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

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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.

So your question should be whether there are any practical applications of the complex numbers. The answer is definitely "yes": complex numbers have many practical applications and even more theoretical applications (many facts can only be conveniently stated using complex numbers); I wouldn't even know where to start if I had to enumerate them.

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  • $\begingroup$ Yes Mr. Leeuwen ,You have correctly pointed out the mistake in the question. I meant to ask the same question which you have suggested. Thank you very much for this. I am grateful for your help. $\endgroup$ – Chaitanya Aug 14 '13 at 16:16
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They're used a lot in electrical engineering.

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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.

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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 $$i(\cos\theta+i\sin\theta)=\cos{(90+\theta)}+i\sin{(90+\theta)}$$

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