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)

### Integration(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).

#### 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_2024_sqrt, author="Wolfram Research", title="{Sqrt}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Sqrt.html}", note=[Accessed: 17-June-2024]}

#### BibLaTeX

@online{reference.wolfram_2024_sqrt, organization={Wolfram Research}, title={Sqrt}, year={1996}, url={https://reference.wolfram.com/language/ref/Sqrt.html}, note=[Accessed: 17-June-2024]}