Problem 27 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{2+3 i}{2+i} $$

Step-by-Step Solution

Answer

The solution to the complex number division is \(0.2 + 0.8i\).

Step 1: Identify the Conjugate

The conjugate of the denominator \(2+i\) is \(2-i\). The conjugate of a complex number \(a+bi\) is \(a-bi\).

Step 2: Multiply by the Conjugate

Multiply the entire fraction by the conjugate of the denominator over itself. The new expression is \(\frac{(2+3i) \cdot (2-i)}{(2+i) \cdot (2-i)}\). This step is necessary because it creates a real number in the denominator.

Step 3: Expand the Numerator and Denominator

Using the distributive property to expand the numerator, we get \((2 \cdot 2) + (2 \cdot -i) + (3i \cdot 2) + (3i \cdot -i)\). When expanded, the numerator equals \(4 - 2i + 6i - 3\). Simplifying, it results in \(1 + 4i\). For the denominator, we get \((2 \cdot 2) + (2 \cdot -i) + (i \cdot 2) + (i \cdot -i)\). After simplifying, the denominator results in \(4+1 = 5\).

Step 4: Express in Standard Form

The final solution is \(\frac{1 + 4i}{5}\) or written in standard form, \(0.2 + 0.8i\). The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.