Simplify \( \frac{3 + 2i}{4 - i} \) and express in rectangular form.
["Simplify ( \frac{3 + 2i}{4 - i} ) and Express in Rectangular Form", "Rational numbers with imaginary components are common in engineering, physics, and advanced mathematics, but complex fractions like ( \frac{3 + 2i}{4 - i} ) can seem daunting—until you simplify them step-by-step. This article walks you through simplifying ( \frac{3 + 2i}{4 - i} ) and expressing the result in rectangular form, helping you master complex number arithmetic with confidence.", "---", "### What is Complex Division and Why Simplify?", "When dividing two complex numbers, the goal is to eliminate the imaginary unit ( i ) from the denominator. This process involves rationalizing the denominator, much like multiplying numerator and denominator by the conjugate of the denominator. Doing so produces a simplified expression in standard rectangular form: ( a + bi ), where ( a ) and ( b ) are real numbers.", "For the expression:", "[\n\frac{3 + 2i}{4 - i}\n]", "the complex conjugate of the denominator ( 4 - i ) is ( 4 + i ). Multiplying both numerator and denominator by this conjugate is the key step.", "---", "### Step-by-Step Simplification", "Begin with:", "[\n\frac{3 + 2i}{4 - i}\n]", "Multiply numerator and denominator by ( 4 + i ):", "[\n\frac{(3 + 2i)(4 + i)}{(4 - i)(4 + i)}\n]", "---", "Step 1: Multiply the denominator\nUse the difference of squares formula:\n( (a - b)(a + b) = a^2 - b^2 )", "[\n(4 - i)(4 + i) = 4^2 - (i)^2 = 16 - i^2\n]", "Since ( i^2 = -1 ):", "[\n16 - (-1) = 16 + 1 = 17\n]", "So the denominator simplifies cleanly to 17.", "---", "Step 2: Multiply the numerator\nExpand ( (3 + 2i)(4 + i) ) using distributive property (FOIL):", "[\n3 \cdot 4 + 3 \cdot i + 2i \cdot 4 + 2i \cdot i = 12 + 3i + 8i + 2i^2\n]", "Simplify and recall ( i^2 = -1 ):", "[\n12 + 11i + 2(-1) = 12 + 11i - 2 = 10 + 11i\n]", "---", "### Final Result", "Now substitute back:", "[\n\frac{10 + 11i}{17} = \frac{10}{17} + \frac{11}{17}i\n]", "---", "### Rectangular Form Summary", "The expression ( \frac{3 + 2i}{4 - i} ) simplifies to:", "[\n\boxed{ \frac{10}{17} + \frac{11}{17}i }\n]", "This rectangular form clearly shows the real part ( \frac{10}{17} ) and the imaginary part ( \frac{11}{17} ), making it easier to visualize, compute, and apply in real-world contexts.", "---", "### Why This Matters", "Understanding how to simplify complex fractions is essential in fields like:", "- Electrical engineering, for analyzing AC circuits\n- Signal processing, for Fourier transforms and phasors\n- Quantum mechanics, where wave functions involve imaginary units\n- Any discipline relying on complex admittances and impedance", "Memorizing steps and practicing with clear examples builds the intuition needed to handle more advanced problems with ease.", "---", "Key Takeaways:", "- Always multiply numerator and denominator by the conjugate of the denominator.\n- Simplify both real and imaginary parts after multiplication.\n- The final rectangular form is ( a + bi ), where ( a ) and ( b ) are real.", "Try simplifying your own complex fractions today—once you master the rationalization method, the process becomes straightforward and rewarding!"]
