A square root of 
is a number 
such that 
.
When written in the form 
or especially 
,
the square root of 
may also be called the radical or surd.
The square root is therefore an nth root with
.
Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are 
and 
,
since 
.
Any nonnegative real number 
has a unique nonnegative square root 
; this is called the principal
square root and is written 
or 
. For example, the principal
square root of 9 is 
,
while the other square root of 9 is 
. In common usage, unless otherwise specified, "the"
square root is generally taken to mean the principal square root. The principal square
root function 
is the inverse function of 
for 
.
Any nonzero complex number 
also has two square roots. For example, using the imaginary
unit i, the two square roots of 
are 
. The principal
square root of a number 
is denoted 
(as in the positive real case) and is returned by the
Wolfram Language function Sqrt[z].
When considering a positive real number 
, the Wolfram Language
function Surd[x,
2] may be used to return the real square root.
For real numbers 
and 
not both zero, the two square roots 
of a complex number 
are given by
![r=+/-(x^2+y^2)^(1/4){cos[1/2tan^(-1)(x,y)]+isin[1/2tan^(-1)(x,y)]}.](/images/equations/SquareRoot/NumberedEquation1.svg) |
(1)
|
In addition,
![sqrt(x+iy)=1/2sqrt(2)[sqrt(sqrt(x^2+y^2)+x)+isgn(y)sqrt(sqrt(x^2+y^2)-x)].](/images/equations/SquareRoot/NumberedEquation2.svg) |
(2)
|
This formula holds for real numbers 
and 
with 
or 
. On the negative real axis,
for 
. As can be seen in the above figure, the imaginary
part of the complex square root function has a branch
cut along the negative real axis.
There are a number of square root algorithms that can be used to approximate the square root of a given (positive real) number.
These include the Bhaskara-Brouncker algorithm and Wolfram's
iteration. The simplest algorithm for 
is Newton's iteration:
 |
(3)
|
with 
.
The square root of 2 is the irrational number 
(OEIS A002193)
sometimes known as Pythagoras's constant,
which has the simple periodic continued fraction
[1, 2, 2, 2, 2, 2, ...] (OEIS A040000). The
square root of 3 is the irrational number 
(OEIS A002194),
sometimes known as Theodorus's constant, which
has the simple periodic continued fraction
[1, 1, 2, 1, 2, 1, 2, ...] (OEIS A040001).
In general, the continued fractions of the
square roots of all positive integers that are
not perfect squares are eventually periodic.
A nested radical of the form 
can sometimes be expressed as a sum or difference of simple square roots by equating
 |
(4)
|
For nonnegative real numbers 
, 
,
and 
with 
, choose 
in the minus-sign case. Squaring gives
 |
(5)
|
A sufficient choice satisfies
 |  |  |
(6)
|
 |  |  |
(7)
|
Solving for 
and 
gives
 |
(8)
|
For example,
 |
(9)
|
 |
(10)
|
The FullSimplify command of the Wolfram Language can apply such simplifications. In general, radical denesting is a difficult problem (Landau 1992a,b, 1994, 1998).
A property of the principal square root is that
![sqrt(z)sqrt(1/z)={-1 for Im[z]=0 and Re[z]<0; undefined for z=0; 1 otherwise,](/images/equations/SquareRoot/NumberedEquation9.svg) |
(11)
|
so the expected identity (i.e., canceling of the 
s) does not hold along the negative real axis.
: Four Different Views." Math. Intell. 20,
55-60, 1998.Sloane, N. J. A. Sequences 
and Its Reciprocal," "The 
Function and Its Reciprocal," and "The
Function." Chs. 12,
14, and 15 in 
." Math. Comput. 36, 593-601, 1981.