To evaluate the given expression $\frac{1 - 3i}{2i}$ and write it in the form $a + bi$, let’s proceed step-by-step:

Step 1: Rewrite the denominator

We rewrite $2i$ as just $2i$, keeping its position intact.

$\frac{1 - 3i}{2i}$

Step 2: Multiply numerator and denominator by the conjugate of $2i$

The conjugate of $2i$ is $-2i$. Multiply both numerator and denominator by $-2i$:

$\frac{1 - 3i}{2i} \cdot \frac{-2i}{-2i} = \frac{(1 - 3i)(-2i)}{(2i)(-2i)}$

Step 3: Simplify the denominator

$(2i)(-2i) = -4i^2 = -4(-1) = 4$.

Step 4: Expand the numerator

Expand $(1 - 3i)(-2i)$:

$(1)(-2i) + (-3i)(-2i) = -2i + 6i^2$

Since $i^2 = -1$, substitute $6i^2 = 6(-1) = -6$. The numerator becomes:

$-2i - 6$

Step 5: Write the fraction

The expression now becomes:

$\frac{-2i - 6}{4}$

Step 6: Split the terms

Separate the real and imaginary parts:

$\frac{-6}{4} + \frac{-2i}{4} = -\frac{3}{2} - \frac{i}{2}$

Step 7: Final result in the form $a + bi$

The result is:

$a = -\frac{3}{2}, \, b = -\frac{1}{2}$

Final Answer:

• $a = -\frac{3}{2}$
• $b = -\frac{1}{2}$

Let me know if you need further clarification! Here are some related questions to expand your understanding:

1. How do you find the conjugate of a complex number?
2. Why is multiplying by the conjugate necessary in these types of problems?
3. Can you simplify $\frac{1 + 2i}{3 - 4i}$ in the same way?
4. What are some applications of complex numbers in engineering or physics?
5. How does $i^n$ simplify for different powers of $n$?

Tip: Always simplify expressions by writing the denominator in terms of a real number using the conjugate method. This avoids dividing directly by $i$.