What is the reciprocal of 2-5i?

Answer 1

Let a + bi be the reciprocal. So (a + bi)(2 - 5i) = 1

(a + bi)(2 - 5i) = 2a - 5ai + 2bi + 5b = (2a + 5b) + (2b - 5a)i

Therefore 2a + 5b = 1 and 2b - 5a = 0. Solving the simultaneous equations, we find that a = 2/29 and b = 5/29.

So the reciprocal of 2 - 5i is 2/29 + 5i/29.