The square root of ii?

Algebra Level 3

Until around the 1500's, 1\sqrt{-1} was , thought of by mathematicians as an evil number. Rafael Bombelli first laid out the basic principles for manipulating these fascinating numbers in 1572. The idea was first though of by Heron of Alexandria (Heron's formula... ring a bell?) The applications for ii are wide-spread. For example, in algebra, some roots of quadratics that don't pass through the xx-axis can be imaginary numbers. The letter ii is the imaginary unit. Complex numbers are numbers that have real and imaginary parts. For example, 2+3i2+3i is complex. The imaginary part is 33 (it's the coefficient of ii) and the real part is 22. Let a+ib=i\frac{a+i}{\sqrt{b}}=\sqrt{i} where aa and bb are coprime, positive integers. Find a+ba+b.

If you found this interesting, you should check out Christopher Boo's problem based on this, The Cube Root of ii?.

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