Square Root

In mathematics, a square root of a number a is a number y such that y2 = a, or, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a. For example, 4 is a square root of 16 because 42 = 16.

Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted, because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.

Every positive number a has two square roots:, which is positive, and, which is negative. Together, these two roots are denoted (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

Square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where m and n are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC. The particular case is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with side length 1.

Read more about Square Root:  Properties, Computation, Square Roots of Negative and Complex Numbers, Square Roots of Matrices and Operators, Uniqueness of Square Roots in General Rings, Geometric Construction of The Square Root, History

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Square Root Of 5 - Relation To The Golden Ratio and Fibonacci Numbers
... This golden ratio φ is the arithmetic mean of 1 and the square root of 5 ... The algebraic relationship between the square root of 5, the golden ratio and the conjugate of the golden ratio are expressed in the following formulae (S ...
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... Geometrically, the square root of 5 corresponds to the diagonal of a rectangle whose sides are of length 1 and 2, as is evident from the Pythagorean theorem ... the geometrical construction of a golden rectangle from a square, and for the construction of a regular pentagon given its side (since the side-to-diagonal ratio in a regular pentagon is φ) ... a dihedral right angle with the two equal squares that halve a 12 rectangle, it can be seen that √5 corresponds also to the ratio between the length of a cube edge and the shortest distance from one of its vertices ...
Square Root - History
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Square Root Of 5
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... Taking the square root to pass to the standard deviation introduces further downward bias, by Jensen's inequality, due to the square root being a concave function ... in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question ... Taking square roots reintroduces bias, and yields the corrected sample standard deviation, denoted by s While s2 is an unbiased estimator for the population variance, s is a ...

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