Simplify 0.5/(4+3i)

Solution:

Simplification Process

Step 1: Rationalize the Denominator

Multiply both the numerator and denominator by the conjugate of the denominator:

$$\frac{0.5}{4 + 3i} \cdot \frac{4 - 3i}{4 - 3i}$$

Step 2: Execute Multiplication

Step 2.1: Expand the Expression

$$\frac{0.5(4 - 3i)}{(4 + 3i)(4 - 3i)}$$

Step 2.2: Work on the Numerator

Step 2.2.1: Distribute the Numerator

$$\frac{0.5 \cdot 4 + 0.5(-3i)}{(4 + 3i)(4 - 3i)}$$

Step 2.2.2: Perform the Multiplication

$$\frac{2 - 1.5i}{(4 + 3i)(4 - 3i)}$$

Step 2.3: Address the Denominator

Step 2.3.1: FOIL the Denominator

$$\frac{2 - 1.5i}{4(4 - 3i) + 3i(4 - 3i)}$$

Step 2.3.2: Simplify the Denominator

$$\frac{2 - 1.5i}{16 - 12i + 12i - 9i^2}$$

Step 2.3.3: Finalize the Denominator

Replace $i^2$ with $-1$ and add the terms:

$$\frac{2 - 1.5i}{25}$$

Step 3: Split the Fraction

Divide the real and imaginary parts:

$$\frac{2}{25} - \frac{1.5i}{25}$$

Step 4: Simplify Each Term

Step 4.1: Simplify the Real Part

$$\frac{2}{25}$$

Step 4.2: Simplify the Imaginary Part

$$- \frac{1.5i}{25}$$

Step 5: Combine the Results

$$\frac{2}{25} - 0.06i$$

Knowledge Notes:

To simplify a complex fraction, we often multiply by the conjugate of the denominator to eliminate the imaginary part from the denominator. The conjugate of a complex number $a + bi$ is $a - bi$. When we multiply a complex number by its conjugate, the result is a real number because $(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2$, and since $i^2 = -1$, this simplifies to $a^2 + b^2$.

In this problem, we used the distributive property, which states that $a(b + c) = ab + ac$, to expand the numerator and the denominator. We also used the fact that $i^2 = -1$ to simplify the denominator after expansion.

After simplifying the denominator to a real number, we split the fraction into real and imaginary parts and simplified each part separately. The final step was to combine these parts to express the simplified form of the original complex fraction.