Sqrt [z]
or gives the square root of z.
Sqrt
Sqrt [z]
or gives the square root of z.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- can be entered using or ∖(∖@z∖).
- Sqrt [z] is converted to .
- Sqrt [z^2] is not automatically converted to z.
- Sqrt [a b] is not automatically converted to Sqrt [a]Sqrt [b].
- These conversions can be done using PowerExpand , but will typically be correct only for positive real arguments.
- For certain special arguments, Sqrt automatically evaluates to exact values.
- Sqrt can be evaluated to arbitrary numerical precision.
- Sqrt automatically threads over lists. »
- In StandardForm , Sqrt [z] is printed as .
- √z can also be used for input. The √ character is entered as sqrt or \[Sqrt] .
Examples
open all close allBasic Examples (6)
Evaluate numerically:
Enter using :
Negative numbers have imaginary square roots:
Plot over a subset of the reals:
Plot over a subset of the complexes:
is not necessarily equal to :
It can be simplified to if one assumes :
Scope (39)
Numerical Evaluation (7)
Evaluate numerically:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Complex number inputs:
Evaluate efficiently at high precision:
Sqrt can deal with real‐valued intervals:
Compute the elementwise values of an array using automatic threading:
Or compute the matrix Sqrt function using MatrixFunction :
Compute average-case statistical intervals using Around :
Specific Values (4)
Visualization (4)
Function Properties (10)
The real domain of Sqrt :
It is defined for all complex values:
Sqrt achieves all non-negative values on the reals:
The range for complex values is the right half-plane, excluding the negative imaginary axis:
Find limits at branch cuts:
Enter a √ character as sqrt or \[Sqrt] , followed by a number:
is not an analytic function:
Nor is it meromorphic:
is neither non-decreasing nor non-increasing:
However, it is increasing where it is real valued:
is injective:
Not surjective:
is non-negative on its domain of definition:
has a branch cut singularity for :
However, it is continuous at the origin:
is neither convex nor concave:
However, it is concave where it is real valued:
Differentiation (3)
The first derivative with respect to z:
Higher derivatives with respect to z:
Plot the higher derivatives with respect to z:
Formula for the ^(th) derivative with respect to z:
Integration (3)
Compute the indefinite integral using Integrate :
Verify the anti-derivative:
Definite integral:
More integrals:
Series Expansions (4)
Find the Taylor expansion using Series :
Plots of the first three approximations around :
The general term in the series expansion using SeriesCoefficient :
The first-order Fourier series:
The Taylor expansion at a generic point:
Function Identities and Simplifications (4)
Primary definition:
is not automatically replaced by :
It can be simplified to if one assumes :
It can be simplified to TemplateBox[{x}, Abs] if one assumes x in TemplateBox[{}, Reals]:
PowerExpand can be used to force cancellation without assumptions:
Expand assuming real variables x and y:
Applications (4)
Properties & Relations (12)
Sqrt [x] and Surd [x,2] are the same for non-negative real values:
For negative reals, Sqrt gives an imaginary result, whereas the real-valued Surd reports an error:
Reduce combinations of square roots:
Evaluate power series involving square roots:
Expand a complex square root assuming variables are real valued:
Factor polynomials with square roots in coefficients:
Simplify handles expressions involving square roots:
There are many subtle issues in handling square roots for arbitrary complex arguments:
PowerExpand expands forms involving square roots:
It generically assumes that all variables are positive:
Finite sums of integers and square roots of integers are algebraic numbers:
Take limits accounting for branch cuts:
Sqrt can be represented as a DifferentialRoot :
The generating function for Sqrt :
Possible Issues (3)
Square root is discontinuous across its branch cut along the negative real axis:
Sqrt [x^2] cannot automatically be reduced to x:
With x assumed positive, the simplification can be done:
Use PowerExpand to do the formal reduction:
Along the branch cut, these are not the same:
Neat Examples (2)
Approximation to GoldenRatio :
Riemann surface for square root:
Tech Notes
History
Introduced in 1988 (1.0) | Updated in 1996 (3.0)
Text
Wolfram Research (1988), Sqrt, Wolfram Language function, https://reference.wolfram.com/language/ref/Sqrt.html (updated 1996).
CMS
Wolfram Language. 1988. "Sqrt." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Sqrt.html.
APA
Wolfram Language. (1988). Sqrt. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sqrt.html
BibTeX
@misc{reference.wolfram_2025_sqrt, author="Wolfram Research", title="{Sqrt}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Sqrt.html}", note=[Accessed: 23-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_sqrt, organization={Wolfram Research}, title={Sqrt}, year={1996}, url={https://reference.wolfram.com/language/ref/Sqrt.html}, note=[Accessed: 23-November-2025]}