Question

Simplify.
1
1+2𝑖
Write your answer in the form a+b𝑖. Reduce all fractions.

Answer 1

To simplify the expression \( \frac{1}{1 + 2i} \) and write it in the form \( a + bi \), we can multiply the numerator and the denominator by the complex conjugate of the denominator.

The complex conjugate of \( 1 + 2i \) is \( 1 - 2i \).

Now, we perform the multiplication:

\[
\frac{1}{1 + 2i} \times \frac{1 - 2i}{1 - 2i} = \frac{1 - 2i}{(1 + 2i)(1 - 2i)}
\]

Next, we calculate the denominator:

\[
(1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5
\]

Now we have:

\[
\frac{1 - 2i}{5}
\]

This can be separated into two fractions:

\[
\frac{1}{5} - \frac{2}{5}i
\]

Thus, the simplified form in the format \( a + bi \) is:

\[
\frac{1}{5} - \frac{2}{5}i
\]

Therefore, the final answer is:

\[
\frac{1}{5} - \frac{2}{5}i
\]