kwadraten

Kwadraten, or square numbers, are the numbers obtained by squaring integers. In mathematical terms, they are numbers of the form n^2 where n is an integer. The set of kwadraten includes 0, 1, 4, 9, 16, 25, 36, and so on. When the discussion focuses on natural numbers, the sequence typically starts at 1, though 0 is often included in broader contexts.

A key property is that a positive integer is a kwadraat (perfect square) if and only if in its prime factorization every exponent is even. The product of two kwadraten is itself a kwadraat, and the square of any integer is a kwadraat. Square numbers also arise as the areas of geometric squares: a square with side length n has area n^2.

The differences between consecutive kwadraten follow a simple rule: the difference between (n+1)^2 and n^2 is 2n+1, which is an increasing sequence of odd numbers. The square root function is the inverse of squaring on nonnegative numbers: sqrt(x) gives the side length of a square with area x.

In modular arithmetic, certain residues modulo m are called quadratic residues if they are congruent to n^2 for some n. In number theory, squares appear in questions about representations of integers as sums of squares and in various algebraic contexts.

Etymology and usage: in Dutch, kwadraat means the square of a number, and kwadraten is the plural. The concept is fundamental in algebra and geometry and is widely used across mathematics.