Difference between revisions of "Square root"

Revision as of 11:38, 30 October 2006

A square root of a number $x$ is a number $y$ such that $y^2 = x$ . Thus $y$ is a square root of $x$ if and only if $x$ is the square of $y$ .

The square root (or the principle square root) of a number $x$ is denoted $\sqrt x$ . For instance, $\sqrt 4 = 2$ . When we consider only positive reals, the square root function is the inverse of the squaring function. However, this does not hold more generally because every positive real has two square roots, one positive and one negative. The notation $\sqrt x$ is used for the positive square root.

Square roots can also be written in exponent notation, so that $x^{\frac 12}$ is equal to the square root of $x$ . Note that this agrees with all the laws of exponentiation, properly interpreted. For example, $\left(x^{\frac12}\right)^2 = x^{\frac12 \cdot 2} = x^1 = 1$ , which is exactly what we would have expected. This notion can also be extended to more general rational, real or complex powers, but some caution is warranted because these do not give functions. In particular, if we require that $x^{\frac 12}$ always gives the positive square root of a positive real number, then the equation $\left(x^2\right)^{\frac 12} = x$ does not hold. For example, replacing $x$ with $-2$ gives $2$ on the left but gives $-2$ on the right.

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-The '''square root''' of a number ''x'' is a number ''y'' such that <math>y^2 = x</math>.  Thus ''y'' is a square root of ''x'' if and only if ''x'' is the square of ''y''.  The square root of a number ''x'' is denoted <math>\sqrt x</math>.  For instance, <math>\sqrt 4 = 2</math>.  When we consider only [[positive number|positive]] [[real number|reals]], the square root function is the [[Function/Introduction#The_Inverse_of_a_Function|inverse]] of the squaring function.  However, this does not hold more generally because every positive real has two square roots, one positive and one negative.  The notation <math>\sqrt x</math> is used for the positive, or principal, square root.
+A '''square root''' of a number <math>x</math> is a number <math>y</math> such that <math>y^2 = x</math>.  Thus <math>y</math> is a square root of <math>x</math> if and only if <math>x</math> is the square of <math>y</math>.
-It is also written as the one half exponent of the argument, so that squaring ''undoes'' this function, just as multiplying by 2 undoes <math>\frac12</math>. Similar functions can be generalized to any real number power, as well as [[complex number|complex]] powers.
+''The square root'' (or the principle square root) of a number <math>x</math> is denoted <math>\sqrt x</math>.  For instance, <math>\sqrt 4 = 2</math>.  When we consider only [[positive number|positive]] [[real number|reals]], the square root [[function]] is the [[Function/Introduction#The_Inverse_of_a_Function|inverse]] of the squaring function.  However, this does not hold more generally because every positive real has two square roots, one positive and one negative.  The notation <math>\sqrt x</math> is used for the positive square root.
+Square roots can also be written in exponent notation, so that <math>x^{\frac 12}</math> is equal to the square root of <math>x</math>.  Note that this agrees with all the laws of exponentiation, properly interpreted.  For example, <math>\left(x^{\frac12}\right)^2 = x^{\frac12 \cdot 2} = x^1 = 1</math>, which is exactly what we would have expected.  This notion can also be extended to more general [[rational]], [[real]] or [[complex]] powers, but some caution is warranted because these do not give functions.  In particular, if we require that <math>x^{\frac 12}</math> always gives the positive square root of a positive real number, then the equation <math>\left(x^2\right)^{\frac 12} = x</math> does not hold.  For example, replacing <math>x</math> with <math>-2</math> gives <math>2</math> on the left but gives <math>-2</math> on the right.
 == See also ==
 * [[Algebra]]
 * [[Exponent]]s

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Difference between revisions of "Square root"

Revision as of 11:38, 30 October 2006

See also