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Sqrt

Sqrt[z]
or gives the square root of z.
MORE INFORMATION
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • can be entered using Ctrl+2 z or .
  • Sqrt[z] is converted to .
  • Sqrt is not automatically converted to z.
  • 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.
  • √z can also be used for input. The character is entered as Esc sqrt Esc or \[Sqrt].
EXAMPLESCLOSE ALL
Basic Examples   (7)
Evaluate numerically to any precision:
Negative numbers have imaginary square roots:
Sqrt threads element-wise over lists:
is not automatically replaced by :
It can be simplified to if one assumes :
Enter using Ctrl+2:
In[1]:=
Click for copyable input
Out[1]=
 
Evaluate numerically to any precision:
In[1]:=
Click for copyable input
Out[1]=
 
Negative numbers have imaginary square roots:
In[1]:=
Click for copyable input
Out[1]=
 
Sqrt threads element-wise over lists:
In[1]:=
Click for copyable input
Out[1]=
 
In[1]:=
Click for copyable input
Out[1]=
 
is not automatically replaced by :
In[1]:=
Click for copyable input
Out[1]=
It can be simplified to if one assumes :
In[2]:=
Click for copyable input
Out[2]=
 
Enter using Ctrl+2:
In[1]:=
Click for copyable input
Out[1]=
Scope   (2)
Exact roots are factored out when possible:
Find square roots of complex numbers:
Applications   (2)
Roots of a quadratic polynomial:
Generate periodic continued fractions:
Properties & Relations   (9)
Reduce combinations of square roots:
Evaluate power series involving square roots:
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:
Take limits accounting for branch cuts:
Possible Issues   (3)
Square root is discontinuous across its branch cut along the negative real axis:
Sqrt 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:
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