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
(1)
|
In addition,
(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
(6)
| |||
(7)
|
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
(11)
|
so the expected identity (i.e., canceling of the s) does not hold along the negative real axis.