square root

[Justa Lol]

can someone explain to me why sqrt(-1) = i ?

[Atheist]

sqrt(-1) isn't possible to calculate. Simply because no number, raised to the power of 2, will ever result in -1.

[jemidiah]

It has to do with the definition of the real numbers.

Some background: let's start with the positive integers, {1, 2, 3, ...}. Let's add 0 to the mix, so that we're dealing with the numbers {0, 1, 2, 3, ...}--think of them as an infinite array. Now let's define addition between numbers in this set in the usual way: 0+2 = 2, 15+18 = 33, etc. Now let's reverse addition, which we'll call subtraction. Do this in the usual way: 15-5 = 10, 12-12 = 0, etc. However, we'll quickly run into a problem. Try doing 4-5. Remember that we only have a concept of the positive integers and 0; we're not dealing with negative numbers yet. In this context, 4-5 is undefined, since it is not in the set {0, 1, 2, 3, ...}. This is a silly situation to be in, so let's make some new numbers up and call them negatives, and extend our set to {..., -3, -2, -1, 0, 1, 2, 3, ...}. Now we may subtract any two numbers and stay in this set. [Note that the negative numbers are formal objects in the above setting; that is, we could call -3 by q, and it won't matter, so long as we have that q+3 = 0; we only wanted subtraction (and addition) to "work".]

The situation with i is analogous. Given a real number, we can square that number and get out a real number: 2^2 = 4, 4^2 = 16, pi^2 = something ugly, (-15)^2 = 225. After doing something to a number, we'd like the ability to undo it if we so choose, so along comes the square root. However, we run into several problems here. Given 225, the square root asks this question: what (real) number, when squared, gives 225? We see that (-15)^2 = 225 = 15^2, so there are two answers, -15 and 15. This is where the +/- comes from when you're taking square roots. The second problem happens given a negative number, say -1. What number, when squared, gives -1? Since all squares of real numbers are positive, the answer is "none".

Mathematicians hate not being able to do something, so they add more numbers with certain properties, just like when we added the negatives to allow subtraction to work earlier. In this case, adding some new number "i" with the property that i^2 = -1, and making sure i behaves as usual when multiplying and adding [i.e. preserving the field properties] works particularly well. Since i^2 = -1, we have i = Sqrt(-1) [to be technical, -i = Sqrt(-1) as well, which is the basis of a lot of complex conjugates showing up in polynomial equations].


One big thing to note, here, is that I'm treating each number just as an object with certain properties. Once you have that, you can add a new object to the mix, i, with other certain properties. In order to allow square roots to operate on negative numbers, we extended the definition of numbers, by adding i (and multiples and sums of i). This inexorably leads us to define the complex numbers, which is a place where multiplication and addition work as you'd expect, but you can also take exponentials and roots with impunity, instead of being told "it can't be done".

[NickThissen]

The short answer: it's just a definition. Well actually, the definition is i^2 = -1. But that implies i = sqrt(-1), but since sqrt(-1) is also -i, you shouldn't use "sqrt(-1) = i" as the definition.

[makis]

-1 equals to i^2.

sqrt(-1) = sqrt(i^2) = i

You have to know about complex numbers to understand.