why do negative values play havoc with square roots?

Answer 1

The square root of y equals x such that x^2 = y

Since x^2 is always positive, y cannot be negative. Therefore you cannot take the square root of a negative number.

Example sqrt(-4) cannot be 2 * 2 because 2*2 = 4
It cannot be 2 * -2 because they are not equal.

However, if you are looking in the world of complex numbers, recall that i^2 = -1 and -1 is a real number so you can have complex solutions of the square root of a negative number.

Answer 2

The definition of a number being a square root is that when you multiply it by itself, the result is the number you found the square root of.

E.g.,
4 is a square root of 16, because 4x4 = 16
-4 is *also* a square root of 16, because (-4)x(-4) = 16

Notice that both products are positive, no matter if the square root was positive or negative. No matter what numbers you try, you cannot get a negative number from multiplying any real number by itself.

Therefore, negative numbers have no real square roots.

However...
negative numbers have *imaginary* square roots. I won't get into those here.

Answer 3

Think about it -- the square of any number, even a negative number is positive. For example -2 times -2 is 4.

So if you are going back the other way, you cannot take the square root of a negative number -- it doesn't exist. Hence the term "imaginary numbers"

Answer 4

Because there is no real number (ie, that you can physically quantify in the normal way) which, when multiplied by itself, will produce a negative number. There is a set of numbers that can be squared to give negative numbers . . . those are complex numbers with imaginary parts to them. What makes them complex is that they can be expressed in the binomial form:
a+b*i, where a and b are coefficients and i is the "imaginary" square root of -1.

Some calculators are capable of giving complex number answers.

Real numbers are also complex numbers, but they have b=0. The reason complex numbers are used for some things is because of some interesting properties they have with respect to real-world applications (I don't remember any of them at the moment, but rest assured they do exist.) Anyway, that's probably more than you needed to know for now. Good night!

Answer 5

because the square root gives you the value that if squared will produce the number you are square-rooting. Its not possible to square a number and get a negative result because (-x)*(-x) = x^2 = (x)*(x).
since you cant square any number to get a negative value, you cant square root any negative number to give you a real value (there is such a thing as imaginary numbers, which arent imaginary, just more complex, but thats a little more math)

Answer 6

If we confine ourselves to the real numbers, there is no number whose square is negative. So there's no possible square root for a negative number.