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i is simply the sqaure root of -1.

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Correct. This is an assumption made about imaginary numbers. If we assume there's a number i which when multiplied together gies a negative number: i x i = -1. Given this assumption, i = sqrt(-1)

But you have to know i is this "imaginary number", or else i could be anything and the relationship i^2 = -1 could be incorrect.

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Yes. Thats why typically lower case i shouldnt be used as a variable to avoid confusion. ME major here. Did 3 calc classes, diff eq., and numerical methods.

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Right, which gets back to my point about "Nope, imaginary numbers. I'm out."

Too many axioms for me to handle. Plus, I've never found a practical use for imaginary numbers because I'm not an electrical engineer.