In mathematics, a **square root** of a number *a* is a number *y* such that *y*2 = *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 *a*1/2.

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

### Other articles related to "square root, square roots, square, squares":

... 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 ... 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 biased estimator for the ...

**Square Root**Of 5 - Geometry

... 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 ... a rectangle can be obtained by halving a

**square**, or by placing two equal

**squares**side by side ... for 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 ...

**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 (See section below ...

**Square Root**Of 5

... The

**square root**of 5 is the positive real number that, when multiplied by itself, gives the prime number 5 ... It is more precisely called the principal

**square root**of 5, to distinguish it from the negative number with the same property ...

**Square Root**- History

... and as 124,51,10 and 4225,35 base 60 numbers on a

**square**crossed by two diagonals ... BC of an even earlier work and shows how the Egyptians extracted

**square roots**... In Ancient India, the knowledge of theoretical and applied aspects of

**square**and

**square root**was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much ...

### Famous quotes containing the words root and/or square:

“But a cultivated man becomes ashamed of his property, out of new respect for his nature. Especially he hates what he has if he see that it is accidental,—came to him by inheritance, or gift, or crime; then he feels that it is not having; it does not belong to him, has no *root* in him and merely lies there because no revolution or no robber takes it away.”

—Ralph Waldo Emerson (1803–1882)

“The *square* dance fiddler’s first concern is to carry a tune, but he must carry it loud enough to be heard over the noise of stamping feet, the cries of the “caller,” and the shouts of the dancers. When he fiddles, he “fiddles all over”; feet, hands, knees, head, and eyes are all busy.”

—State of Oklahoma, U.S. public relief program (1935-1943)