Define Principal Square Root

In mathematics, the principal square root of a non-negative real number ( x ) is the non-negative number ( y ) that, when multiplied by itself, equals ( x ). It is denoted as ( \sqrt{x} ). For example, the principal square root of 9 is 3, because ( 3 \times 3 = 9 ).

Key Properties and Definitions:

1. Domain and Range:

   • The principal square root function ( f(x) = \sqrt{x} ) is defined for all non-negative real numbers ( x \geq 0 ).
   • Its range is also all non-negative real numbers ( y \geq 0 ).

2. Non-Negativity:

   • By definition, the principal square root is always non-negative. For instance, ( \sqrt{4} = 2 ), not (-2), even though ((-2)^2 = 4).

3. Special Cases:

   • For ( x = 0 ), the principal square root is ( \sqrt{0} = 0 ).
   • For ( x = 1 ), the principal square root is ( \sqrt{1} = 1 ).

4. Relationship to the General Square Root:

   • Every non-negative number ( x ) has exactly one principal square root and one negative square root (except for ( x = 0 ), which only has one root).
   • The general square root refers to both the principal and negative roots, but the principal square root specifically refers to the non-negative one.

Mathematical Context:

The concept of the principal square root is fundamental in algebra, geometry, and calculus. It is used in solving equations, calculating distances, and analyzing functions. For example, in the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), the principal square root is used to ensure the solution is real and non-negative when applicable.

Example:

Consider the equation ( y^2 = 16 ). The solutions are ( y = 4 ) and ( y = -4 ). However, the principal square root of 16 is ( \sqrt{16} = 4 ).