Squaring

Squaring is the arithmetic operation of multiplying a number by itself, yielding its square. It is denoted as x^2 or n^2. For example, 3^2 = 9 and (-4)^2 = 16. The square of a nonnegative value is nonnegative, and squaring a negative number gives a positive result. Squaring is a basic form of exponentiation with exponent 2.

In algebra, squaring follows standard rules of exponents. (a + b)^2 = a^2 + 2ab + b^2 and (ab)^2 = a^2 b^2. More generally, (x^m)^n = x^{mn}. The square of zero is zero. For complex numbers, squaring is the operation z^2; the squared magnitude satisfies |z|^2 = z z̄, linking squaring to length in the complex plane.

Geometrically, squaring connects to area: a square with side length s has area s^2. This interpretation anchors many relationships in geometry, including the Pythagorean theorem, which relates the squares on the sides of a right triangle.

In applied contexts, squaring emphasizes larger deviations. Squared errors, the variance of a distribution, and the L2 norm use sums of squared quantities, providing sensitivity to outliers and a smooth optimization landscape.

Computationally, squaring can raise concerns about overflow or precision when dealing with very large or very small numbers, and some algorithms implement strategies to manage such issues in finite-precision environments.

Historically, “squaring” also appears in the phrase squaring the circle—the classical problem of constructing a square with the same area as a given circle using only a compass and straightedge. It is proven impossible, tied to the transcendence of pi. Related ideas include square numbers and tiling concepts such as squared squares. See also exponent, power, and geometry.