If (a + bi)^2 = b + ai, where i^2 = -1, and a and b are positive, then find (a,b).
(a + bi)^2 = b + ai
a^2 + 2abi + (bi)^2 = b + ai
a^2 + 2abi - b^2 = b + ai
Equating terms we have that
2abi = ai
2b = 1
b = 1/2
And
a^2 - b^2 = b
a^2 - (1/4) = (1/2)
a^2 = 3/4
a = √3 / 2
So
(a,b) = ( √3 / 2 , 1 / 2 )