The reason for taking the principal square root as the value of the square root of a positive real number is the following. For the function f(x) = x², the inverse map, call it g, is g(x) = ±√x. This means g maps x to two values. Then g is not a function. We can make g a function by ensuring that it maps x to a unique value. This can be achieved by assuming a convention where the value of g at x is either always the principal square root of x or always the negative of the principal square root of x. The world of mathematics has adopted the former. Thus g, the square-root map, becomes a function for postive real number x. Now, an inverse map g of f must satisfy the following property: g(f(x))= f(g(x))= x. With f(x) = x² and g(x) = √x, this gives √(x²) = x. That is, for positive real number N, we shall mean by the square root of N, the positive square root of N.