Difference between revisions of "Square root"
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− | A '''square root''' of a number | + | A '''square root''' of a number <math>x</math> is a number <math>y</math> such that <math>y^2 = x</math>. Generally, the square root only takes the positive value of <math>y</math>. This can be altered by placing a <math>\pm</math> before the root. Thus <math>y</math> is a square root of <math>x</math> if <math>x</math> is the square of <math>y</math>. |
− | + | ==Notation== | |
+ | The square root (or the principal 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. | ||
+ | |||
+ | ==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 == | ||
+ | * [[Algebra]] | ||
+ | * [[Root (operation)]] | ||
+ | |||
+ | [[Category:Operation]] |
Latest revision as of 11:14, 18 July 2024
A square root of a number is a number
such that
. Generally, the square root only takes the positive value of
. This can be altered by placing a
before the root. Thus
is a square root of
if
is the square of
.
Notation
The square root (or the principal square root) of a number is denoted
. For instance,
. 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 is equal to the square root of
. Note that this agrees with all the laws of exponentiation, properly interpreted. For example,
, 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
always gives the positive square root of a positive real number, then the equation
does not hold. For example, replacing
with
gives
on the left but gives
on the right.