>Your answer is quite similar to the one I got from a college math professor when I asked about i. He said it doesn't sound like it makes sense but it makes possible a number of elegant proofs and has practical applications.
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>I still don't understand how -1 can have a square root given that a positive number times itself yields a positive result and a negative number times iteslf yields a positive result. So exactly what numbers can be multiplied by themselves to give negative results? (Yeah, I know: i).

Exactly. The imaginary and complex numbers are to be understood as extensions to the real numbers (just as negative numbers are an extension to positive-only numbers).

This extension is sometimes described as the possibility to have solutions to more and more equations.

For instance, if you add or multiply two positive integers, you always get a positive integer.

If you want to do unlimited subtraction between positive integers, you get equations that you can't solve, e.g.: x = 3 - 5. This has no solution if you only accept positive integers. Therefore, negative integers are "invented".

Still, you can't divide: x = 1 / 2 has no integer-only solution. Therefore, you have to "invent" rational numbers. (A rational number can be expressed as a ratio between two integers).

Since you still don't have a solution for equations like x^2 = 2 (i.e., x = sqrt(2)), you "invent" real numbers. Real numbers include rational and irrational numbers, and can all be located within a straight line.

Finally, to solve equations like x^2 + 1 = 0 (i.e., x = sqrt(-1)), you have to "invent" the complex number plane. The new numbers occupy an entire plane, not just a line.


Of course, just as for some quantities (e.g., number of people) it doesn't make sense to use fractional and/or negative numbers, it also doesn't make sense to express many physical quantities in complex numbers. But for some - like speed or mass, etc. - fractional quantities make sense, for some - like altitude above sea level, or the amount of money you have, negative numbers make sense (a debt is a negative possession), and for some - like electrical quantities in AC circuits - complex numbers make sense.

Regards, Hilmar.