Sqrt[z]
or gives the square root of z.
- 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].
Basic Examples(6)
Scope(38)
Numerical Evaluation(6)
Specific Values(4)
Values of Sqrt at fixed points:
Visualization(4)
Function Properties(10)
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:
Enter a √ character as sqrt or \[Sqrt], followed by a number:
is neither non-decreasing nor non-increasing:
However, it is increasing where it is real valued:
is non-negative on its domain of definition:
has a branch cut singularity for :
However, it is continuous at the origin:
Differentiation(3)
Series Expansions(4)
Function Identities and Simplifications(4)
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:
Possible Issues(3)
Power CubeRoot Surd PowerExpand SqrtBox
Characters: \[Sqrt]
- ▪
- Some Mathematical Functions ▪
- Operators ▪
- Typing Square Roots
- ▪
- Arithmetic Functions ▪
- Elementary Functions ▪
- ▪
- Mathematical Functions
- MathWorld
- The Wolfram Functions Site
- An Elementary Introduction to the Wolfram Language : More about Numbers
- NKS|Online (A New Kind of Science)
Introduced in 1988 (1.0) | Updated in 1996 (3.0)
Wolfram Research (1988), Sqrt, Wolfram Language function, https://reference.wolfram.com/language/ref/Sqrt.html (updated 1996). Wolfram Research (1988), Sqrt, Wolfram Language function, https://reference.wolfram.com/language/ref/Sqrt.html (updated 1996). Wolfram Language. 1988. "Sqrt." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Sqrt.html. Wolfram Language. (1988). Sqrt. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Sqrt.html @misc{reference.wolfram_2024_sqrt, author="Wolfram Research", title="{Sqrt}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Sqrt.html}", note=[Accessed: 18-June-2024]} @online{reference.wolfram_2024_sqrt, organization={Wolfram Research}, title={Sqrt}, year={1996}, url={https://reference.wolfram.com/language/ref/Sqrt.html}, note=[Accessed: 18-June-2024]}Text
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