The square root of \(i\)?

Algebra Level 3

Until around the 1500's, \(\sqrt{-1}\) 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 \(i\) are wide-spread. For example, in algebra, some roots of quadratics that don't pass through the \(x\)-axis can be imaginary numbers. The letter \(i\) is the imaginary unit. Complex numbers are numbers that have real and imaginary parts. For example, \(2+3i\) is complex. The imaginary part is \(3\) (it's the coefficient of \(i\)) and the real part is \(2\). Let \(\frac{a+i}{\sqrt{b}}=\sqrt{i}\) where \(a\) and \(b\) are coprime, positive integers. Find \(a+b\).

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

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