# Regarding uses of $i$ (square root of $-1$) [duplicate]

Are there any uses of 'square root of $-1$' in practical life ; like in Physics ?

## marked as duplicate by M Turgeon, mrf, Branimir Ćaćić, Amzoti, Dan RustAug 14 '13 at 15:51

• Fourier transform? AC Electricity? – peterwhy Aug 14 '13 at 15:18
• $$i = \sqrt{-1} \Rightarrow \times$$ $$i^2 = - 1 \Rightarrow right$$ – what'sup Aug 14 '13 at 15:20
• @what'sup: $\Rightarrow$ means "implies"; it does not mean what you think it means ("is"?) – ShreevatsaR Aug 14 '13 at 15:22
• $e^{i\pi} + 1 = 0$ comes to mind... especially as it relates to finding alternate ways of representing $\sin$ and $\cos$. – abiessu Aug 14 '13 at 15:27
• – M Turgeon Aug 14 '13 at 15:31

This question is not well formulated; it makes no sense to ask for uses of a mathematical quantity in isolation. I would be hard pressed to give any practical applications of the number $\frac{173}{41}$, but that doesn't mean that I think one could do without it. It is not the individual numbers/vectors/functions/whatever that have useful applications, it is the structures in which they live, and which make it possible to express relations, that are potentially useful.
The Schrodinger Equation: $\frac{-\hbar}{2m}\frac{d^2\psi}{ dx^2} + V\psi = i\hbar\frac{d\psi}{dt}$ is just one of very many.
Modeling rotations, since multiplication by $i$ induces a $90$ degree counterclockwise rotation. Extending the idea to hypercomplex numbers helps simplify calculations involving three axis rotation of a rigid body through spacetime. $$i(\cos\theta+i\sin\theta)=i\cos\theta+i^2\sin\theta=i\cos\theta-\sin\theta$$ Since sine is an odd function and cosine is an even function we get $$\sin{(-\theta)}+i\cos{(-\theta)}=\cos{(90-(-\theta))}+i\sin{(90-(-\theta))}=\cos{(90+\theta)}+i\sin{(90+\theta)}$$ This shows multiplication by $i$ induces a $90$ degree rotation since $$i(\cos\theta+i\sin\theta)=\cos{(90+\theta)}+i\sin{(90+\theta)}$$