Square Root

In mathematics, a square root of a number a is a number y such that y2 = a, or, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a. For example, 4 is a square root of 16 because 42 = 16.

Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted, because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.

Every positive number a has two square roots:, which is positive, and, which is negative. Together, these two roots are denoted (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

Square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where m and n are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC. The particular case is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with side length 1.