The square of a number is the result obtained when that number is multiplied by itself. In algebraic notation this is written as x squared or x^2; for example, 6 × 6 = 36, so 6^2 = 36. Squaring is the special case of exponentiation with exponent two and is one of the most frequently used operations in elementary and advanced mathematics. For background on the broader field that studies this operation see mathematics.

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.