174. Find the necessary and sufficient condition for the sum of two complex numbers $a+bi$ and $c+di$ to be 1) a real number, 2) a purely imaginary number.
1) The sum of the given complex numbers is expressed as $(a+c)+(b+d) i$. For this complex number to be real, the equality $b+d=0$ or $b=-d$ must hold. If $b=-d$, then the complex number will be real. Therefore, the condition $b=-d$ is necessary and sufficient for the sum of the given complex numbers to be a real number.

2) The conditions $a=-c$ and $b \neq-d$ are necessary and sufficient for the sum of the given complex numbers to be a purely imaginary number.