ( i^2 = -1 )?\n- How are complex numbers used in real life?", "For further learning, explore complex conjugates, modulus, and polar form—tools that expand on multiplication to encompass phase and magnitude in the complex plane."]