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.
The square roots of a complex number 
are given by
![sqrt(x+iy)=+/-(x^2+y^2)^(1/4){cos[1/2tan^(-1)(x,y)]+isin[1/2tan^(-1)(x,y)]}.](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FSquareRoot%2FNumberedEquation1.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)].](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FSquareRoot%2FNumberedEquation2.svg) |
(2) |
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 are periodic.
A nested radical of the form 
can sometimes be simplified into a simple square root by equating
 |
(4) |
Squaring gives
 |
(5) |
so
Solving for 
and 
gives
 |
(8) |
For example,
 |
(9) |
 |
(10) |
The Simplify command of the Wolfram Language does not apply such simplifications, but FullSimplify does. In general, radical denesting is a difficult problem (Landau 1992ab, 1994, 1998).
A counterintuitive property of inverse functions is that
![sqrt(z)sqrt(1/z)={-1 for I[z]=0 and R[z]<0; undefined for z=0; 1 otherwise,](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FSquareRoot%2FNumberedEquation9.svg) |
(11) |
so the expected identity (i.e., canceling of the 
s) does not hold along the negative real axis.
See also
Cube Root, nth Root, Nested Radical, Newton's Iteration, Principal Square Root, Pythagoras's Constant, Quadratic Surd, Radical, Root, Root of Unity, Square Number, Square Root Algorithms, Square Root Inequality, Square Triangular Number, Surd, Theodorus's Constant Explore this topic in the MathWorld classroom
Related Wolfram sites
http://functions.wolfram.com/ElementaryFunctions/Sqrt/
Explore with Wolfram|Alpha
References
Landau, S. "A Note on 'Zippel Denesting.' " J. Symb. Comput. 13, 31-45, 1992a.Landau, S. "Simplification
of Nested Radicals." SIAM J. Comput. 21, 85-110, 1992b.Landau,
S. "How to Tangle with a Nested Radical." Math. Intell. 16,
49-55, 1994.Landau, S. "
: Four Different Views." Math. Intell. 20,
55-60, 1998.Sloane, N. J. A. Sequences A002193/M3195,
A002194/M4326, A040000,
and A040001 in "The On-Line Encyclopedia
of Integer Sequences."Spanier, J. and Oldham, K. B. "The
Square-Root Function 
and Its Reciprocal," "The 
Function and Its Reciprocal," and "The
Function." Chs. 12,
14, and 15 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 91-99, 107-115,
and 115-122, 1987.Williams, H. C. "A Numerical Investigation
into the Length of the Period of the Continued Fraction Expansion of 
." Math. Comput. 36, 593-601, 1981.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Square Root." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SquareRoot.html