All right ladies and gentlemen, take out your calculators. Punch in the number 4. Whack the square root button. What do you get?
Unless I am badly mistaken, you’re going to get the number 2. As you probably remember, the square of a number is just that number multiplied by itself: 3 squared gives 9, 4 squared gives 16, 1/2 squared gives 1/4, etc. The square root is the inverse of this process. It tells you what you have to square to get the given number. Two times two is four, so the square root of four is 2. You can plot the graph of the function to see what the square roots of various numbers are:
Now I invite you to try another experiment with your calculator. Type in negative 2 and multiply it by itself. Negative times negative is positive, and so you’ll see that -2 * -2 = 4. So why isn’t the square root of 4 equal to -2?
That’s not even so bad. We can also define a 4th-root function, which tells us which number multiplied by itself four times gives us whatever we start with. For instance, the fourth root of 81 is 3, because 3*3*3*3 = 81. If you take the 4th root of the number 1, you’ll see that the answer is 1, because 1 multiplied by itself four times gives you 1. But so does -1. And so does the number i. And so does the number -i. (The letter “i” is the symbol for the complex number which gives -1 when squared.)
What about 100th roots? It’s more work, but there’s in general one hundred 100th roots of whatever number you care to name. You can guess the pattern for the various other nth roots.
But there can be only one. A function can only associate a given number in the domain with one number in the range. We have to pick. The usual choice is to say that “the” nth root of x is the one that happens to be positive and real. This tends to be the most convenient and physically meaningful choice. It’s not the only choice though, and our somewhat arbitrary selection of that (or any) particular choice will leave us with a gaping slice of discontinuous function on the complex plane. These slices are called branch cuts, and you have to deal with them carefully. We’re smart people, we can do it. And one lovely Sunday sometime soon, we will.