To solve the expression $\frac{3 + 4i}{1 + i}$, we need to divide these complex numbers and write the result in standard form $a + bi$, where $a$ and $b$ are real numbers.

Step-by-step solution:

1. Multiply by the conjugate: To simplify the division, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $1 + i$ is $1 - i$. $\frac{3 + 4i}{1 + i} \cdot \frac{1 - i}{1 - i} = \frac{(3 + 4i)(1 - i)}{(1 + i)(1 - i)}$

2. Simplify the denominator: Use the difference of squares formula: $(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2$ So, the denominator simplifies to $2$.

3. Multiply the numerator: Now, distribute $(3 + 4i)(1 - i)$: $3(1 - i) + 4i(1 - i) = 3 - 3i + 4i - 4i^2$ Simplifying: $3 - 3i + 4i - 4(-1) = 3 - 3i + 4i + 4 = 7 + i$

4. Final division: Now divide the result by $2$: $\frac{7 + i}{2} = \frac{7}{2} + \frac{1}{2}i$

Thus, the final answer in standard form is: $\frac{7}{2} + \frac{1}{2}i \quad \text{or} \quad 3.5 + 0.5i$

Would you like more details on any step or have any questions? Here are some related questions to explore further:

1. What is the significance of the conjugate when dividing complex numbers?
2. How do you multiply two complex numbers in general?
3. What are some real-world applications of complex numbers?
4. Can you divide complex numbers without using conjugates?
5. How do we interpret complex numbers geometrically on the complex plane?

Tip: Always remember to rationalize the denominator by multiplying with the conjugate when dividing complex numbers.