and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers, the latter being a superset of the rational numbers). For all real numbers x, (see absolute value) For all nonnegative real numbers x and y, and The square root function is continuous for all nonnegative x, and differentiable for all positive x. If f denotes the square root function, whose derivative is given by: The Taylor series of about x = 0 converges for ≤ 1, and is given by The square root of a nonnegative number is used in the definition of Euclidean norm (and distance), as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws. Square roots of positive integers A positive number has two square roots, one positive, and one negative, which are opposite to each other. When talking of the square root of a positive integer, it is usually the positive square root that is meant. The square roots of an integer are algebraic integers—more specifically quadratic integers. The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since only roots of those primes having an odd power in the factorization are necessary. More precisely, the square root of a prime factorization is As decimal expansions The square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and hence have non-repeating decimals in their decimal representations. Decimal approximations of the square roots of the first few natural numbers are given in the following table. {|class="wikitable" ! !! truncated to 50 decimal places |- |align="right" | 0 || 0 |- |align="right" | 1 || 1 |- |align="right" | 2 || |- |align="right" | 3 || |- |align="right" | 4 || 2 |- |align="right" | 5 || |- |align="right" | 6 ||