Question 16 Simplify the following Expression \( \frac{1+2 i}{3+7 i}=\square \)

To simplify the expression \( \frac{1+2 i}{3+7 i} \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 3 - 7 i \): \[ \frac{(1 + 2 i)(3 - 7 i)}{(3 + 7 i)(3 - 7 i)}. \] This results in the denominator as \( 3^2 + 7^2 = 9 + 49 = 58 \). For the numerator, distribute: \[ (1)(3) + (1)(-7i) + (2i)(3) + (2i)(-7i) = 3 - 7i + 6i + 14 = 17 - i. \] Thus, we have: \[ \frac{17 - i}{58} = \frac{17}{58} - \frac{1}{58}i. \] So the simplified expression is \( \frac{17}{58} - \frac{1}{58}i \). Now, wasn't that a ride through the complex numbers? Want to dive into the world of complex planes or unearth some fascinating historical tidbits about complex numbers' journey into mathematics?