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I imagine there have been many questions about imaginary numbers, so if I am asking a question already answered my apologies. I understand that it is perfectly correct to create new number systems that may not have any physical realty such as the real numbers, to facilitate certain mathematical processes. My question is why the square root of $-1$ was selected in particular for so many purposes and are there other forms of imaginary numbers in use?

miracle173
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George Schiro
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  • A couple of links: http://math.stackexchange.com/questions/199676/what-are-imaginary-numbers, http://math.stackexchange.com/questions/154/do-complex-numbers-really-exist – Dmoreno Aug 23 '14 at 11:05
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    Since when do real numbers have physical reality? The real numbers are every bit as artificial as imaginary numbers! – David H Aug 23 '14 at 11:52
  • A millennium ago you might have said the same thing about negative numbers. Let's say you're a farmer and you have three cows. You sell one cow and another one dies. You now have one cow. There is no way for you to sell five cows and thus wind up with minus four cows. – Robert Soupe Aug 23 '14 at 18:26
  • I'm surprised no one's mentioned Cardano's cubic formula… – Akiva Weinberger Aug 08 '16 at 19:23

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Originally, imaginary numbers were introduced to allow us to deal with numbers that square to give negative numbers. In other words, we want to be able to handle $\sqrt{-x}$ for any $x>0$. By the laws of surds we can rewrite this as $\sqrt{x}\sqrt{-1}$ from which it's natural to define the imaginary unit $i=\sqrt{-1}$ which serves very much like the number 1. So we can write $xi$ for any imaginary number which makes dealing with them a lot simpler.

The imaginary numbers can be extended, for example, to more dimensions. See quaternions.

lemon
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number systems that may not have any physical reality

Come again!? $($Seriously, I can't speak for all technical branches out there, but engineering without complex numbers is like swimming without water$)$.

Lucian
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  • I’m a quality engineer but I haven’t come across a use for complex numbers. I learned about using complex numbers to solve ODEs, but I haven’t found a connection of that to a real world application. – Radial Arm Saw Jun 12 '20 at 01:11
  • @RadialArmSaw: Quality engineering is not the only type of engineering. – Lucian Jun 12 '20 at 01:16