Difference between revisions of "Square root"

Latest revision as of 21:14, 4 February 2025

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 principal 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.

Exponential notation

Square roots can also be written in exponential 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, $\left(x^{\frac12}\right)^2 = x^{\frac12 \cdot 2} = x^1 = x$ , 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.

Properties

No square roots are negative. This is because the positive root is always taken.

@@ Line 5: / Line 5: @@
 ==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.
+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.
+==Properties==
+* No square roots are negative. This is because the positive root is always taken.
 == See also ==
@@ Line 12: / Line 15: @@
 [[Category:Operation]]
+{{stub}}

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Difference between revisions of "Square root"

Latest revision as of 21:14, 4 February 2025

Contents

Notation

Exponential notation

Properties

See also