We consider here only real numbers. The square root of a positive real number x is a number r satisfying r2 = x.
But there is a problem. If r2 = x then (-r)2 = x as well. So the square root of a positive real number has two values, differing only in sign.
This is exemplified by the formula for the root of a quadratic equation, say ax2 + bx + c = 0. The roots are
(-b ±√(b2 - 4ac))/2a.
If b = 0 and a = -1, the root is indeed ±√c.
The positive square root of a positive real number is called a principal square root. In common usage, the square root of a positive real number is taken as the principal square root. This is why in school, the square root of, say, 4 is taken as +2.
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.
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Last updated - January 25, 2005