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square root∗ Wkbj79† 2013-03-21 12:46:17 √ The square root of a nonnegative real number x, written√as x, is the 2 2 unique √ √nonnegative real number y such that y = x. Thus, ( x) ≡ x. Or, x × x ≡ x. √ Example. 9 = 3 because 3 ≥ 0 and 32 = 3 × 3 = 9. √ Example. x2 + 2x + 1 = |x + 1| (see absolute value and even-even-odd rule) because (x + 1)2 = (x + 1)(x + 1) = x2 + x + x + 1 = x2 + 2x + 1. √ it is better to allow two values for x. For example, √ In some situations 4 = ±2 because 22 = 4 and (−2)2 = 4. Over nonnegative real numbers, the square root operation is left distributive over multiplication and division, but not over addition or subtraction.rThat √ is, if √ √ x x √ x and y are nonnegative real numbers, then x × y = x × y and = √ . y y p Example. x2 y 2 = xy because (xy)2 = xy × xy = x × x × y × y = x2 × y 2 = x2 y 2 . r 2 3 3 32 9 9 Example. = because = 2 = . 25 5 5 5 25 √ √ √ √ √ √ On the other hand, in general, x + y 6= x + y and x − y 6= x − y. This error is an instance of the freshman’s dream error. The square root notation is actually an alternative to exponentiation. That √ 1 is, x ≡ x 2 . When it is defined, √ the asquare√ root operation is commutative with exponentiation. That is, xa = x 2 = ( x)a whenever both xa > 0 and x > 0. The restrictions can be lifted if we extend the domain and codomain of the square root function to the complex numbers. √ Negative real numbers do not have real square roots. For example, −4 is not by contradiction as follows: Suppose √ a real number. This fact can be proven −4 = x ∈ R. If x is negative, then x2 is positive, and if x is positive, then ∗ hSquareRooti created: h2013-03-21i by: hWkbj79i version: h30747i Privacy setting: h1i hDefinitioni h11A25i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 x2 is also positive. Therefore, x cannot be positive or negative. Moreover, x √ / R. cannot be zero either, because 02 = 0. Hence, −4 ∈ For additional discussion of the square root and negative numbers, see the discussion of complex numbers. The √ square root function generally maps rational numbers to algebraic numwhich, after cancelling, bers; x is rational if and only if x is a rational number √ is a fraction of two perfect squares. In particular, 2 is irrational. The function is continuous for all nonnegative x, and differentiable for all positive x (it is not differentiable for x = 0). Its derivative is given by: 1 d √ x = √ dx 2 x It is possible to consider square roots in rings other than the integers or the rationals. For any ring R, with x, y ∈ R, we say that y is a square root of x if y 2 = x. When working in the ring of integers modulo n, we give a special name to members of the ring that have a square root. We say x is a quadratic residue modulo n if there exists y coprime to x such that y 2 ≡ x (mod n). Rabin’s cryptosystem is based on the difficulty of finding square roots modulo an integer n. 2