kvadreringen

Kvadrering is the mathematical operation of raising a number to the second power, commonly expressed as x^2. It means multiplying the number by itself: x × x.

Geometrically, x^2 is the area of a square with side length x. For real numbers, x^2 is always non-negative, and squaring a number gives the same result for x and −x.

Key algebraic identities include (a+b)^2 = a^2 + 2ab + b^2 and (a−b)^2 = a^2 − 2ab + b^2. In general, (ab)^2 = a^2 b^2, while squaring does not distribute over addition: (a+b)^2 ≠ a^2 + b^2 unless one of the terms is zero.

The square root operation undoes kvadrering for non-negative real numbers: sqrt(x^2) = |x|. In the complex numbers, square roots come in pairs.

In linear algebra, squaring a matrix A yields A^2 = A·A. Properties depend on A; for example, A^2 may have different eigenvalues and is not simply related to A times another matrix.

Common uses include solving quadratic equations, calculating areas, and modeling quadratic relations in physics and statistics. The concept also underpins many numerical methods that involve squaring errors or distances.