Square (algebra): definition, properties, and uses

Basic characteristics

For real numbers, the square of any number is nonnegative: x^2 ≥ 0. Squares of integers form the sequence of perfect squares: 0, 1, 4, 9, 16, 25, … . Important algebraic identities involving squares include the expansion of a binomial and the factorization of a difference of squares:

• (a + b)^2 = a^2 + 2ab + b^2
• a^2 - b^2 = (a - b)(a + b)

Geometric connection

The term "square" comes from plane geometry: squaring a number corresponds to computing the area of a geometric square whose side length equals that number. A square of side length s has area s^2, which links numeric squaring to two-dimensional measurement. For the geometric square and its area properties see square (shape). Similarly, many area units are expressed as squares of length units (for example, square meters); see area units.

Algebraic and functional roles

As a function, f(x) = x^2 is a basic quadratic function whose graph is a parabola symmetric about the vertical axis. Squaring appears throughout algebra: in solving quadratic equations, manipulating polynomials, computing norms in vector spaces, and defining quadratic forms. The inverse operation is taking a square root; over the nonnegative reals the principal square root is usually denoted √x. For a discussion of exponent rules that include squaring, consult sources on exponentiation.

Examples and notable facts

• Squares preserve parity: an even integer squared is even, an odd integer squared is odd.
• Negative numbers squared yield positive results: (-3)^2 = 9.
• Many problem-solving techniques exploit square identities—for instance, completing the square to solve quadratics.

Because of its simplicity and ubiquity, the square operation serves as a bridge between arithmetic, geometry, and higher algebra, appearing in formulas for area, physical laws, statistics (variance), and many computational algorithms.