Square root: Difference between revisions

Revision as of 19:57, 8 January 2024

A square root of a number $x$ is a number $y$ such that $y^2 = x$ . Generally, the square root only takes the positive value of $y$ . This can be altered by placing a $\pm$ before the root. Thus $y$ is a square root of $x$ if $x$ is the square of $y$ .

Notation

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.

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 ==Exponential notation==
-Square roots can also be written in [[exponentiation | exponential]] 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 = x</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 ==

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Square root: Difference between revisions

Revision as of 19:57, 8 January 2024

Notation

Exponential notation

See also