If a+bi = 0, does a=0 and b=0?

Question

when a real number and an imaginary number added together equals 0, does both the real number and the real part of the imaginary number have to equal 0?

Answer 1

Yes. In the complex plane, these are orthogonal directions (complex values is on the y axis and the real value is on the x axis), so they are linearly independent. You can also prove this by taking the absolute value of both sides.